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---|---|---|---|---|---|---|
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
hs : IsClosed s
⊢ mk ⁻¹' (mk '' s) ⊆ s | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
| rintro x ⟨y, hys, hxy⟩ | theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
| Mathlib.Topology.Inseparable.474_0.2NeLzt0mQ64QlfB | theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s | Mathlib_Topology_Inseparable |
case intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
hs : IsClosed s
x y : X
hys : y ∈ s
hxy : mk y = mk x
⊢ x ∈ s | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
| exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys | theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
| Mathlib.Topology.Inseparable.474_0.2NeLzt0mQ64QlfB | theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
⊢ IsClosed (Set.range mk) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by | rw [range_mk] | theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by | Mathlib.Topology.Inseparable.485_0.2NeLzt0mQ64QlfB | theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
⊢ IsClosed univ | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; | exact isClosed_univ | theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; | Mathlib.Topology.Inseparable.485_0.2NeLzt0mQ64QlfB | theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
⊢ Filter.map mk (𝓝 x) = 𝓝 (mk x) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
| rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk] | theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
| Mathlib.Topology.Inseparable.499_0.2NeLzt0mQ64QlfB | theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
⊢ Filter.map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
| rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk] | theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
| Mathlib.Topology.Inseparable.503_0.2NeLzt0mQ64QlfB | theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
⊢ comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
| conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image] | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
| Mathlib.Topology.Inseparable.507_0.2NeLzt0mQ64QlfB | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
| comap mk (𝓝ˢ t) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => | rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image] | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => | Mathlib.Topology.Inseparable.507_0.2NeLzt0mQ64QlfB | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
| comap mk (𝓝ˢ t) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => | rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image] | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => | Mathlib.Topology.Inseparable.507_0.2NeLzt0mQ64QlfB | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
| comap mk (𝓝ˢ t) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => | rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image] | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => | Mathlib.Topology.Inseparable.507_0.2NeLzt0mQ64QlfB | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
x : X
y : Y
⊢ Filter.map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
| rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq] | theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
| Mathlib.Topology.Inseparable.528_0.2NeLzt0mQ64QlfB | theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y z : X
s✝ : Set X
f g : X → Y
t s : Set (SeparationQuotient X)
x : X
⊢ Filter.map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
| rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds] | theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
| Mathlib.Topology.Inseparable.533_0.2NeLzt0mQ64QlfB | theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y z : X
s : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → α
hf : ∀ (x y : X), (x ~ᵢ y) → f x = f y
x : X
l : Filter α
⊢ Tendsto (lift f hf) (𝓝 (mk x)) l ↔ Tendsto f (𝓝 x) l | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
| simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk] | @[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
| Mathlib.Topology.Inseparable.554_0.2NeLzt0mQ64QlfB | @[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → α
hf : ∀ (x y : X), (x ~ᵢ y) → f x = f y
x : X
s : Set (SeparationQuotient X)
l : Filter α
⊢ Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
| simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk] | @[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
| Mathlib.Topology.Inseparable.560_0.2NeLzt0mQ64QlfB | @[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y
hf : ∀ (x y : X), (x ~ᵢ y) → f x = f y
s : Set (SeparationQuotient X)
⊢ ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
| simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage] | @[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
| Mathlib.Topology.Inseparable.580_0.2NeLzt0mQ64QlfB | @[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y
hf : ∀ (x y : X), (x ~ᵢ y) → f x = f y
⊢ Continuous (lift f hf) ↔ Continuous f | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
#align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
@[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
| simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ] | @[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
| Mathlib.Topology.Inseparable.586_0.2NeLzt0mQ64QlfB | @[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
x : X
y : Y
l : Filter α
⊢ Tendsto (uncurry (lift₂ f hf)) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
#align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
@[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]
#align separation_quotient.continuous_lift SeparationQuotient.continuous_lift
/-- Lift a map `f : X → Y → α` such that `Inseparable a b → Inseparable c d → f a c = f b d` to a
map `SeparationQuotient X → SeparationQuotient Y → α`. -/
def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) :
SeparationQuotient X → SeparationQuotient Y → α := fun x y => Quotient.liftOn₂' x y f hf
#align separation_quotient.lift₂ SeparationQuotient.lift₂
@[simp]
theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) (x : X)
(y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
rfl
#align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
@[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
| rw [← map_prod_map_mk_nhds, tendsto_map'_iff] | @[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
| Mathlib.Topology.Inseparable.604_0.2NeLzt0mQ64QlfB | @[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
x : X
y : Y
l : Filter α
⊢ Tendsto (uncurry (lift₂ f hf) ∘ Prod.map mk mk) (𝓝 (x, y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
#align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
@[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]
#align separation_quotient.continuous_lift SeparationQuotient.continuous_lift
/-- Lift a map `f : X → Y → α` such that `Inseparable a b → Inseparable c d → f a c = f b d` to a
map `SeparationQuotient X → SeparationQuotient Y → α`. -/
def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) :
SeparationQuotient X → SeparationQuotient Y → α := fun x y => Quotient.liftOn₂' x y f hf
#align separation_quotient.lift₂ SeparationQuotient.lift₂
@[simp]
theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) (x : X)
(y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
rfl
#align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
@[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
| rfl | @[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
| Mathlib.Topology.Inseparable.604_0.2NeLzt0mQ64QlfB | @[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
x : X
y : Y
s : Set (SeparationQuotient X × SeparationQuotient Y)
l : Filter α
⊢ Tendsto (uncurry (lift₂ f hf)) (𝓝[s] (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
#align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
@[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]
#align separation_quotient.continuous_lift SeparationQuotient.continuous_lift
/-- Lift a map `f : X → Y → α` such that `Inseparable a b → Inseparable c d → f a c = f b d` to a
map `SeparationQuotient X → SeparationQuotient Y → α`. -/
def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) :
SeparationQuotient X → SeparationQuotient Y → α := fun x y => Quotient.liftOn₂' x y f hf
#align separation_quotient.lift₂ SeparationQuotient.lift₂
@[simp]
theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) (x : X)
(y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
rfl
#align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
@[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
rfl
#align separation_quotient.tendsto_lift₂_nhds SeparationQuotient.tendsto_lift₂_nhds
@[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l := by
| rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal] | @[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l := by
| Mathlib.Topology.Inseparable.612_0.2NeLzt0mQ64QlfB | @[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
x : X
y : Y
s : Set (SeparationQuotient X × SeparationQuotient Y)
l : Filter α
⊢ Tendsto (uncurry (lift₂ f hf)) (Filter.map (Prod.map mk mk) (𝓝 (x, y) ⊓ 𝓟 (Prod.map mk mk ⁻¹' s))) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
#align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
@[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]
#align separation_quotient.continuous_lift SeparationQuotient.continuous_lift
/-- Lift a map `f : X → Y → α` such that `Inseparable a b → Inseparable c d → f a c = f b d` to a
map `SeparationQuotient X → SeparationQuotient Y → α`. -/
def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) :
SeparationQuotient X → SeparationQuotient Y → α := fun x y => Quotient.liftOn₂' x y f hf
#align separation_quotient.lift₂ SeparationQuotient.lift₂
@[simp]
theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) (x : X)
(y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
rfl
#align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
@[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
rfl
#align separation_quotient.tendsto_lift₂_nhds SeparationQuotient.tendsto_lift₂_nhds
@[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l := by
rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]
| rfl | @[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l := by
rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]
| Mathlib.Topology.Inseparable.612_0.2NeLzt0mQ64QlfB | @[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → Z
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
s : Set (SeparationQuotient X × SeparationQuotient Y)
⊢ ContinuousOn (uncurry (lift₂ f hf)) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
#align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
@[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]
#align separation_quotient.continuous_lift SeparationQuotient.continuous_lift
/-- Lift a map `f : X → Y → α` such that `Inseparable a b → Inseparable c d → f a c = f b d` to a
map `SeparationQuotient X → SeparationQuotient Y → α`. -/
def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) :
SeparationQuotient X → SeparationQuotient Y → α := fun x y => Quotient.liftOn₂' x y f hf
#align separation_quotient.lift₂ SeparationQuotient.lift₂
@[simp]
theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) (x : X)
(y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
rfl
#align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
@[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
rfl
#align separation_quotient.tendsto_lift₂_nhds SeparationQuotient.tendsto_lift₂_nhds
@[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l := by
rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]
rfl
#align separation_quotient.tendsto_lift₂_nhds_within SeparationQuotient.tendsto_lift₂_nhdsWithin
@[simp]
theorem continuousAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} :
ContinuousAt (uncurry <| lift₂ f hf) (mk x, mk y) ↔ ContinuousAt (uncurry f) (x, y) :=
tendsto_lift₂_nhds
#align separation_quotient.continuous_at_lift₂ SeparationQuotient.continuousAt_lift₂
@[simp] theorem continuousWithinAt_lift₂ {f : X → Y → Z}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {x : X} {y : Y} :
ContinuousWithinAt (uncurry <| lift₂ f hf) s (mk x, mk y) ↔
ContinuousWithinAt (uncurry f) (Prod.map mk mk ⁻¹' s) (x, y) :=
tendsto_lift₂_nhdsWithin
#align separation_quotient.continuous_within_at_lift₂ SeparationQuotient.continuousWithinAt_lift₂
@[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) := by
| simp_rw [ContinuousOn, (surjective_mk.Prod_map surjective_mk).forall, Prod.forall, Prod.map,
continuousWithinAt_lift₂] | @[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) := by
| Mathlib.Topology.Inseparable.636_0.2NeLzt0mQ64QlfB | @[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → Z
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
s : Set (SeparationQuotient X × SeparationQuotient Y)
⊢ (∀ (a : X) (b : Y), (mk a, mk b) ∈ s → ContinuousWithinAt (uncurry f) (Prod.map mk mk ⁻¹' s) (a, b)) ↔
∀ (a : X) (b : Y), (a, b) ∈ Prod.map mk mk ⁻¹' s → ContinuousWithinAt (uncurry f) (Prod.map mk mk ⁻¹' s) (a, b) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
#align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
@[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]
#align separation_quotient.continuous_lift SeparationQuotient.continuous_lift
/-- Lift a map `f : X → Y → α` such that `Inseparable a b → Inseparable c d → f a c = f b d` to a
map `SeparationQuotient X → SeparationQuotient Y → α`. -/
def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) :
SeparationQuotient X → SeparationQuotient Y → α := fun x y => Quotient.liftOn₂' x y f hf
#align separation_quotient.lift₂ SeparationQuotient.lift₂
@[simp]
theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) (x : X)
(y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
rfl
#align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
@[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
rfl
#align separation_quotient.tendsto_lift₂_nhds SeparationQuotient.tendsto_lift₂_nhds
@[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l := by
rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]
rfl
#align separation_quotient.tendsto_lift₂_nhds_within SeparationQuotient.tendsto_lift₂_nhdsWithin
@[simp]
theorem continuousAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} :
ContinuousAt (uncurry <| lift₂ f hf) (mk x, mk y) ↔ ContinuousAt (uncurry f) (x, y) :=
tendsto_lift₂_nhds
#align separation_quotient.continuous_at_lift₂ SeparationQuotient.continuousAt_lift₂
@[simp] theorem continuousWithinAt_lift₂ {f : X → Y → Z}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {x : X} {y : Y} :
ContinuousWithinAt (uncurry <| lift₂ f hf) s (mk x, mk y) ↔
ContinuousWithinAt (uncurry f) (Prod.map mk mk ⁻¹' s) (x, y) :=
tendsto_lift₂_nhdsWithin
#align separation_quotient.continuous_within_at_lift₂ SeparationQuotient.continuousWithinAt_lift₂
@[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) := by
simp_rw [ContinuousOn, (surjective_mk.Prod_map surjective_mk).forall, Prod.forall, Prod.map,
continuousWithinAt_lift₂]
| rfl | @[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) := by
simp_rw [ContinuousOn, (surjective_mk.Prod_map surjective_mk).forall, Prod.forall, Prod.map,
continuousWithinAt_lift₂]
| Mathlib.Topology.Inseparable.636_0.2NeLzt0mQ64QlfB | @[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → Z
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
⊢ Continuous (uncurry (lift₂ f hf)) ↔ Continuous (uncurry f) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
#align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
@[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]
#align separation_quotient.continuous_lift SeparationQuotient.continuous_lift
/-- Lift a map `f : X → Y → α` such that `Inseparable a b → Inseparable c d → f a c = f b d` to a
map `SeparationQuotient X → SeparationQuotient Y → α`. -/
def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) :
SeparationQuotient X → SeparationQuotient Y → α := fun x y => Quotient.liftOn₂' x y f hf
#align separation_quotient.lift₂ SeparationQuotient.lift₂
@[simp]
theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) (x : X)
(y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
rfl
#align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
@[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
rfl
#align separation_quotient.tendsto_lift₂_nhds SeparationQuotient.tendsto_lift₂_nhds
@[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l := by
rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]
rfl
#align separation_quotient.tendsto_lift₂_nhds_within SeparationQuotient.tendsto_lift₂_nhdsWithin
@[simp]
theorem continuousAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} :
ContinuousAt (uncurry <| lift₂ f hf) (mk x, mk y) ↔ ContinuousAt (uncurry f) (x, y) :=
tendsto_lift₂_nhds
#align separation_quotient.continuous_at_lift₂ SeparationQuotient.continuousAt_lift₂
@[simp] theorem continuousWithinAt_lift₂ {f : X → Y → Z}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {x : X} {y : Y} :
ContinuousWithinAt (uncurry <| lift₂ f hf) s (mk x, mk y) ↔
ContinuousWithinAt (uncurry f) (Prod.map mk mk ⁻¹' s) (x, y) :=
tendsto_lift₂_nhdsWithin
#align separation_quotient.continuous_within_at_lift₂ SeparationQuotient.continuousWithinAt_lift₂
@[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) := by
simp_rw [ContinuousOn, (surjective_mk.Prod_map surjective_mk).forall, Prod.forall, Prod.map,
continuousWithinAt_lift₂]
rfl
#align separation_quotient.continuous_on_lift₂ SeparationQuotient.continuousOn_lift₂
@[simp]
theorem continuous_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} :
Continuous (uncurry <| lift₂ f hf) ↔ Continuous (uncurry f) := by
| simp only [continuous_iff_continuousOn_univ, continuousOn_lift₂, preimage_univ] | @[simp]
theorem continuous_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} :
Continuous (uncurry <| lift₂ f hf) ↔ Continuous (uncurry f) := by
| Mathlib.Topology.Inseparable.645_0.2NeLzt0mQ64QlfB | @[simp]
theorem continuous_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} :
Continuous (uncurry <| lift₂ f hf) ↔ Continuous (uncurry f) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
h : ∀ (x : X), f x ~ᵢ g x
⊢ Continuous f ↔ Continuous g | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
#align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
@[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]
#align separation_quotient.continuous_lift SeparationQuotient.continuous_lift
/-- Lift a map `f : X → Y → α` such that `Inseparable a b → Inseparable c d → f a c = f b d` to a
map `SeparationQuotient X → SeparationQuotient Y → α`. -/
def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) :
SeparationQuotient X → SeparationQuotient Y → α := fun x y => Quotient.liftOn₂' x y f hf
#align separation_quotient.lift₂ SeparationQuotient.lift₂
@[simp]
theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) (x : X)
(y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
rfl
#align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
@[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
rfl
#align separation_quotient.tendsto_lift₂_nhds SeparationQuotient.tendsto_lift₂_nhds
@[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l := by
rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]
rfl
#align separation_quotient.tendsto_lift₂_nhds_within SeparationQuotient.tendsto_lift₂_nhdsWithin
@[simp]
theorem continuousAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} :
ContinuousAt (uncurry <| lift₂ f hf) (mk x, mk y) ↔ ContinuousAt (uncurry f) (x, y) :=
tendsto_lift₂_nhds
#align separation_quotient.continuous_at_lift₂ SeparationQuotient.continuousAt_lift₂
@[simp] theorem continuousWithinAt_lift₂ {f : X → Y → Z}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {x : X} {y : Y} :
ContinuousWithinAt (uncurry <| lift₂ f hf) s (mk x, mk y) ↔
ContinuousWithinAt (uncurry f) (Prod.map mk mk ⁻¹' s) (x, y) :=
tendsto_lift₂_nhdsWithin
#align separation_quotient.continuous_within_at_lift₂ SeparationQuotient.continuousWithinAt_lift₂
@[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) := by
simp_rw [ContinuousOn, (surjective_mk.Prod_map surjective_mk).forall, Prod.forall, Prod.map,
continuousWithinAt_lift₂]
rfl
#align separation_quotient.continuous_on_lift₂ SeparationQuotient.continuousOn_lift₂
@[simp]
theorem continuous_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} :
Continuous (uncurry <| lift₂ f hf) ↔ Continuous (uncurry f) := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift₂, preimage_univ]
#align separation_quotient.continuous_lift₂ SeparationQuotient.continuous_lift₂
end SeparationQuotient
theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
Continuous f ↔ Continuous g := by
| simp_rw [SeparationQuotient.inducing_mk.continuous_iff (β := Y)] | theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
Continuous f ↔ Continuous g := by
| Mathlib.Topology.Inseparable.653_0.2NeLzt0mQ64QlfB | theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
Continuous f ↔ Continuous g | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
h : ∀ (x : X), f x ~ᵢ g x
⊢ Continuous (SeparationQuotient.mk ∘ f) ↔ Continuous (SeparationQuotient.mk ∘ g) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
/-!
# Inseparable points in a topological space
In this file we define
* `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
* `Inseparable`: a relation saying that two points in a topological space have the same
neighbourhoods; equivalently, they can't be separated by an open set;
* `InseparableSetoid X`: same relation, as a `Setoid`;
* `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
We also prove various basic properties of the relation `Inseparable`.
## Notations
- `x ⤳ y`: notation for `Specializes x y`;
- `x ~ᵢ y` is used as a local notation for `Inseparable x y`;
- `𝓝 x` is the neighbourhoods filter `nhds x` of a point `x`, defined elsewhere.
## Tags
topological space, separation setoid
-/
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
/-!
### `Specializes` relation
-/
/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈ s → y ∈ s`;
* for any open set `s` we have `y ∈ s → x ∈ s`;
* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
#align specializes Specializes
@[inherit_doc]
infixl:300 " ⤳ " => Specializes
/-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. -/
theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] := by
tfae_have 1 → 2
· exact (pure_le_nhds _).trans
tfae_have 2 → 3
· exact fun h s hso hy => h (hso.mem_nhds hy)
tfae_have 3 → 4
· exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
tfae_have 4 → 5
· exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
tfae_have 6 ↔ 5
· exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
tfae_have 5 ↔ 7
· rw [mem_closure_iff_clusterPt, principal_singleton]
tfae_have 5 → 1
· refine' fun h => (nhds_basis_opens _).ge_iff.2 _
rintro s ⟨hy, ho⟩
rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
exact ho.mem_nhds hxs
tfae_finish
#align specializes_tfae specializes_TFAE
theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
Iff.rfl
#align specializes_iff_nhds specializes_iff_nhds
theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
(specializes_TFAE x y).out 0 1
#align specializes_iff_pure specializes_iff_pure
alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
#align specializes.nhds_le_nhds Specializes.nhds_le_nhds
alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
#align specializes.pure_le_nhds Specializes.pure_le_nhds
theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
ext; simp [specializes_iff_pure, le_def]
theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
(specializes_TFAE x y).out 0 2
#align specializes_iff_forall_open specializes_iff_forall_open
theorem Specializes.mem_open (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s :=
specializes_iff_forall_open.1 h s hs hy
#align specializes.mem_open Specializes.mem_open
theorem IsOpen.not_specializes (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y := fun h =>
hx <| h.mem_open hs hy
#align is_open.not_specializes IsOpen.not_specializes
theorem specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : Set X, IsClosed s → x ∈ s → y ∈ s :=
(specializes_TFAE x y).out 0 3
#align specializes_iff_forall_closed specializes_iff_forall_closed
theorem Specializes.mem_closed (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s :=
specializes_iff_forall_closed.1 h s hs hx
#align specializes.mem_closed Specializes.mem_closed
theorem IsClosed.not_specializes (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y := fun h =>
hy <| h.mem_closed hs hx
#align is_closed.not_specializes IsClosed.not_specializes
theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
(specializes_TFAE x y).out 0 4
#align specializes_iff_mem_closure specializes_iff_mem_closure
alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
#align specializes.mem_closure Specializes.mem_closure
theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
(specializes_TFAE x y).out 0 5
#align specializes_iff_closure_subset specializes_iff_closure_subset
alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
#align specializes.closure_subset Specializes.closure_subset
-- porting note: new lemma
theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
(specializes_TFAE x y).out 0 6
theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
specializes_iff_pure.trans h.ge_iff
#align filter.has_basis.specializes_iff Filter.HasBasis.specializes_iff
theorem specializes_rfl : x ⤳ x := le_rfl
#align specializes_rfl specializes_rfl
@[refl]
theorem specializes_refl (x : X) : x ⤳ x :=
specializes_rfl
#align specializes_refl specializes_refl
@[trans]
theorem Specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z :=
le_trans
#align specializes.trans Specializes.trans
theorem specializes_of_eq (e : x = y) : x ⤳ y :=
e ▸ specializes_refl x
#align specializes_of_eq specializes_of_eq
theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
specializes_iff_pure.2 <|
calc
pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
_ ≤ 𝓝[s] y := h₁
_ ≤ 𝓝 y := inf_le_left
#align specializes_of_nhds_within specializes_of_nhdsWithin
theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
specializes_iff_pure.2 fun _s hs =>
mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
#align specializes.map_of_continuous_at Specializes.map_of_continuousAt
theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt
#align specializes.map Specializes.map
theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage]
#align inducing.specializes_iff Inducing.specializes_iff
theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
inducing_subtype_val.specializes_iff.symm
#align subtype_specializes_iff subtype_specializes_iff
@[simp]
theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
simp only [Specializes, nhds_prod_eq, prod_le_prod]
#align specializes_prod specializes_prod
theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) :=
specializes_prod.2 ⟨hx, hy⟩
#align specializes.prod Specializes.prod
@[simp]
theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
simp only [Specializes, nhds_pi, pi_le_pi]
#align specializes_pi specializes_pi
theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
rw [specializes_iff_forall_open]
push_neg
rfl
#align not_specializes_iff_exists_open not_specializes_iff_exists_open
theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
rw [specializes_iff_forall_closed]
push_neg
rfl
#align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
Continuous (s.piecewise f g) := by
simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
variable (X)
/-- Specialization forms a preorder on the topological space. -/
def specializationPreorder : Preorder X :=
{ Preorder.lift (OrderDual.toDual ∘ 𝓝) with
le := fun x y => y ⤳ x
lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
#align specialization_preorder specializationPreorder
variable {X}
/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem Continuous.specialization_monotone (hf : Continuous f) :
@Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun _ _ h => h.map hf
#align continuous.specialization_monotone Continuous.specialization_monotone
/-!
### `Inseparable` relation
-/
/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
-/
def Inseparable (x y : X) : Prop :=
𝓝 x = 𝓝 y
#align inseparable Inseparable
local infixl:0 " ~ᵢ " => Inseparable
theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
Iff.rfl
#align inseparable_def inseparable_def
theorem inseparable_iff_specializes_and : (x ~ᵢ y) ↔ x ⤳ y ∧ y ⤳ x :=
le_antisymm_iff
#align inseparable_iff_specializes_and inseparable_iff_specializes_and
theorem Inseparable.specializes (h : x ~ᵢ y) : x ⤳ y := h.le
#align inseparable.specializes Inseparable.specializes
theorem Inseparable.specializes' (h : x ~ᵢ y) : y ⤳ x := h.ge
#align inseparable.specializes' Inseparable.specializes'
theorem Specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ᵢ y :=
le_antisymm h₁ h₂
#align specializes.antisymm Specializes.antisymm
theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and, ← iff_def,
Iff.comm]
#align inseparable_iff_forall_open inseparable_iff_forall_open
theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
#align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def]
#align inseparable_iff_forall_closed inseparable_iff_forall_closed
theorem inseparable_iff_mem_closure :
(x ~ᵢ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
#align inseparable_iff_mem_closure inseparable_iff_mem_closure
theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
#align inseparable_iff_closure_eq inseparable_iff_closure_eq
theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ᵢ y :=
(specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
#align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ᵢ f y) ↔ (x ~ᵢ y) := by
simp only [inseparable_iff_specializes_and, hf.specializes_iff]
#align inducing.inseparable_iff Inducing.inseparable_iff
theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y) ↔ ((x : X) ~ᵢ y) :=
inducing_subtype_val.inseparable_iff.symm
#align subtype_inseparable_iff subtype_inseparable_iff
@[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
simp only [Inseparable, nhds_prod_eq, prod_inj]
#align inseparable_prod inseparable_prod
theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
(x₁, y₁) ~ᵢ (x₂, y₂) :=
inseparable_prod.2 ⟨hx, hy⟩
#align inseparable.prod Inseparable.prod
@[simp]
theorem inseparable_pi {f g : ∀ i, π i} : (f ~ᵢ g) ↔ ∀ i, f i ~ᵢ g i := by
simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
#align inseparable_pi inseparable_pi
namespace Inseparable
@[refl]
theorem refl (x : X) : x ~ᵢ x :=
Eq.refl (𝓝 x)
#align inseparable.refl Inseparable.refl
theorem rfl : x ~ᵢ x :=
refl x
#align inseparable.rfl Inseparable.rfl
theorem of_eq (e : x = y) : Inseparable x y :=
e ▸ refl x
#align inseparable.of_eq Inseparable.of_eq
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
#align inseparable.symm Inseparable.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
#align inseparable.trans Inseparable.trans
theorem nhds_eq (h : x ~ᵢ y) : 𝓝 x = 𝓝 y := h
#align inseparable.nhds_eq Inseparable.nhds_eq
theorem mem_open_iff (h : x ~ᵢ y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_open.1 h s hs
#align inseparable.mem_open_iff Inseparable.mem_open_iff
theorem mem_closed_iff (h : x ~ᵢ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
inseparable_iff_forall_closed.1 h s hs
#align inseparable.mem_closed_iff Inseparable.mem_closed_iff
theorem map_of_continuousAt (h : x ~ᵢ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
f x ~ᵢ f y :=
(h.specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
#align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
theorem map (h : x ~ᵢ y) (hf : Continuous f) : f x ~ᵢ f y :=
h.map_of_continuousAt hf.rst.imntinuousAt hf.rst.imntinuousAt
#align inseparable.map Inseparable.map
end Inseparable
theorem IsClosed.not_inseparable (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_closed_iff hs).1 hx
#align is_closed.not_inseparable IsClosed.not_inseparable
theorem IsOpen.not_inseparable (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬(x ~ᵢ y) := fun h =>
hy <| (h.mem_open_iff hs).1 hx
#align is_open.not_inseparable IsOpen.not_inseparable
/-!
### Separation quotient
In this section we define the quotient of a topological space by the `Inseparable` relation.
-/
variable (X)
/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
#align inseparable_setoid inseparableSetoid
/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
be a T₀ space. -/
def SeparationQuotient := Quotient (inseparableSetoid X)
#align separation_quotient SeparationQuotient
instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
variable {X}
variable {t : Set (SeparationQuotient X)}
namespace SeparationQuotient
/-- The natural map from a topological space to its separation quotient. -/
def mk : X → SeparationQuotient X := Quotient.mk''
#align separation_quotient.mk SeparationQuotient.mk
theorem quotientMap_mk : QuotientMap (mk : X → SeparationQuotient X) :=
quotientMap_quot_mk
#align separation_quotient.quotient_map_mk SeparationQuotient.quotientMap_mk
theorem continuous_mk : Continuous (mk : X → SeparationQuotient X) :=
continuous_quot_mk
#align separation_quotient.continuous_mk SeparationQuotient.continuous_mk
@[simp]
theorem mk_eq_mk : mk x = mk y ↔ (x ~ᵢ y) :=
Quotient.eq''
#align separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
theorem surjective_mk : Surjective (mk : X → SeparationQuotient X) :=
surjective_quot_mk _
#align separation_quotient.surjective_mk SeparationQuotient.surjective_mk
@[simp]
theorem range_mk : range (mk : X → SeparationQuotient X) = univ :=
surjective_mk.range_eq
#align separation_quotient.range_mk SeparationQuotient.range_mk
instance [Nonempty X] : Nonempty (SeparationQuotient X) :=
Nonempty.map mk ‹_›
instance [Inhabited X] : Inhabited (SeparationQuotient X) :=
⟨mk default⟩
instance [Subsingleton X] : Subsingleton (SeparationQuotient X) :=
surjective_mk.subsingleton
theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
#align separation_quotient.preimage_image_mk_open SeparationQuotient.preimage_image_mk_open
theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
#align separation_quotient.is_open_map_mk SeparationQuotient.isOpenMap_mk
theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
#align separation_quotient.preimage_image_mk_closed SeparationQuotient.preimage_image_mk_closed
theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk) fun s hs =>
⟨mk '' s, isOpenMap_mk s hs, preimage_image_mk_open hs⟩⟩
#align separation_quotient.inducing_mk SeparationQuotient.inducing_mk
theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; exact isClosed_univ
#align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
@[simp]
theorem comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x :=
(inducing_mk.nhds_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_mk SeparationQuotient.comap_mk_nhds_mk
@[simp]
theorem comap_mk_nhdsSet_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s :=
(inducing_mk.nhdsSet_eq_comap _).symm
#align separation_quotient.comap_mk_nhds_set_image SeparationQuotient.comap_mk_nhdsSet_image
theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds SeparationQuotient.map_mk_nhds
theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]
#align separation_quotient.map_mk_nhds_set SeparationQuotient.map_mk_nhdsSet
theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]
#align separation_quotient.comap_mk_nhds_set SeparationQuotient.comap_mk_nhdsSet
theorem preimage_mk_closure : mk ⁻¹' closure t = closure (mk ⁻¹' t) :=
isOpenMap_mk.preimage_closure_eq_closure_preimage continuous_mk t
#align separation_quotient.preimage_mk_closure SeparationQuotient.preimage_mk_closure
theorem preimage_mk_interior : mk ⁻¹' interior t = interior (mk ⁻¹' t) :=
isOpenMap_mk.preimage_interior_eq_interior_preimage continuous_mk t
#align separation_quotient.preimage_mk_interior SeparationQuotient.preimage_mk_interior
theorem preimage_mk_frontier : mk ⁻¹' frontier t = frontier (mk ⁻¹' t) :=
isOpenMap_mk.preimage_frontier_eq_frontier_preimage continuous_mk t
#align separation_quotient.preimage_mk_frontier SeparationQuotient.preimage_mk_frontier
theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
(image_closure_subset_closure_image continuous_mk).antisymm <|
isClosedMap_mk.closure_image_subset _
#align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
#align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
#align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
/-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
`SeparationQuotient X → α`. -/
def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>
Quotient.liftOn' x f hf
#align separation_quotient.lift SeparationQuotient.lift
@[simp]
theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
rfl
#align separation_quotient.lift_mk SeparationQuotient.lift_mk
@[simp]
theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : lift f hf ∘ mk = f :=
rfl
#align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
@[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
@[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
#align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
@[simp]
theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
tendsto_lift_nhds_mk
#align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
@[simp]
theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} {x : X} :
ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
tendsto_lift_nhdsWithin_mk
#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
@[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
#align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
@[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]
#align separation_quotient.continuous_lift SeparationQuotient.continuous_lift
/-- Lift a map `f : X → Y → α` such that `Inseparable a b → Inseparable c d → f a c = f b d` to a
map `SeparationQuotient X → SeparationQuotient Y → α`. -/
def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) :
SeparationQuotient X → SeparationQuotient Y → α := fun x y => Quotient.liftOn₂' x y f hf
#align separation_quotient.lift₂ SeparationQuotient.lift₂
@[simp]
theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d) (x : X)
(y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
rfl
#align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
@[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
rfl
#align separation_quotient.tendsto_lift₂_nhds SeparationQuotient.tendsto_lift₂_nhds
@[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l := by
rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]
rfl
#align separation_quotient.tendsto_lift₂_nhds_within SeparationQuotient.tendsto_lift₂_nhdsWithin
@[simp]
theorem continuousAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} :
ContinuousAt (uncurry <| lift₂ f hf) (mk x, mk y) ↔ ContinuousAt (uncurry f) (x, y) :=
tendsto_lift₂_nhds
#align separation_quotient.continuous_at_lift₂ SeparationQuotient.continuousAt_lift₂
@[simp] theorem continuousWithinAt_lift₂ {f : X → Y → Z}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {x : X} {y : Y} :
ContinuousWithinAt (uncurry <| lift₂ f hf) s (mk x, mk y) ↔
ContinuousWithinAt (uncurry f) (Prod.map mk mk ⁻¹' s) (x, y) :=
tendsto_lift₂_nhdsWithin
#align separation_quotient.continuous_within_at_lift₂ SeparationQuotient.continuousWithinAt_lift₂
@[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) := by
simp_rw [ContinuousOn, (surjective_mk.Prod_map surjective_mk).forall, Prod.forall, Prod.map,
continuousWithinAt_lift₂]
rfl
#align separation_quotient.continuous_on_lift₂ SeparationQuotient.continuousOn_lift₂
@[simp]
theorem continuous_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} :
Continuous (uncurry <| lift₂ f hf) ↔ Continuous (uncurry f) := by
simp only [continuous_iff_continuousOn_univ, continuousOn_lift₂, preimage_univ]
#align separation_quotient.continuous_lift₂ SeparationQuotient.continuous_lift₂
end SeparationQuotient
theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
Continuous f ↔ Continuous g := by
simp_rw [SeparationQuotient.inducing_mk.continuous_iff (β := Y)]
| exact continuous_congr fun x ↦ SeparationQuotient.mk_eq_mk.mpr (h x) | theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
Continuous f ↔ Continuous g := by
simp_rw [SeparationQuotient.inducing_mk.continuous_iff (β := Y)]
| Mathlib.Topology.Inseparable.653_0.2NeLzt0mQ64QlfB | theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
Continuous f ↔ Continuous g | Mathlib_Topology_Inseparable |
x : ℂ
⊢ Real.sin (arg x) = x.im / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
| unfold arg | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.36_0.CflASCTDE9UCom5 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ Real.sin
(if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then arcsin ((-x).im / abs x) + π else arcsin ((-x).im / abs x) - π) =
x.im / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; | split_ifs | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; | Mathlib.Analysis.SpecialFunctions.Complex.Arg.36_0.CflASCTDE9UCom5 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
h✝ : 0 ≤ x.re
⊢ Real.sin (arcsin (x.im / abs x)) = x.im / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
| simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.36_0.CflASCTDE9UCom5 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
h✝¹ : ¬0 ≤ x.re
h✝ : 0 ≤ x.im
⊢ Real.sin (arcsin ((-x).im / abs x) + π) = x.im / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
| simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.36_0.CflASCTDE9UCom5 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x : ℂ
h✝¹ : ¬0 ≤ x.re
h✝ : ¬0 ≤ x.im
⊢ Real.sin (arcsin ((-x).im / abs x) - π) = x.im / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
| simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.36_0.CflASCTDE9UCom5 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
hx : x ≠ 0
⊢ Real.cos (arg x) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
| rw [arg] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
hx : x ≠ 0
⊢ Real.cos
(if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then arcsin ((-x).im / abs x) + π else arcsin ((-x).im / abs x) - π) =
x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
| split_ifs with h₁ h₂ | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
hx : x ≠ 0
h₁ : 0 ≤ x.re
⊢ Real.cos (arcsin (x.im / abs x)) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· | rw [Real.cos_arcsin] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
hx : x ≠ 0
h₁ : 0 ≤ x.re
⊢ sqrt (1 - (x.im / abs x) ^ 2) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
| field_simp [Real.sqrt_sq, (abs.pos hx).le, *] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
hx : x ≠ 0
h₁ : ¬0 ≤ x.re
h₂ : 0 ≤ x.im
⊢ Real.cos (arcsin ((-x).im / abs x) + π) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· | rw [Real.cos_add_pi, Real.cos_arcsin] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
hx : x ≠ 0
h₁ : ¬0 ≤ x.re
h₂ : 0 ≤ x.im
⊢ -sqrt (1 - ((-x).im / abs x) ^ 2) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
| field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x : ℂ
hx : x ≠ 0
h₁ : ¬0 ≤ x.re
h₂ : ¬0 ≤ x.im
⊢ Real.cos (arcsin ((-x).im / abs x) - π) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· | rw [Real.cos_sub_pi, Real.cos_arcsin] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x : ℂ
hx : x ≠ 0
h₁ : ¬0 ≤ x.re
h₂ : ¬0 ≤ x.im
⊢ -sqrt (1 - ((-x).im / abs x) ^ 2) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
| field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ ↑(abs x) * cexp (↑(arg x) * I) = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
| rcases eq_or_ne x 0 with (rfl | hx) | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl
⊢ ↑(abs 0) * cexp (↑(arg 0) * I) = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· | simp | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
x : ℂ
hx : x ≠ 0
⊢ ↑(abs x) * cexp (↑(arg x) * I) = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· | have : abs x ≠ 0 := abs.ne_zero hx | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
x : ℂ
hx : x ≠ 0
this : abs x ≠ 0
⊢ ↑(abs x) * cexp (↑(arg x) * I) = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
| apply Complex.ext | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.a
x : ℂ
hx : x ≠ 0
this : abs x ≠ 0
⊢ (↑(abs x) * cexp (↑(arg x) * I)).re = x.re | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> | field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> | Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.a
x : ℂ
hx : x ≠ 0
this : abs x ≠ 0
⊢ (↑(abs x) * cexp (↑(arg x) * I)).im = x.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> | field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> | Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ ↑(abs x) * (cos ↑(arg x) + sin ↑(arg x) * I) = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
| rw [← exp_mul_I, abs_mul_exp_arg_mul_I] | @[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.65_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ abs x * Real.cos (arg x) = x.re | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
| simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) | @[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.71_0.CflASCTDE9UCom5 | @[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ abs x * Real.sin (arg x) = x.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
| simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) | @[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.75_0.CflASCTDE9UCom5 | @[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ abs z = 1 ↔ ∃ θ, cexp (↑θ * I) = z | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
| refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.79_0.CflASCTDE9UCom5 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_1
z : ℂ
hz : abs z = 1
⊢ cexp (↑(arg z) * I) = z | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· | calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.79_0.CflASCTDE9UCom5 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
hz : abs z = 1
⊢ cexp (↑(arg z) * I) = ↑(abs z) * cexp (↑(arg z) * I) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by | rw [hz, ofReal_one, one_mul] | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.79_0.CflASCTDE9UCom5 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_2
z : ℂ
⊢ (∃ θ, cexp (↑θ * I) = z) → abs z = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· | rintro ⟨θ, rfl⟩ | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.79_0.CflASCTDE9UCom5 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_2.intro
θ : ℝ
⊢ abs (cexp (↑θ * I)) = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
| exact Complex.abs_exp_ofReal_mul_I θ | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.79_0.CflASCTDE9UCom5 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
⊢ (Set.range fun x => cexp (↑x * I)) = Metric.sphere 0 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
| ext x | @[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.89_0.CflASCTDE9UCom5 | @[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case h
x : ℂ
⊢ (x ∈ Set.range fun x => cexp (↑x * I)) ↔ x ∈ Metric.sphere 0 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
| simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] | @[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.89_0.CflASCTDE9UCom5 | @[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
⊢ arg (↑r * (cos ↑θ + sin ↑θ * I)) = θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
| simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
⊢ (if 0 ≤ (↑r * (cos ↑θ + sin ↑θ * I)).re then arcsin ((↑r * (cos ↑θ + sin ↑θ * I)).im / r)
else
if 0 ≤ (↑r * (cos ↑θ + sin ↑θ * I)).im then arcsin ((-(↑r * (cos ↑θ + sin ↑θ * I))).im / r) + π
else arcsin ((-(↑r * (cos ↑θ + sin ↑θ * I))).im / r) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
| simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
| by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· | rw [if_pos] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
⊢ arcsin (Real.sin θ) = θ
case pos.hc r : ℝ hr : 0 < r θ : ℝ hθ : θ ∈ Set.Ioc (-π) π h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) ⊢ 0 ≤ Real.cos θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
| exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : θ ∉ Set.Icc (-(π / 2)) (π / 2)
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· | rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : θ < -(π / 2) ∨ π / 2 < θ
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
| cases' h₁ with h₁ h₁ | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : θ < -(π / 2)
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· | replace hθ := hθ.1 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
| have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
⊢ Real.cos θ < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
| rw [← neg_pos, ← Real.cos_add_pi] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
⊢ 0 < Real.cos (θ + π) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
| refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_1
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
⊢ -(π / 2) < θ + π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_2
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
⊢ θ + π < π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
| have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
⊢ θ < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
| rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
| rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl.hx₁
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ -(π / 2) ≤ θ + π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [ | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [ | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl.hx₂
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ θ + π ≤ π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
| linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl.hnc
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ ¬0 ≤ Real.sin θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; | exact hsin.not_le | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl.hnc
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ ¬0 ≤ Real.cos θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; | exact hcos.not_le | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : π / 2 < θ
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· | replace hθ := hθ.2 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
| have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
⊢ θ < π + π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
| have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
⊢ 0 ≤ θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
| rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
| rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr.hx₁
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ -(π / 2) ≤ θ - π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [ | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [ | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr.hx₂
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ θ - π ≤ π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
| linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr.hc
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ 0 ≤ Real.sin θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; | exact hsin | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr.hnc
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ ¬0 ≤ Real.cos θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; | exact hcos.not_le | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
⊢ arg (cos ↑θ + sin ↑θ * I) = θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
| rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] | theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.121_0.CflASCTDE9UCom5 | theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
⊢ arg 0 = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by | simp [arg, le_refl] | @[simp]
theorem arg_zero : arg 0 = 0 := by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.126_0.CflASCTDE9UCom5 | @[simp]
theorem arg_zero : arg 0 = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x y : ℂ
h₁ : abs x = abs y
h₂ : arg x = arg y
⊢ x = y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
| rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] | theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.130_0.CflASCTDE9UCom5 | theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
| have hπ : 0 < π := Real.pi_pos | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
hπ : 0 < π
⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
| rcases eq_or_ne z 0 with (rfl | hz) | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl
hπ : 0 < π
⊢ arg 0 ∈ Set.Ioc (-π) π
case inr z : ℂ hπ : 0 < π hz : z ≠ 0 ⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); | simp [hπ, hπ.le] | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); | Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
z : ℂ
hπ : 0 < π
hz : z ≠ 0
⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
| rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.intro.intro
z : ℂ
hπ : 0 < π
hz : z ≠ 0
N : ℤ
hN : arg z + N • (2 * π) ∈ Set.Ioc (-π) (-π + 2 * π)
⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
| rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.intro.intro
z : ℂ
hπ : 0 < π
hz : z ≠ 0
N : ℤ
hN : arg z + ↑N * (2 * π) ∈ Set.Ioc (-π) π
⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
| rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.intro.intro
z : ℂ
hπ : 0 < π
hz : z ≠ 0
N : ℤ
hN : arg z + ↑N * (2 * π) ∈ Set.Ioc (-π) π
⊢ arg (↑(abs z) * (cos (↑(arg z) + ↑N * (2 * ↑π)) + sin (↑(arg z) + ↑N * (2 * ↑π)) * I)) ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N]
| have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.intro.intro
z : ℂ
hπ : 0 < π
hz : z ≠ 0
N : ℤ
hN : arg z + ↑N * (2 * π) ∈ Set.Ioc (-π) π
this : arg (↑(abs z) * (cos ↑(arg z + ↑N * (2 * π)) + sin ↑(arg z + ↑N * (2 * π)) * I)) = arg z + ↑N * (2 * π)
⊢ arg (↑(abs z) * (cos (↑(arg z) + ↑N * (2 * ↑π)) + sin (↑(arg z) + ↑N * (2 * ↑π)) * I)) ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N]
have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN
| push_cast at this | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N]
have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.intro.intro
z : ℂ
hπ : 0 < π
hz : z ≠ 0
N : ℤ
hN : arg z + ↑N * (2 * π) ∈ Set.Ioc (-π) π
this : arg (↑(abs z) * (cos (↑(arg z) + ↑N * (2 * ↑π)) + sin (↑(arg z) + ↑N * (2 * ↑π)) * I)) = arg z + ↑N * (2 * π)
⊢ arg (↑(abs z) * (cos (↑(arg z) + ↑N * (2 * ↑π)) + sin (↑(arg z) + ↑N * (2 * ↑π)) * I)) ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N]
have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN
push_cast at this
| rwa [this] | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N]
have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN
push_cast at this
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ 0 ≤ arg z ↔ 0 ≤ z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N]
have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN
push_cast at this
rwa [this]
#align complex.arg_mem_Ioc Complex.arg_mem_Ioc
@[simp]
theorem range_arg : Set.range arg = Set.Ioc (-π) π :=
(Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩
#align complex.range_arg Complex.range_arg
theorem arg_le_pi (x : ℂ) : arg x ≤ π :=
(arg_mem_Ioc x).2
#align complex.arg_le_pi Complex.arg_le_pi
theorem neg_pi_lt_arg (x : ℂ) : -π < arg x :=
(arg_mem_Ioc x).1
#align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg
theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π :=
abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩
#align complex.abs_arg_le_pi Complex.abs_arg_le_pi
@[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
| rcases eq_or_ne z 0 with (rfl | h₀) | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.166_0.CflASCTDE9UCom5 | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl
⊢ 0 ≤ arg 0 ↔ 0 ≤ 0.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# The argument of a complex number.
We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π],
such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
while `arg 0` defaults to `0`
-/
noncomputable section
namespace Complex
open ComplexConjugate Real Topology
open Filter Set
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re then Real.arcsin (x.im / abs x)
else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π
#align complex.arg Complex.arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
#align complex.sin_arg Complex.sin_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
#align complex.cos_arg Complex.cos_arg
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x)
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x)
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
#align complex.abs_eq_one_iff Complex.abs_eq_one_iff
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
set_option linter.uppercaseLean3 false in
#align complex.range_exp_mul_I Complex.range_exp_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_pos]
exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁]
· rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁
cases' h₁ with h₁ h₁
· replace hθ := hθ.1
have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith
have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ
rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le]
· replace hθ := hθ.2
have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith)
have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩
rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le]
set_option linter.uppercaseLean3 false in
#align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
set_option linter.uppercaseLean3 false in
#align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I
@[simp]
theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl]
#align complex.arg_zero Complex.arg_zero
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]
#align complex.ext_abs_arg Complex.ext_abs_arg
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y :=
⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩
#align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N]
have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN
push_cast at this
rwa [this]
#align complex.arg_mem_Ioc Complex.arg_mem_Ioc
@[simp]
theorem range_arg : Set.range arg = Set.Ioc (-π) π :=
(Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩
#align complex.range_arg Complex.range_arg
theorem arg_le_pi (x : ℂ) : arg x ≤ π :=
(arg_mem_Ioc x).2
#align complex.arg_le_pi Complex.arg_le_pi
theorem neg_pi_lt_arg (x : ℂ) : -π < arg x :=
(arg_mem_Ioc x).1
#align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg
theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π :=
abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩
#align complex.abs_arg_le_pi Complex.abs_arg_le_pi
@[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
rcases eq_or_ne z 0 with (rfl | h₀); · | simp | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
rcases eq_or_ne z 0 with (rfl | h₀); · | Mathlib.Analysis.SpecialFunctions.Complex.Arg.166_0.CflASCTDE9UCom5 | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |