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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