Loogle!

Result

Found 212 declarations mentioning Homeomorph.symm. Of these, only the first 200 are shown.

• Homeomorph.symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : Y ≃ₜ X
• Homeomorph.refl_symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} [TopologicalSpace X] : (Homeomorph.refl X).symm = Homeomorph.refl X
• Homeomorph.symm_bijective 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] : Function.Bijective Homeomorph.symm
• Homeomorph.symm_symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : h.symm.symm = h
• Homeomorph.self_trans_symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : h.trans h.symm = Homeomorph.refl X
• Homeomorph.symm_trans_self 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : h.symm.trans h = Homeomorph.refl Y
• Homeomorph.inv_def 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} [TopologicalSpace X] (h : X ≃ₜ X) : h⁻¹ = h.symm
• Homeomorph.continuous_symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : Continuous ⇑h.symm
• Homeomorph.homeomorphOfUnique_symm_apply 📋 Mathlib.Topology.Homeomorph.Defs
  (X : Type u_1) (Y : Type u_2) [TopologicalSpace X] [TopologicalSpace Y] [Unique X] [Unique Y] (a✝ : Y) : (Homeomorph.homeomorphOfUnique X Y).symm a✝ = default
• Homeomorph.apply_symm_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (y : Y) : h (h.symm y) = y
• Homeomorph.coe_symm_toEquiv 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : ⇑h.symm = ⇑h.symm
• Homeomorph.symm_apply_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (x : X) : h.symm (h x) = x
• IsHomeomorph.homeomorph_symm_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (hf : IsHomeomorph f) (b : Y) : (IsHomeomorph.homeomorph f hf).symm b = Function.surjInv ⋯ b
• Homeomorph.self_comp_symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : ⇑h ∘ ⇑h.symm = id
• Homeomorph.symm_comp_self 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : ⇑h.symm ∘ ⇑h = id
• Homeomorph.eq_symm_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) {x : X} {y : Y} : x = h.symm y ↔ h x = y
• Homeomorph.image_symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : Set.image ⇑h.symm = Set.preimage ⇑h
• Homeomorph.preimage_symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : Set.preimage ⇑h.symm = Set.image ⇑h
• Homeomorph.symm_apply_eq 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) {x : X} {y : Y} : h.symm y = x ↔ y = h x
• Homeomorph.image_eq_preimage_symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (s : Set X) : ⇑h '' s = ⇑h.symm ⁻¹' s
• Equiv.symm_toHomeomorph 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ Y) (he : ∀ (s : Set Y), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s) : (e.toHomeomorph he).symm = e.symm.toHomeomorph ⋯
• Homeomorph.comap_nhds_eq 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (y : Y) : Filter.comap (⇑h) (nhds y) = nhds (h.symm y)
• Homeomorph.symm_map_nhds_eq 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (x : X) : Filter.map (⇑h.symm) (nhds (h x)) = nhds x
• Homeomorph.homeomorph_mk_coe_symm 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (a : X ≃ Y) (b : Continuous a.toFun) (c : Continuous a.invFun) : ⇑{ toEquiv := a, continuous_toFun := b, continuous_invFun := c }.symm = ⇑a.symm
• Homeomorph.inv_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} [TopologicalSpace X] (f : X ≃ₜ X) (x : X) : f⁻¹ x = f.symm x
• Equiv.toHomeomorphOfIsInducing_symm_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X ≃ Y) (hf : Topology.IsInducing ⇑f) : ⇑(f.toHomeomorphOfIsInducing hf).symm = ⇑f.symm
• Equiv.toHomeomorphOfContinuousClosed_symm_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ Y) (h₁ : Continuous ⇑e) (h₂ : IsClosedMap ⇑e) : ⇑(e.toHomeomorphOfContinuousClosed h₁ h₂).symm = ⇑e.symm
• Equiv.toHomeomorphOfContinuousOpen_symm_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ Y) (h₁ : Continuous ⇑e) (h₂ : IsOpenMap ⇑e) : ⇑(e.toHomeomorphOfContinuousOpen h₁ h₂).symm = ⇑e.symm
• Homeomorph.symm_trans_apply 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : X ≃ₜ Y) (g : Y ≃ₜ Z) (z : Z) : (f.trans g).symm z = f.symm (g.symm z)
• Homeomorph.zpow_def 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} [TopologicalSpace X] (a✝ : ℤ) (a✝¹ : X ≃ₜ X) : ZPow.zpow a✝ a✝¹ = zpowRec npowRec a✝ a✝¹
• Homeomorph.div_def 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} [TopologicalSpace X] (a b : X ≃ₜ X) : a / b = DivInvMonoid.div' a b
• Homeomorph.prodComm_symm 📋 Mathlib.Topology.Constructions.SumProd
  (X : Type u) (Y : Type v) [TopologicalSpace X] [TopologicalSpace Y] : (Homeomorph.prodComm X Y).symm = Homeomorph.prodComm Y X
• Homeomorph.sumComm_symm 📋 Mathlib.Topology.Constructions.SumProd
  (X : Type u) (Y : Type v) [TopologicalSpace X] [TopologicalSpace Y] : (Homeomorph.sumComm X Y).symm = Homeomorph.sumComm Y X
• Homeomorph.prodCongr_symm 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {X' : Type u_5} {Y' : Type u_6} [TopologicalSpace X'] [TopologicalSpace Y'] (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') : (h₁.prodCongr h₂).symm = h₁.symm.prodCongr h₂.symm
• Homeomorph.sumCongr_symm 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {X' : Type u_5} {Y' : Type u_6} [TopologicalSpace X'] [TopologicalSpace Y'] (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') : (h₁.sumCongr h₂).symm = h₁.symm.sumCongr h₂.symm
• Homeomorph.prodProdProdComm_symm 📋 Mathlib.Topology.Constructions.SumProd
  (X : Type u) (Y : Type v) (W : Type u_1) (Z : Type u_2) [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W] : (Homeomorph.prodProdProdComm X Y W Z).symm = Homeomorph.prodProdProdComm X W Y Z
• Homeomorph.sumSumSumComm_symm 📋 Mathlib.Topology.Constructions.SumProd
  (X : Type u) (Y : Type v) (W : Type u_1) (Z : Type u_2) [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace W] [TopologicalSpace Z] : (Homeomorph.sumSumSumComm X Y W Z).symm = Homeomorph.sumSumSumComm X W Y Z
• Homeomorph.sumProdDistrib_symm_apply 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (a✝ : X × Z ⊕ Y × Z) : Homeomorph.sumProdDistrib.symm a✝ = (Equiv.sumProdDistrib X Y Z).symm a✝
• Homeomorph.piCurry_symm_apply 📋 Mathlib.Topology.Constructions
  {X : Type u_6} {Y : Type u_7} {Z : Type u_8} [TopologicalSpace Z] (a✝ : X → Y → Z) (a✝¹ : X × Y) : Homeomorph.piCurry.symm a✝ a✝¹ = Function.uncurry a✝ a✝¹
• Homeomorph.funUnique_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  (ι : Type u_5) (X : Type u_6) [Unique ι] [TopologicalSpace X] : ⇑(Homeomorph.funUnique ι X).symm = uniqueElim
• Homeomorph.piCongrRight_symm 📋 Mathlib.Topology.Homeomorph.Lemmas
  {ι : Type u_5} {Y₁ : ι → Type u_6} {Y₂ : ι → Type u_7} [(i : ι) → TopologicalSpace (Y₁ i)] [(i : ι) → TopologicalSpace (Y₂ i)] (F : (i : ι) → Y₁ i ≃ₜ Y₂ i) : (Homeomorph.piCongrRight F).symm = Homeomorph.piCongrRight fun i => (F i).symm
• Homeomorph.prodUnique_symm_apply_snd 📋 Mathlib.Topology.Homeomorph.Lemmas
  (X : Type u_1) (Y : Type u_2) [TopologicalSpace X] [TopologicalSpace Y] [Unique Y] (a✝ : X) : ((Homeomorph.prodUnique X Y).symm a✝).2 = default
• Homeomorph.Set.univ_symm_apply_coe 📋 Mathlib.Topology.Homeomorph.Lemmas
  (X : Type u_5) [TopologicalSpace X] (a : X) : ↑((Homeomorph.Set.univ X).symm a) = a
• Topology.IsCoinducing.connectedComponentsHomeomorph_symm_mk_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsCoinducing f) (hf' : ∀ (y : Y), IsConnected (f ⁻¹' {y})) (x : X) : (hf.connectedComponentsHomeomorph hf').symm (ConnectedComponents.mk (f x)) = ConnectedComponents.mk x
• Homeomorph.piUnique_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {α : Type u_5} [Unique α] (f : α → Type u_6) [(x : α) → TopologicalSpace (f x)] : ⇑(Homeomorph.piUnique f).symm = uniqueElim
• Homeomorph.uniqueProd_symm_apply_snd 📋 Mathlib.Topology.Homeomorph.Lemmas
  (X : Type u_5) (Y : Type u_6) [TopologicalSpace X] [TopologicalSpace Y] [Unique X] (a✝ : Y) : ((Homeomorph.uniqueProd X Y).symm a✝).2 = ((Homeomorph.prodUnique Y X).symm a✝).1
• Topology.IsEmbedding.toHomeomorph_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Topology.IsEmbedding f) (x : X) : hf.toHomeomorph.symm ⟨f x, ⋯⟩ = x
• Homeomorph.finTwoArrow_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} [TopologicalSpace X] : ⇑Homeomorph.finTwoArrow.symm = fun x => ![x.1, x.2]
• Homeomorph.subtype_symm_apply_coe 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {p : X → Prop} {q : Y → Prop} (h : X ≃ₜ Y) (h_iff : ∀ (x : X), p x ↔ q (h x)) (b : { b // q b }) : ↑((h.subtype h_iff).symm b) = h.symm ↑b
• Homeomorph.sumArrowHomeomorphProdArrow_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} [TopologicalSpace X] {ι : Type u_5} {ι' : Type u_6} (p : (ι → X) × (ι' → X)) (a✝ : ι ⊕ ι') : Homeomorph.sumArrowHomeomorphProdArrow.symm p a✝ = Sum.elim p.1 p.2 a✝
• Homeomorph.sigmaProdDistrib_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {Y : Type u_2} [TopologicalSpace Y] {ι : Type u_5} {X : ι → Type u_6} [(i : ι) → TopologicalSpace (X i)] (a✝ : (i : ι) × X i × Y) : Homeomorph.sigmaProdDistrib.symm a✝ = (⟨a✝.fst, a✝.snd.1⟩, a✝.snd.2)
• Homeomorph.funSplitAt_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  (Y : Type u_2) [TopologicalSpace Y] {ι : Type u_5} [DecidableEq ι] (i : ι) (f : (fun a => Y) i × ((j : { j // j ≠ i }) → (fun a => Y) ↑j)) (j : ι) : (Homeomorph.funSplitAt Y i).symm f j = if h : j = i then f.1 else f.2 ⟨j, ⋯⟩
• Homeomorph.piCongrLeft_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {ι : Type u_5} {ι' : Type u_6} {Y : ι' → Type u_7} [(j : ι') → TopologicalSpace (Y j)] (e : ι ≃ ι') : ⇑(Homeomorph.piCongrLeft e).symm = fun x1 x2 => x1 (e x2)
• Fin.appendHomeomorph_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} [TopologicalSpace X] (m n : ℕ) (f : Fin (m + n) → X) : (Fin.appendHomeomorph m n).symm f = (fun i => f (Fin.castAdd n i), fun i => f (Fin.natAdd m i))
• Homeomorph.piSplitAt_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {ι : Type u_5} [DecidableEq ι] (i : ι) (Y : ι → Type u_6) [(j : ι) → TopologicalSpace (Y j)] (f : Y i × ((j : { j // j ≠ i }) → Y ↑j)) (j : ι) : (Homeomorph.piSplitAt i Y).symm f j = if h : j = i then ⋯ ▸ f.1 else f.2 ⟨j, h⟩
• Homeomorph.image_symm_apply_coe 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (s : Set X) (y : ↑(⇑e.toEquiv '' s)) : ↑((e.image s).symm y) = e.symm ↑y
• Homeomorph.Set.prod_symm_apply_coe 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (s : Set X) (t : Set Y) (x : { a // (fun x => x ∈ s) a } × { b // (fun x => x ∈ t) b }) : ↑((Homeomorph.Set.prod s t).symm x) = (↑x.1, ↑x.2)
• Homeomorph.piEquivPiSubtypeProd_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  {ι : Type u_5} (p : ι → Prop) (Y : ι → Type u_6) [(i : ι) → TopologicalSpace (Y i)] [DecidablePred p] (f : ((i : { x // p x }) → Y ↑i) × ((i : { x // ¬p x }) → Y ↑i)) (x : ι) : (Homeomorph.piEquivPiSubtypeProd p Y).symm f x = if h : p x then f.1 ⟨x, h⟩ else f.2 ⟨x, h⟩
• Homeomorph.piFinTwo_symm_apply 📋 Mathlib.Topology.Homeomorph.Lemmas
  (X : Fin 2 → Type u) [(i : Fin 2) → TopologicalSpace (X i)] : ⇑(Homeomorph.piFinTwo X).symm = fun p => Fin.cons p.1 (Fin.cons p.2 finZeroElim)
• AddOpposite.opHomeomorph_symm_apply 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] (a✝ : Mᵃᵒᵖ) : AddOpposite.opHomeomorph.symm a✝ = AddOpposite.unop a✝
• MulOpposite.opHomeomorph_symm_apply 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] (a✝ : Mᵐᵒᵖ) : MulOpposite.opHomeomorph.symm a✝ = MulOpposite.unop a✝
• Homeomorph.smul_symm 📋 Mathlib.Topology.Algebra.ConstMulAction
  {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] {g : G} : (Homeomorph.smul g).symm = Homeomorph.smul g⁻¹
• Homeomorph.vadd_symm 📋 Mathlib.Topology.Algebra.ConstMulAction
  {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] {g : G} : (Homeomorph.vadd g).symm = Homeomorph.vadd (-g)
• Homeomorph.smul_symm_apply 📋 Mathlib.Topology.Algebra.ConstMulAction
  {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] (γ : G) (x : α) : (Homeomorph.smul γ).symm x = γ⁻¹ • x
• Homeomorph.vadd_symm_apply 📋 Mathlib.Topology.Algebra.ConstMulAction
  {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] (γ : G) (x : α) : (Homeomorph.vadd γ).symm x = -γ +ᵥ x
• Homeomorph.smulOfNeZero_symm_apply 📋 Mathlib.Topology.Algebra.ConstMulAction
  {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {c : G₀} (hc : c ≠ 0) : ⇑(Homeomorph.smulOfNeZero c hc).symm = fun x => c⁻¹ • x
• Homeomorph.symm_comp_toContinuousMap 📋 Mathlib.Topology.ContinuousMap.Basic
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) : (↑f.symm).comp ↑f = ContinuousMap.id α
• Homeomorph.toContinuousMap_comp_symm 📋 Mathlib.Topology.ContinuousMap.Basic
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) : (↑f).comp ↑f.symm = ContinuousMap.id β
• Homeomorph.continuousMapCongr_apply 📋 Mathlib.Topology.ContinuousMap.Basic
  {X₁ : Type u_5} {X₂ : Type u_6} {Y₁ : Type u_7} {Y₂ : Type u_8} [TopologicalSpace X₁] [TopologicalSpace X₂] [TopologicalSpace Y₁] [TopologicalSpace Y₂] (e : X₁ ≃ₜ X₂) (e' : Y₁ ≃ₜ Y₂) (f : C(X₁, Y₁)) : (e.continuousMapCongr e') f = { toFun := ⇑e', continuous_toFun := ⋯ }.comp (f.comp { toFun := ⇑e.symm, continuous_toFun := ⋯ })
• Homeomorph.continuousMapCongr_symm_apply 📋 Mathlib.Topology.ContinuousMap.Basic
  {X₁ : Type u_5} {X₂ : Type u_6} {Y₁ : Type u_7} {Y₂ : Type u_8} [TopologicalSpace X₁] [TopologicalSpace X₂] [TopologicalSpace Y₁] [TopologicalSpace Y₂] (e : X₁ ≃ₜ X₂) (e' : Y₁ ≃ₜ Y₂) (g : C(X₂, Y₂)) : (e.continuousMapCongr e').symm g = { toFun := ⇑e'.symm, continuous_toFun := ⋯ }.comp (g.comp { toFun := ⇑e, continuous_toFun := ⋯ })
• Topology.IsQuotientMap.homeomorph_symm_apply 📋 Mathlib.Topology.ContinuousMap.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} (hf : Topology.IsQuotientMap ⇑f) (b : Y) : hf.homeomorph.symm b = Quotient.mk'' (Function.surjInv ⋯ b)
• Function.RightInverse.homeomorph_symm_apply 📋 Mathlib.Topology.ContinuousMap.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} {f' : C(Y, X)} (hf : Function.RightInverse ⇑f' ⇑f) (b : Y) : hf.homeomorph.symm b = Quotient.mk'' (f' b)
• Topology.IsQuotientMap.lift_apply 📋 Mathlib.Topology.ContinuousMap.Basic
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : Topology.IsQuotientMap ⇑f) (g : C(X, Z)) (h : Function.FactorsThrough ⇑g ⇑f) (a✝ : Y) : (hf.lift g h) a✝ = ((fun i => i.liftOn' ⇑g ⋯) ∘ ⇑hf.homeomorph.symm) a✝
• UniformEquiv.toHomeomorph_symm_apply 📋 Mathlib.Topology.UniformSpace.Equiv
  {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (e : α ≃ᵤ β) : ⇑e.toHomeomorph.symm = ⇑e.symm
• Homeomorph.symm_inv 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
  {G : Type u_4} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] : (Homeomorph.inv G).symm = Homeomorph.inv G
• Homeomorph.symm_neg 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
  {G : Type u_4} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G] : (Homeomorph.neg G).symm = Homeomorph.neg G
• Homeomorph.addLeft_symm 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} [TopologicalSpace G] [AddGroup G] [SeparatelyContinuousAdd G] (a : G) : (Homeomorph.addLeft a).symm = Homeomorph.addLeft (-a)
• Homeomorph.addRight_symm 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} [TopologicalSpace G] [AddGroup G] [SeparatelyContinuousAdd G] (a : G) : (Homeomorph.addRight a).symm = Homeomorph.addRight (-a)
• Homeomorph.mulLeft_symm 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} [TopologicalSpace G] [Group G] [SeparatelyContinuousMul G] (a : G) : (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft a⁻¹
• Homeomorph.mulRight_symm 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} [TopologicalSpace G] [Group G] [SeparatelyContinuousMul G] (a : G) : (Homeomorph.mulRight a).symm = Homeomorph.mulRight a⁻¹
• Homeomorph.shearAddRight_symm_coe 📋 Mathlib.Topology.Algebra.Group.Basic
  (G : Type u_1) [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] : ⇑(Homeomorph.shearAddRight G).symm = fun z => (z.1, -z.1 + z.2)
• Homeomorph.shearMulRight_symm_coe 📋 Mathlib.Topology.Algebra.Group.Basic
  (G : Type u_1) [TopologicalSpace G] [Group G] [IsTopologicalGroup G] : ⇑(Homeomorph.shearMulRight G).symm = fun z => (z.1, z.1⁻¹ * z.2)
• Homeomorph.divRight_symm_apply 📋 Mathlib.Topology.Algebra.Group.ContinuousDiv
  {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] (x b : G) : (Homeomorph.divRight x).symm b = b * x
• Homeomorph.subRight_symm_apply 📋 Mathlib.Topology.Algebra.Group.ContinuousDiv
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] (x b : G) : (Homeomorph.subRight x).symm b = b + x
• Homeomorph.divLeft_symm_apply 📋 Mathlib.Topology.Algebra.Group.ContinuousDiv
  {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] (x b : G) : (Homeomorph.divLeft x).symm b = b⁻¹ * x
• Homeomorph.subLeft_symm_apply 📋 Mathlib.Topology.Algebra.Group.ContinuousDiv
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] (x b : G) : (Homeomorph.subLeft x).symm b = -b + x
• Homeomorph.mulLeft₀_symm_apply 📋 Mathlib.Topology.Algebra.GroupWithZero
  {α : Type u_1} [TopologicalSpace α] [GroupWithZero α] [SeparatelyContinuousMul α] (c : α) (hc : c ≠ 0) : ⇑(Homeomorph.mulLeft₀ c hc).symm = fun x => c⁻¹ * x
• Homeomorph.mulRight₀_symm_apply 📋 Mathlib.Topology.Algebra.GroupWithZero
  {α : Type u_1} [TopologicalSpace α] [GroupWithZero α] [SeparatelyContinuousMul α] (c : α) (hc : c ≠ 0) : ⇑(Homeomorph.mulRight₀ c hc).symm = fun x => x * c⁻¹
• ContinuousAddEquiv.coe_toHomeomorph_symm 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {M : Type u_1} {N : Type u_2} [TopologicalSpace M] [TopologicalSpace N] [Add M] [Add N] (f : M ≃ₜ+ N) : (↑f).symm = ↑f.symm
• ContinuousMulEquiv.coe_toHomeomorph_symm 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {M : Type u_1} {N : Type u_2} [TopologicalSpace M] [TopologicalSpace N] [Mul M] [Mul N] (f : M ≃ₜ* N) : (↑f).symm = ↑f.symm
• affineHomeomorph_symm_apply 📋 Mathlib.Topology.Algebra.Field
  {𝕜 : Type u_2} [Field 𝕜] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] (a b : 𝕜) (h : a ≠ 0) (y : 𝕜) : (affineHomeomorph a b h).symm y = (y - b) / a
• Nonneg.val_unitsHomeomorphPos_symm_apply_coe 📋 Mathlib.Topology.Algebra.Field
  (R : Type u_2) [DivisionSemiring R] [PartialOrder R] [IsStrictOrderedRing R] [PosMulReflectLT R] [TopologicalSpace R] [ContinuousInv₀ R] (r : { r // 0 < r }) : ↑↑((Nonneg.unitsHomeomorphPos R).symm r) = ↑r
• Nonneg.val_inv_unitsHomeomorphPos_symm_apply_coe 📋 Mathlib.Topology.Algebra.Field
  (R : Type u_2) [DivisionSemiring R] [PartialOrder R] [IsStrictOrderedRing R] [PosMulReflectLT R] [TopologicalSpace R] [ContinuousInv₀ R] (r : { r // 0 < r }) : ↑↑((Nonneg.unitsHomeomorphPos R).symm r)⁻¹ = (↑r)⁻¹
• Homeomorph.toPartialHomeomorph_symm_apply 📋 Mathlib.Topology.PartialHomeomorph.Defs
  {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) : ↑e.toPartialHomeomorph.symm = ⇑e.symm
• Homeomorph.toPartialHomeomorphOfImageEq_symm_apply 📋 Mathlib.Topology.PartialHomeomorph.Defs
  {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (s : Set X) (t : Set Y) (h : ⇑e '' s = t) : ↑(e.toPartialHomeomorphOfImageEq s t h).symm = ⇑e.symm
• Homeomorph.toOpenPartialHomeomorph_symm_apply 📋 Mathlib.Topology.OpenPartialHomeomorph.Defs
  {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) : ↑e.toOpenPartialHomeomorph.symm = ⇑e.symm
• Homeomorph.toOpenPartialHomeomorphOfImageEq_symm_apply 📋 Mathlib.Topology.OpenPartialHomeomorph.Defs
  {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s) (t : Set Y) (h : ⇑e '' s = t) : ↑(e.toOpenPartialHomeomorphOfImageEq s hs t h).symm = ⇑e.symm
• AddCircle.homeomorphAddCircle_symm_apply_mk 📋 Mathlib.Topology.Instances.AddCircle.Defs
  {𝕜 : Type u_1} [Field 𝕜] (p q : 𝕜) [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) : (AddCircle.homeomorphAddCircle p q hp hq).symm ↑x = ↑(x * (q⁻¹ * p))
• Homeomorph.opensCongr_symm 📋 Mathlib.Topology.Sets.Opens
  {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) : f.opensCongr.symm = f.symm.opensCongr
• Homeomorph.opensCongr_apply 📋 Mathlib.Topology.Sets.Opens
  {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) : ⇑f.opensCongr = ⇑(TopologicalSpace.Opens.comap ↑f.symm)
• TopologicalSpace.Compacts.equiv_symm 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) : TopologicalSpace.Compacts.equiv f.symm = (TopologicalSpace.Compacts.equiv f).symm
• TopologicalSpace.Compacts.coe_equiv_apply_eq_preimage 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) (K : TopologicalSpace.Compacts α) : ↑((TopologicalSpace.Compacts.equiv f) K) = ⇑f.symm ⁻¹' ↑K
• TopologicalSpace.Compacts.equiv_symm_apply 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) (K : TopologicalSpace.Compacts β) : (TopologicalSpace.Compacts.equiv f).symm K = TopologicalSpace.Compacts.map ⇑f.symm ⋯ K
• TopCat.isoOfHomeo_inv 📋 Mathlib.Topology.Category.TopCat.Basic
  {X Y : TopCat} (f : ↑X ≃ₜ ↑Y) : (TopCat.isoOfHomeo f).inv = TopCat.ofHom ↑f.symm
• TopCat.homeoOfIso_symm_apply 📋 Mathlib.Topology.Category.TopCat.Basic
  {X Y : TopCat} (f : X ≅ Y) (a : ↑Y) : (TopCat.homeoOfIso f).symm a = (CategoryTheory.ConcreteCategory.hom f.inv) a
• ContinuousLinearMap.homeomorphOfUnit_symm_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (T : (M₁ →L[R₁] M₁)ˣ) (a : M₁) : (ContinuousLinearMap.homeomorphOfUnit T).symm a = ↑T⁻¹ a
• ContinuousLinearEquiv.toHomeomorph_symm 📋 Mathlib.Topology.Algebra.Module.Equiv
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.toHomeomorph = e.toHomeomorph.symm
• ContinuousLinearEquiv.coe_symm_toHomeomorph 📋 Mathlib.Topology.Algebra.Module.Equiv
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (e : M₁ ≃SL[σ₁₂] M₂) : ⇑e.toHomeomorph.symm = ⇑e.symm
• CommRingCat.HomTopology.precompHomeomorph_symm_apply 📋 Mathlib.Algebra.Category.Ring.Topology
  {R A B : CommRingCat} [TopologicalSpace ↑R] (f : A ≅ B) (φ : A ⟶ R) : (CommRingCat.HomTopology.precompHomeomorph f).symm φ = CategoryTheory.CategoryStruct.comp f.inv φ
• CommRingCat.HomTopology.mvPolynomialHomeomorph_symm_apply_hom 📋 Mathlib.Algebra.Category.Ring.Topology
  (σ : Type v) (R A : CommRingCat) [TopologicalSpace ↑R] [IsTopologicalRing ↑R] (fx : (A ⟶ R) × (σ → ↑R)) : CommRingCat.Hom.hom ((CommRingCat.HomTopology.mvPolynomialHomeomorph σ R A).symm fx) = MvPolynomial.eval₂Hom (CommRingCat.Hom.hom fx.1) fx.2
• PrimeSpectrum.coe_preimageHomeomorphFiber_symm_apply_coe_asIdeal 📋 Mathlib.RingTheory.LocalRing.ResidueField.Fiber
  (R : Type u_3) (S : Type u_4) [CommRing R] [CommRing S] [Algebra R S] (p : PrimeSpectrum R) (q : PrimeSpectrum (p.asIdeal.Fiber S)) : ↑(↑((PrimeSpectrum.preimageHomeomorphFiber R S p).symm q)).asIdeal = ⇑Algebra.TensorProduct.includeRight ⁻¹' ↑q.asIdeal
• LocallyConstant.congrLeft_apply 📋 Mathlib.Topology.LocallyConstant.Basic
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (g : LocallyConstant X Z) : (LocallyConstant.congrLeft e) g = LocallyConstant.comap (↑e.symm) g
• OrderIso.coe_toHomeomorph_symm 📋 Mathlib.Topology.Order.MonotoneContinuity
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [OrderTopology α] [OrderTopology β] (e : α ≃o β) : ⇑e.toHomeomorph.symm = ⇑e.symm
• IsometryEquiv.coe_toHomeomorph_symm 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) : ⇑h.toHomeomorph.symm = ⇑h.symm
• DilationEquiv.toHomeomorph_symm 📋 Mathlib.Topology.MetricSpace.DilationEquiv
  {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y) : e.symm.toHomeomorph = e.toHomeomorph.symm
• DilationEquiv.coe_symm_toHomeomorph 📋 Mathlib.Topology.MetricSpace.DilationEquiv
  {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] (e : X ≃ᵈ Y) : ⇑e.toHomeomorph.symm = ⇑e.symm
• Homeomorph.toMeasurableEquiv_symm_coe 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
  {γ : Type u_3} {γ₂ : Type u_4} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace γ₂] [MeasurableSpace γ₂] [BorelSpace γ₂] (h : γ ≃ₜ γ₂) : ⇑h.toMeasurableEquiv.symm = ⇑h.symm
• unitInterval.symmHomeomorph_symm_apply 📋 Mathlib.Topology.UnitInterval
  (a✝ : ↑unitInterval) : unitInterval.symmHomeomorph.symm a✝ = unitInterval.symm a✝
• iccHomeoI_symm_apply_coe 📋 Mathlib.Topology.UnitInterval
  {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] (a b : 𝕜) (h : a < b) (x : ↑(Set.Icc 0 1)) : ↑((iccHomeoI a b h).symm x) = (b - a) * ↑x + a
• Metric.PiNatEmbed.toPiNatHomeo_symm_apply 📋 Mathlib.Topology.MetricSpace.PiNat
  {ι : Type u_2} (X : Type u_3) (Y : ι → Type u_4) (f : (i : ι) → X → Y i) [Encodable ι] [(i : ι) → MetricSpace (Y i)] [TopologicalSpace X] [CompactSpace X] (continuous_f : ∀ (i : ι), Continuous (f i)) (separating_f : Pairwise fun x y => ∃ i, f i x ≠ f i y) (self : Metric.PiNatEmbed X Y f) : (Metric.PiNatEmbed.toPiNatHomeo X Y f continuous_f separating_f).symm self = self.ofPiNat
• LinearIsometryEquiv.toHomeomorph_symm 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) : e.symm.toHomeomorph = e.toHomeomorph.symm
• LinearIsometryEquiv.coe_symm_toHomeomorph 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) : ⇑e.toHomeomorph.symm = ⇑e.symm
• ContinuousAlgEquiv.symm_toHomeomorph 📋 Mathlib.Topology.Algebra.Algebra.Equiv
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [TopologicalSpace A] [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (e : A ≃A[R] B) : e.symm.toHomeomorph = e.toHomeomorph.symm
• PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv_symm_apply 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (h : e.source = Set.univ) (h' : e.target = Set.univ) : ⇑(e.toHomeomorphOfSourceEqUnivTargetEqUniv h h').symm = ↑e.symm
• PartialHomeomorph.homeomorphOfImageSubsetSource_symm_apply 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : Set X} {t : Set Y} (hs : s ⊆ e.source) (ht : ↑e '' s = t) (a✝ : ↑t) : (e.homeomorphOfImageSubsetSource hs ht).symm a✝ = Set.MapsTo.restrict (↑e.symm) t s ⋯ a✝
• PartialHomeomorph.toHomeomorphSourceTarget_symm_apply_coe 📋 Mathlib.Topology.PartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (a✝ : ↑e.target) : ↑(e.toHomeomorphSourceTarget.symm a✝) = ↑e.symm ↑a✝
• OpenPartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv_symm_apply 📋 Mathlib.Topology.OpenPartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : OpenPartialHomeomorph X Y) (h : e.source = Set.univ) (h' : e.target = Set.univ) : ⇑(e.toHomeomorphOfSourceEqUnivTargetEqUniv h h').symm = ↑e.symm
• OpenPartialHomeomorph.homeomorphOfImageSubsetSource_symm_apply_coe 📋 Mathlib.Topology.OpenPartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : OpenPartialHomeomorph X Y) {s : Set X} {t : Set Y} (hs : s ⊆ e.source) (ht : ↑e '' s = t) (a✝ : ↑t) : ↑((e.homeomorphOfImageSubsetSource hs ht).symm a✝) = ↑e.symm ↑a✝
• OpenPartialHomeomorph.toHomeomorphSourceTarget_symm_apply_coe 📋 Mathlib.Topology.OpenPartialHomeomorph.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : OpenPartialHomeomorph X Y) (a✝ : ↑e.target) : ↑(e.toHomeomorphSourceTarget.symm a✝) = ↑e.symm ↑a✝
• Homeomorph.isBigO_congr 📋 Mathlib.Analysis.Asymptotics.Lemmas
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {E : Type u_3} [Norm E] {F : Type u_4} [Norm F] (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} : f =O[nhds b] g ↔ (f ∘ ⇑e) =O[nhds (e.symm b)] (g ∘ ⇑e)
• Homeomorph.isLittleO_congr 📋 Mathlib.Analysis.Asymptotics.Lemmas
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {E : Type u_3} [Norm E] {F : Type u_4} [Norm F] (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} : f =o[nhds b] g ↔ (f ∘ ⇑e) =o[nhds (e.symm b)] (g ∘ ⇑e)
• Homeomorph.isBigOWith_congr 📋 Mathlib.Analysis.Asymptotics.Lemmas
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {E : Type u_3} [Norm E] {F : Type u_4} [Norm F] (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} {C : ℝ} : Asymptotics.IsBigOWith C (nhds b) f g ↔ Asymptotics.IsBigOWith C (nhds (e.symm b)) (f ∘ ⇑e) (g ∘ ⇑e)
• Homeomorph.continuousMapOfUnique_symm_apply 📋 Mathlib.Topology.CompactOpen
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Unique X] (f : C(X, Y)) : Homeomorph.continuousMapOfUnique.symm f = f default
• ContinuousMap.homeoFnOfDiscrete_symm_apply 📋 Mathlib.Topology.CompactOpen
  {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [DiscreteTopology X] (f : X → Y) : ⇑(ContinuousMap.homeoFnOfDiscrete.symm f) = f
• ContinuousMap.homeoFnOfDiscrete_symm_apply_apply 📋 Mathlib.Topology.CompactOpen
  {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [DiscreteTopology X] (f : X → Y) (a✝ : X) : (ContinuousMap.homeoFnOfDiscrete.symm f) a✝ = f a✝
• Homeomorph.preimage_pathComponent 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (y : Y) : ⇑h ⁻¹' pathComponent y = pathComponent (h.symm y)
• stdSimplexHomeomorphUnitInterval_symm_apply_coe 📋 Mathlib.Analysis.Convex.StdSimplex
  (x : ↑(Set.Icc 0 1)) : ↑(stdSimplexHomeomorphUnitInterval.symm x) = ![1 - ↑x, ↑x]
• Homeomorph.smulConst_symm_apply 📋 Mathlib.Topology.Algebra.Group.Torsor
  {V : Type u_1} {P : Type u_2} [Group V] [TopologicalSpace V] [Torsor V P] [TopologicalSpace P] [IsTopologicalTorsor P] (p p' : P) : (Homeomorph.smulConst p).symm p' = p' /ₛ p
• Homeomorph.vaddConst_symm_apply 📋 Mathlib.Topology.Algebra.Group.Torsor
  {V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] (p p' : P) : (Homeomorph.vaddConst p).symm p' = p' -ᵥ p
• Homeomorph.constSDiv_symm_apply 📋 Mathlib.Topology.Algebra.Group.Torsor
  {V : Type u_1} {P : Type u_2} [Group V] [TopologicalSpace V] [Torsor V P] [TopologicalSpace P] [IsTopologicalTorsor P] (p : P) (x✝ : V) : (Homeomorph.constSDiv p).symm x✝ = x✝⁻¹ • p
• Homeomorph.constVSub_symm_apply 📋 Mathlib.Topology.Algebra.Group.Torsor
  {V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] (p : P) (x✝ : V) : (Homeomorph.constVSub p).symm x✝ = -x✝ +ᵥ p
• Homeomorph.Quot.congrLeft 📋 Mathlib.Topology.Homeomorph.Quotient
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {r : X → X → Prop} (e : X ≃ₜ Y) : Quot r ≃ₜ Quot fun y₁ y₂ => r (e.symm y₁) (e.symm y₂)
• AffineIsometryEquiv.toHomeomorph_symm 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : e.symm.toHomeomorph = e.toHomeomorph.symm
• AffineIsometryEquiv.coe_symm_toHomeomorph 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : ⇑e.toHomeomorph.symm = ⇑e.symm
• AffineEquiv.coe_toHomeomorphOfFiniteDimensional_symm 📋 Mathlib.Analysis.Normed.Module.FiniteDimension
  {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {PE : Type u_1} {PF : Type u_2} [MetricSpace PE] [NormedAddTorsor E PE] [MetricSpace PF] [NormedAddTorsor F PF] [FiniteDimensional 𝕜 E] (f : PE ≃ᵃ[𝕜] PF) : ⇑f.toHomeomorphOfFiniteDimensional.symm = ⇑f.symm
• Homeomorph.compStarAlgEquiv'_symm_apply 📋 Mathlib.Topology.ContinuousMap.Star
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (𝕜 : Type u_3) [CommSemiring 𝕜] (A : Type u_4) [TopologicalSpace A] [Semiring A] [IsTopologicalSemiring A] [StarRing A] [ContinuousStar A] [Algebra 𝕜 A] (f : X ≃ₜ Y) (a : C(X, A)) : (Homeomorph.compStarAlgEquiv' 𝕜 A f).symm a = (ContinuousMap.compStarAlgHom' 𝕜 A ↑f.symm) a
• AlgebraicGeometry.Scheme.homeoOfIso_symm 📋 Mathlib.AlgebraicGeometry.Scheme
  {X Y : AlgebraicGeometry.Scheme} (e : X ≅ Y) : (AlgebraicGeometry.Scheme.homeoOfIso e).symm = AlgebraicGeometry.Scheme.homeoOfIso e.symm
• AlgebraicGeometry.coprodSpec_apply 📋 Mathlib.AlgebraicGeometry.Limits
  (R S : Type u) [CommRing R] [CommRing S] (x : ↥(AlgebraicGeometry.Spec (CommRingCat.of R) ⨿ AlgebraicGeometry.Spec (CommRingCat.of S))) : (AlgebraicGeometry.coprodSpec R S) x = (PrimeSpectrum.primeSpectrumProd R S).symm ((AlgebraicGeometry.coprodMk (AlgebraicGeometry.Spec (CommRingCat.of R)) (AlgebraicGeometry.Spec (CommRingCat.of S))).symm x)
• AlgebraicGeometry.Scheme.Hom.fiberι_fiberHomeo_symm 📋 Mathlib.AlgebraicGeometry.Fiber
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (y : ↥Y) (x : ↑(⇑f ⁻¹' {y})) : (AlgebraicGeometry.Scheme.Hom.fiberι f y) ((AlgebraicGeometry.Scheme.Hom.fiberHomeo f y).symm x) = ↑x
• ContinuousMapZero.starAlgEquivPrecomp_symm_apply_toFun 📋 Mathlib.Topology.ContinuousMap.ContinuousMapZero
  {X : Type u_1} {Y : Type u_2} (R : Type u_4) [Zero X] [Zero Y] [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace R] [CommSemiring R] [StarRing R] [IsTopologicalSemiring R] [ContinuousStar R] (f : X ≃ₜ Y) (hf : f 0 = 0) (a : ContinuousMapZero X R) (a✝ : Y) : ((ContinuousMapZero.starAlgEquivPrecomp R f hf).symm a) a✝ = a (f.symm a✝)
• Homeomorph.symm_toOpenPartialHomeomorph 📋 Mathlib.Topology.OpenPartialHomeomorph.IsImage
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) : e.symm.toOpenPartialHomeomorph = e.toOpenPartialHomeomorph.symm
• Homeomorph.contDiff_symm 📋 Mathlib.Analysis.Calculus.ContDiff.Operations
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} [CompleteSpace E] (f : E ≃ₜ F) {f₀' : E → E ≃L[𝕜] F} (hf₀' : ∀ (a : E), HasFDerivAt (⇑f) (↑(f₀' a)) a) (hf : ContDiff 𝕜 n ⇑f) : ContDiff 𝕜 n ⇑f.symm
• Homeomorph.contDiff_symm_deriv 📋 Mathlib.Analysis.Calculus.ContDiff.Operations
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} [CompleteSpace 𝕜] (f : 𝕜 ≃ₜ 𝕜) {f' : 𝕜 → 𝕜} (h₀ : ∀ (x : 𝕜), f' x ≠ 0) (hf' : ∀ (x : 𝕜), HasDerivAt (⇑f) (f' x) x) (hf : ContDiff 𝕜 n ⇑f) : ContDiff 𝕜 n ⇑f.symm
• TopCat.uliftFunctorObjHomeo_symm_naturality_apply 📋 Mathlib.Topology.Category.TopCat.ULift
  {X Y : TopCat} (f : X ⟶ Y) (x : ↑(TopCat.uliftFunctor.obj X)) : Y.uliftFunctorObjHomeo.symm ((CategoryTheory.ConcreteCategory.hom (TopCat.uliftFunctor.map f)) x) = (CategoryTheory.ConcreteCategory.hom f) (X.uliftFunctorObjHomeo.symm x)
• Homeomorph.symm_toHomotopyEquiv 📋 Mathlib.Topology.Homotopy.Equiv
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : h.symm.toHomotopyEquiv = h.toHomotopyEquiv.symm
• TopCat.stdSimplexHomeomorphI_symm_one 📋 Mathlib.AlgebraicTopology.SimplicialSet.TopAdj
  : TopCat.stdSimplexHomeomorphI.symm 1 = stdSimplex.vertex 1
• TopCat.stdSimplexHomeomorphI_symm_zero 📋 Mathlib.AlgebraicTopology.SimplicialSet.TopAdj
  : TopCat.stdSimplexHomeomorphI.symm 0 = stdSimplex.vertex 0
• SimplexCategory.toTopHomeo_symm_naturality_apply 📋 Mathlib.AlgebraicTopology.SimplicialSet.TopAdj
  {n m : SimplexCategory} (f : n ⟶ m) (x : ↑(stdSimplex ℝ (Fin (n.len + 1)))) : m.toTopHomeo.symm (stdSimplex.map (⇑(CategoryTheory.ConcreteCategory.hom f)) x) = (CategoryTheory.ConcreteCategory.hom (SSet.toTop.map (SSet.stdSimplex.map f))) (n.toTopHomeo.symm x)
• SimplexCategory.toTopHomeo_symm_naturality 📋 Mathlib.AlgebraicTopology.SimplicialSet.TopAdj
  {n m : SimplexCategory} (f : n ⟶ m) : ⇑m.toTopHomeo.symm ∘ stdSimplex.map ⇑(CategoryTheory.ConcreteCategory.hom f) = ⇑(TopCat.Hom.hom (SSet.toTop.map (SSet.stdSimplex.map f))) ∘ ⇑n.toTopHomeo.symm
• TopCat.pathEquiv_apply_apply 📋 Mathlib.Topology.Homotopy.TopCat.Path
  {X : TopCat} {x y : ↑X} (p : X.Path x y) (a✝ : ↑unitInterval) : (TopCat.pathEquiv p) a✝ = (TopCat.Hom.hom p.hom) (TopCat.I.homeomorph.symm a✝)
• ENNReal.logHomeomorph_symm 📋 Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp
  : ENNReal.logHomeomorph.symm = EReal.expHomeomorph
• EReal.expHomeomorph_symm 📋 Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp
  : EReal.expHomeomorph.symm = ENNReal.logHomeomorph
• polynomialFunctions.comap_compRightAlgHom_iccHomeoI 📋 Mathlib.Topology.ContinuousMap.Polynomial
  (a b : ℝ) (h : a < b) : Subalgebra.comap (ContinuousMap.compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm) (polynomialFunctions unitInterval) = polynomialFunctions (Set.Icc a b)
• cfcHomTransfer_apply 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Transfer
  {R : Type u_1} {A : Type u_2} {B : Type u_3} {p : A → Prop} {q : B → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [Ring B] [StarRing B] [Algebra R B] [instCFC : ContinuousFunctionalCalculus R A p] (e : A ≃⋆ₐ[R] B) (hpq : ∀ (x : A), p x ↔ q (e x)) (b : B) (hb : q b) (x : C(↑(spectrum R b), R)) : (cfcHomTransfer e hpq b hb) x = e ((cfcHom ⋯) (x.comp ↑(Homeomorph.setCongr ⋯).symm))
• Homeomorph.transOpenPartialHomeomorph_symm_apply 📋 Mathlib.Topology.OpenPartialHomeomorph.Composition
  {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : X ≃ₜ Y) (f' : OpenPartialHomeomorph Y Z) : ↑(e.transOpenPartialHomeomorph f').symm = ⇑e.symm ∘ ↑f'.symm
• OpenPartialHomeomorph.transHomeomorph_symm_apply 📋 Mathlib.Topology.OpenPartialHomeomorph.Constructions
  {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : OpenPartialHomeomorph X Y) (f' : Y ≃ₜ Z) : ↑(e.transHomeomorph f').symm = ↑e.symm ∘ ⇑f'.symm
• OpenPartialHomeomorph.transHomeomorph_target 📋 Mathlib.Topology.OpenPartialHomeomorph.Constructions
  {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : OpenPartialHomeomorph X Y) (f' : Y ≃ₜ Z) : (e.transHomeomorph f').target = ⇑f'.symm ⁻¹' e.target
• Bundle.Trivialization.preimageSingletonHomeomorph_symm_apply 📋 Mathlib.Topology.FiberBundle.Trivialization
  {B : Type u_1} {F : Type u_2} {Z : Type u_4} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Bundle.Trivialization F proj) {b : B} (hb : b ∈ e.baseSet) (p : F) : (e.preimageSingletonHomeomorph hb).symm p = ⟨↑e.symm (b, p), ⋯⟩
• Bundle.Trivialization.preimageHomeomorph_symm_apply 📋 Mathlib.Topology.FiberBundle.Trivialization
  {B : Type u_1} {F : Type u_2} {Z : Type u_4} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Bundle.Trivialization F proj) {s : Set B} (hb : s ⊆ e.baseSet) (p : ↑s × F) : (e.preimageHomeomorph hb).symm p = ⟨↑e.symm (↑p.1, p.2), ⋯⟩
• Bundle.Trivialization.sourceHomeomorphBaseSetProd_symm_apply 📋 Mathlib.Topology.FiberBundle.Trivialization
  {B : Type u_1} {F : Type u_2} {Z : Type u_4} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [TopologicalSpace Z] (e : Bundle.Trivialization F proj) (p : ↑e.baseSet × F) : e.sourceHomeomorphBaseSetProd.symm p = ⟨↑e.symm (↑p.1, p.2), ⋯⟩
• IsCoveringMapOn.homeomorph_comp 📋 Mathlib.Topology.Covering.Basic
  {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {s : Set X} (hf : IsCoveringMapOn f s) {Y : Type u_3} [TopologicalSpace Y] (g : X ≃ₜ Y) : IsCoveringMapOn (⇑g ∘ f) (⇑g.symm ⁻¹' s)
• IsCoveringMapOn.homeomorph_comp_iff 📋 Mathlib.Topology.Covering.Basic
  {E : Type u_1} {X : Type u_2} [TopologicalSpace E] [TopologicalSpace X] {f : E → X} {s : Set X} {Y : Type u_3} [TopologicalSpace Y] (g : X ≃ₜ Y) : IsCoveringMapOn (⇑g ∘ f) (⇑g.symm ⁻¹' s) ↔ IsCoveringMapOn f s
• AddCircle.homeomorphCircle'_symm_apply 📋 Mathlib.Analysis.SpecialFunctions.Complex.Circle
  (x : Circle) : AddCircle.homeomorphCircle'.symm x = ↑(↑x).arg
• Homeomorph.unitBall_symm_apply 📋 Mathlib.Analysis.Normed.Module.Ball.Homeomorph
  {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (a✝ : ↑(Metric.ball 0 1)) : Homeomorph.unitBall.symm a✝ = ↑(OpenPartialHomeomorph.univUnitBall.toHomeomorphSourceTarget.symm a✝)
• ModelWithCorners.toHomeomorph_symm_apply 📋 Mathlib.Geometry.Manifold.IsManifold.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) [I.Boundaryless] (a✝ : E) : I.toHomeomorph.symm a✝ = ↑I.symm a✝
• Bundle.Trivial.homeomorphProd_symm_apply_proj 📋 Mathlib.Topology.FiberBundle.Constructions
  (B : Type u_1) (F : Type u_2) [TopologicalSpace B] [TopologicalSpace F] (x : B × F) : ((Bundle.Trivial.homeomorphProd B F).symm x).proj = x.1
• Bundle.Trivial.homeomorphProd_symm_apply_snd 📋 Mathlib.Topology.FiberBundle.Constructions
  (B : Type u_1) (F : Type u_2) [TopologicalSpace B] [TopologicalSpace F] (x : B × F) : ((Bundle.Trivial.homeomorphProd B F).symm x).snd = x.2
• tangentBundleModelSpaceHomeomorph_coe_symm 📋 Mathlib.Geometry.Manifold.VectorBundle.Tangent
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} : ⇑(tangentBundleModelSpaceHomeomorph I).symm = ⇑(Bundle.TotalSpace.toProd H E).symm
• contMDiff_tangentBundleModelSpaceHomeomorph_symm 📋 Mathlib.Geometry.Manifold.VectorBundle.Tangent
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} : ContMDiff I.tangent I.tangent n ⇑(tangentBundleModelSpaceHomeomorph I).symm
• Diffeomorph.symm_toHomeomorph 📋 Mathlib.Geometry.Manifold.Diffeomorph
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : WithTop ℕ∞} (h : Diffeomorph I J M N n) : h.symm.toHomeomorph = h.toHomeomorph.symm
• Diffeomorph.coe_toHomeomorph_symm 📋 Mathlib.Geometry.Manifold.Diffeomorph
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : WithTop ℕ∞} (h : Diffeomorph I J M N n) : ⇑h.toHomeomorph.symm = ⇑h.symm
• Real.sinhHomeomorph_symm_apply 📋 Mathlib.Analysis.SpecialFunctions.Arsinh
  : ⇑Real.sinhHomeomorph.symm = EquivLike.inv Real.sinhOrderIso
• CompHausLike.homeoOfIso_symm_apply 📋 Mathlib.Topology.Category.CompHausLike.Basic
  {P : TopCat → Prop} {X Y : CompHausLike P} (f : X ≅ Y) (a : ↑Y.toTop) : (CompHausLike.homeoOfIso f).symm a = (CategoryTheory.ConcreteCategory.hom f.inv.hom) a
• CompHausLike.isoOfHomeo_inv_hom_hom_apply 📋 Mathlib.Topology.Category.CompHausLike.Basic
  {P : TopCat → Prop} {X Y : CompHausLike P} (f : ↑X.toTop ≃ₜ ↑Y.toTop) (a : ↑((CompHausLike.compHausLikeToTop P).obj Y)) : (TopCat.Hom.hom (CompHausLike.isoOfHomeo f).inv.hom) a = f.symm a
• Homeomorph.onePointCongr_symm_apply 📋 Mathlib.Topology.Compactification.OnePoint.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (a✝ : OnePoint Y) : h.onePointCongr.symm a✝ = OnePoint.map (⇑h.symm) a✝
• ContinuousAffineMap.decompHomeomorph_symm_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
  (R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (p : Q × (V →L[R] W)) : ((ContinuousAffineMap.decompHomeomorph R V Q).symm p).contLinear = p.2
• ContinuousAffineMap.decompHomeomorph_symm_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
  (R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (p : Q × (V →L[R] W)) (x : V) : ((ContinuousAffineMap.decompHomeomorph R V Q).symm p) x = p.2 x +ᵥ p.1
• homeomorphSphereProd_symm_apply_coe 📋 Mathlib.Analysis.Normed.Module.Ball.RadialEquiv
  (E : Type u_2) [NormedAddCommGroup E] [NormedSpace ℝ E] (r : ℝ) (hr : 0 < r) (x : ↑(Metric.sphere 0 r) × ↑(Set.Ioi 0)) : ↑((homeomorphSphereProd E r hr).symm x) = ↑x.2 • ↑x.1
• homeomorphUnitSphereProd_symm_apply_coe 📋 Mathlib.Analysis.Normed.Module.Ball.RadialEquiv
  (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] (x : ↑(Metric.sphere 0 1) × ↑(Set.Ioi 0)) : ↑((homeomorphUnitSphereProd E).symm x) = ↑x.2 • ↑x.1
• LocallyConstant.congrLeftₐ_apply_apply 📋 Mathlib.Topology.LocallyConstant.Algebra
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Z] [Algebra R Z] (e : X ≃ₜ Y) (g : LocallyConstant X Z) (a✝ : Y) : ((LocallyConstant.congrLeftₐ R e) g) a✝ = g (e.symm a✝)
• LocallyConstant.congrLeftRingEquiv_apply_apply 📋 Mathlib.Topology.LocallyConstant.Algebra
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [Semiring Z] (e : X ≃ₜ Y) (g : LocallyConstant X Z) (a✝ : Y) : ((LocallyConstant.congrLeftRingEquiv e) g) a✝ = g (e.symm a✝)
• LocallyConstant.congrLeftₗ_apply_apply 📋 Mathlib.Topology.LocallyConstant.Algebra
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Z] [Module R Z] (e : X ≃ₜ Y) (g : LocallyConstant X Z) (a✝ : Y) : ((LocallyConstant.congrLeftₗ R e) g) a✝ = g (e.symm a✝)
• Sequential.isoOfHomeo_inv 📋 Mathlib.Topology.Category.Sequential
  {X Y : Sequential} (f : ↑X.toTop ≃ₜ ↑Y.toTop) : (Sequential.isoOfHomeo f).inv = CategoryTheory.InducedCategory.homMk (TopCat.ofHom { toFun := ⇑f.symm, continuous_toFun := ⋯ })
• Sequential.homeoOfIso_symm_apply 📋 Mathlib.Topology.Category.Sequential
  {X Y : Sequential} (f : X ≅ Y) (a : ↑Y.toTop) : (Sequential.homeoOfIso f).symm a = (CategoryTheory.ConcreteCategory.hom f.inv) a
• CompactlyGenerated.isoOfHomeo_inv 📋 Mathlib.Topology.Category.CompactlyGenerated
  {X Y : CompactlyGenerated} (f : ↑X.toTop ≃ₜ ↑Y.toTop) : (CompactlyGenerated.isoOfHomeo f).inv = CompactlyGenerated.ofHom { toFun := ⇑f.symm, continuous_toFun := ⋯ }
• CompactlyGenerated.homeoOfIso_symm_apply 📋 Mathlib.Topology.Category.CompactlyGenerated
  {X Y : CompactlyGenerated} (f : X ≅ Y) (a : ↑Y.toTop) : (CompactlyGenerated.homeoOfIso f).symm a = (CategoryTheory.ConcreteCategory.hom f.inv) a
• Flow.toHomeomorph_symm_apply 📋 Mathlib.Dynamics.Flow
  {τ : Type u_1} {α : Type u_2} [TopologicalSpace τ] [TopologicalSpace α] [AddGroup τ] (ϕ : Flow τ α) (t : τ) (x : α) : (ϕ.toHomeomorph t).symm x = ϕ.toFun (-t) x
• Convexity.StdSimplex.homeomorphI_symm_one 📋 Mathlib.Geometry.Convex.ConvexSpace.PathConnectedSpaceStdSimplex
  : Convexity.StdSimplex.homeomorphI.symm 1 = Convexity.StdSimplex.single 1
• Convexity.StdSimplex.homeomorphI_symm_zero 📋 Mathlib.Geometry.Convex.ConvexSpace.PathConnectedSpaceStdSimplex
  : Convexity.StdSimplex.homeomorphI.symm 0 = Convexity.StdSimplex.single 0
• DomAddAct.coe_mkHomeomorph_symm 📋 Mathlib.Topology.Algebra.Constructions.DomMulAct
  {M : Type u_1} [TopologicalSpace M] : ⇑DomAddAct.mkHomeomorph.symm = ⇑DomAddAct.mk.symm

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59