fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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continuous_op : Continuous (op : M → Mᵐᵒᵖ) :=
continuous_induced_rng.2 continuous_id | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | continuous_op | null |
@[to_additive (attr := simps!) /-- `AddOpposite.op` as a homeomorphism. -/]
opHomeomorph : M ≃ₜ Mᵐᵒᵖ where
toEquiv := opEquiv
continuous_toFun := continuous_op
continuous_invFun := continuous_unop
@[to_additive] | def | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | opHomeomorph | `MulOpposite.op` as a homeomorphism. |
instT2Space [T2Space M] : T2Space Mᵐᵒᵖ := opHomeomorph.t2Space
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | instT2Space | null |
instDiscreteTopology [DiscreteTopology M] : DiscreteTopology Mᵐᵒᵖ :=
opHomeomorph.symm.isEmbedding.discreteTopology
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | instDiscreteTopology | null |
instCompactSpace [CompactSpace M] : CompactSpace Mᵐᵒᵖ :=
opHomeomorph.compactSpace
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | instCompactSpace | null |
instWeaklyLocallyCompactSpace [WeaklyLocallyCompactSpace M] :
WeaklyLocallyCompactSpace Mᵐᵒᵖ :=
opHomeomorph.symm.isClosedEmbedding.weaklyLocallyCompactSpace
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | instWeaklyLocallyCompactSpace | null |
instLocallyCompactSpace [LocallyCompactSpace M] :
LocallyCompactSpace Mᵐᵒᵖ :=
opHomeomorph.symm.isClosedEmbedding.locallyCompactSpace
@[to_additive (attr := simp)] | instance | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | instLocallyCompactSpace | null |
map_op_nhds (x : M) : map (op : M → Mᵐᵒᵖ) (𝓝 x) = 𝓝 (op x) :=
opHomeomorph.map_nhds_eq x
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | map_op_nhds | null |
map_unop_nhds (x : Mᵐᵒᵖ) : map (unop : Mᵐᵒᵖ → M) (𝓝 x) = 𝓝 (unop x) :=
opHomeomorph.symm.map_nhds_eq x
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | map_unop_nhds | null |
comap_op_nhds (x : Mᵐᵒᵖ) : comap (op : M → Mᵐᵒᵖ) (𝓝 x) = 𝓝 (unop x) :=
opHomeomorph.comap_nhds_eq x
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | comap_op_nhds | null |
comap_unop_nhds (x : M) : comap (unop : Mᵐᵒᵖ → M) (𝓝 x) = 𝓝 (op x) :=
opHomeomorph.symm.comap_nhds_eq x | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | comap_unop_nhds | null |
@[to_additive
/-- The additive units of a monoid are equipped with a topology, via the embedding into `M × M`. -/]
instTopologicalSpaceUnits : TopologicalSpace Mˣ :=
TopologicalSpace.induced (embedProduct M) inferInstance
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | instTopologicalSpaceUnits | The units of a monoid are equipped with a topology, via the embedding into `M × M`. |
isInducing_embedProduct : IsInducing (embedProduct M) := ⟨rfl⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | isInducing_embedProduct | null |
isEmbedding_embedProduct : IsEmbedding (embedProduct M) :=
⟨isInducing_embedProduct, embedProduct_injective M⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | isEmbedding_embedProduct | null |
instT2Space [T2Space M] : T2Space Mˣ := isEmbedding_embedProduct.t2Space
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | instT2Space | null |
instDiscreteTopology [DiscreteTopology M] : DiscreteTopology Mˣ :=
isEmbedding_embedProduct.discreteTopology
@[to_additive] lemma topology_eq_inf :
instTopologicalSpaceUnits =
.induced (val : Mˣ → M) ‹_› ⊓ .induced (fun u ↦ ↑u⁻¹ : Mˣ → M) ‹_› := by
simp only [isInducing_embedProduct.1, instTopologicalSpaceProd, induced_inf,
instTopologicalSpaceMulOpposite, induced_compose]; rfl | instance | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | instDiscreteTopology | null |
@[to_additive /-- An auxiliary lemma that can be used to prove that coercion `AddUnits M → M` is a
topological embedding. Use `AddUnits.isEmbedding_val` or `toAddUnits_homeomorph` instead. -/]
isEmbedding_val_mk' {M : Type*} [Monoid M] [TopologicalSpace M] {f : M → M}
(hc : ContinuousOn f {x : M | IsUnit x}) (hf : ∀ u : Mˣ, f u.1 = ↑u⁻¹) :
IsEmbedding (val : Mˣ → M) := by
refine ⟨⟨?_⟩, val_injective⟩
rw [topology_eq_inf, inf_eq_left, ← continuous_iff_le_induced,
@continuous_iff_continuousAt _ _ (.induced _ _)]
intro u s hs
simp only [← hf, nhds_induced, Filter.mem_map] at hs ⊢
exact ⟨_, mem_inf_principal.1 (hc u u.isUnit hs), fun u' hu' ↦ hu' u'.isUnit⟩ | lemma | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | isEmbedding_val_mk' | An auxiliary lemma that can be used to prove that coercion `Mˣ → M` is a topological embedding.
Use `Units.isEmbedding_val₀`, `Units.isEmbedding_val`, or `toUnits_homeomorph` instead. |
@[to_additive /-- An auxiliary lemma that can be used to prove that coercion `AddUnits M → M` is a
topological embedding. Use `AddUnits.isEmbedding_val` or `toAddUnits_homeomorph` instead. -/]
embedding_val_mk {M : Type*} [DivisionMonoid M] [TopologicalSpace M]
(h : ContinuousOn Inv.inv {x : M | IsUnit x}) : IsEmbedding (val : Mˣ → M) :=
isEmbedding_val_mk' h fun u ↦ (val_inv_eq_inv_val u).symm
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | embedding_val_mk | An auxiliary lemma that can be used to prove that coercion `Mˣ → M` is a topological embedding.
Use `Units.isEmbedding_val₀`, `Units.isEmbedding_val`, or `toUnits_homeomorph` instead. |
continuous_embedProduct : Continuous (embedProduct M) :=
continuous_induced_dom
@[to_additive (attr := fun_prop)] | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | continuous_embedProduct | null |
continuous_val : Continuous ((↑) : Mˣ → M) :=
(@continuous_embedProduct M _ _).fst
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | continuous_val | null |
protected continuous_iff {f : X → Mˣ} :
Continuous f ↔ Continuous (val ∘ f) ∧ Continuous (fun x => ↑(f x)⁻¹ : X → M) := by
simp only [isInducing_embedProduct.continuous_iff, embedProduct_apply, Function.comp_def,
continuous_prodMk, opHomeomorph.symm.isInducing.continuous_iff, opHomeomorph_symm_apply,
unop_op]
@[to_additive (attr := fun_prop)] | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | continuous_iff | null |
continuous_coe_inv : Continuous (fun u => ↑u⁻¹ : Mˣ → M) :=
(Units.continuous_iff.1 continuous_id).2
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | continuous_coe_inv | null |
continuous_map {f : M →* N} (hf : Continuous f) : Continuous (map f) :=
Units.continuous_iff.mpr ⟨hf.comp continuous_val, hf.comp continuous_coe_inv⟩
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | continuous_map | null |
isOpenMap_map {f : M →* N} (hf_inj : Function.Injective f) (hf : IsOpenMap f) :
IsOpenMap (map f) := by
rintro _ ⟨U, hU, rfl⟩
have hg_openMap := hf.prodMap <| opHomeomorph.isOpenMap.comp (hf.comp opHomeomorph.symm.isOpenMap)
refine ⟨_, hg_openMap U hU, Set.ext fun y ↦ ?_⟩
simp only [embedProduct, OneHom.coe_mk, Set.mem_preimage, Set.mem_image, Prod.mk.injEq,
Prod.map, Prod.exists, MulOpposite.exists, MonoidHom.coe_mk]
refine ⟨fun ⟨a, b, h, ha, hb⟩ ↦ ⟨⟨a, b, hf_inj ?_, hf_inj ?_⟩, ?_⟩,
fun ⟨x, hxV, hx⟩ ↦ ⟨x, x.inv, by simp [hxV, ← hx]⟩⟩
all_goals simp_all | lemma | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | isOpenMap_map | null |
ContinuousAffineEquiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [Ring k]
[AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
[AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₂]
extends P₁ ≃ᵃ[k] P₂ where
continuous_toFun : Continuous toFun := by fun_prop
continuous_invFun : Continuous invFun := by fun_prop
@[inherit_doc]
notation:25 P₁ " ≃ᴬ[" k:25 "] " P₂:0 => ContinuousAffineEquiv k P₁ P₂
variable {k P₁ P₂ P₃ P₄ V₁ V₂ V₃ V₄ : Type*} [Ring k]
[AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
[AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₂]
[AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] [TopologicalSpace P₃]
[AddCommGroup V₄] [Module k V₄] [AddTorsor V₄ P₄] [TopologicalSpace P₄] | structure | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | ContinuousAffineEquiv | A continuous affine equivalence, denoted `P₁ ≃ᴬ[k] P₂`, between two affine topological spaces
is an affine equivalence such that forward and inverse maps are continuous. |
toHomeomorph (e : P₁ ≃ᴬ[k] P₂) : P₁ ≃ₜ P₂ where
__ := e | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | toHomeomorph | A continuous affine equivalence is a homeomorphism. |
toAffineEquiv_injective : Injective (toAffineEquiv : (P₁ ≃ᴬ[k] P₂) → P₁ ≃ᵃ[k] P₂) := by
rintro ⟨e, econt, einv_cont⟩ ⟨e', e'cont, e'inv_cont⟩ H
congr | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | toAffineEquiv_injective | null |
instEquivLike : EquivLike (P₁ ≃ᴬ[k] P₂) P₁ P₂ where
coe f := f.toFun
inv f := f.invFun
left_inv f := f.left_inv
right_inv f := f.right_inv
coe_injective' _ _ h _ := toAffineEquiv_injective (DFunLike.coe_injective h) | instance | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | instEquivLike | null |
coe : Coe (P₁ ≃ᴬ[k] P₂) (P₁ ≃ᵃ[k] P₂) := ⟨toAffineEquiv⟩
@[deprecated (since := "2025-08-15")] alias coe_injective := toAffineEquiv_injective | instance | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | coe | Coerce continuous affine equivalences to affine equivalences. |
instFunLike : FunLike (P₁ ≃ᴬ[k] P₂) P₁ P₂ where
coe f := f.toAffineEquiv
coe_injective' _ _ h := toAffineEquiv_injective (DFunLike.coe_injective h)
@[simp, norm_cast] | instance | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | instFunLike | null |
coe_coe (e : P₁ ≃ᴬ[k] P₂) : ⇑(e : P₁ ≃ᵃ[k] P₂) = e :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | coe_coe | null |
coe_toEquiv (e : P₁ ≃ᴬ[k] P₂) : ⇑e.toEquiv = e :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | coe_toEquiv | null |
Simps.apply (e : P₁ ≃ᴬ[k] P₂) : P₁ → P₂ :=
e | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | Simps.apply | See Note [custom simps projection].
We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. |
Simps.symm_apply (e : P₁ ≃ᴬ[k] P₂) : P₂ → P₁ :=
e.symm
initialize_simps_projections ContinuousAffineEquiv (toFun → apply, invFun → symm_apply)
@[ext] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | Simps.symm_apply | See Note [custom simps projection]. |
ext {e e' : P₁ ≃ᴬ[k] P₂} (h : ∀ x, e x = e' x) : e = e' :=
DFunLike.ext _ _ h
@[continuity] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | ext | null |
protected continuous (e : P₁ ≃ᴬ[k] P₂) : Continuous e :=
e.2 | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | continuous | null |
toContinuousAffineMap (e : P₁ ≃ᴬ[k] P₂) : P₁ →ᴬ[k] P₂ where
__ := e
cont := e.continuous_toFun
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | toContinuousAffineMap | A continuous affine equivalence is a continuous affine map. |
coe_toContinuousAffineMap (e : P₁ ≃ᴬ[k] P₂) : ⇑e.toContinuousAffineMap = e :=
rfl | lemma | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | coe_toContinuousAffineMap | null |
toContinuousAffineMap_injective :
Function.Injective (toContinuousAffineMap : (P₁ ≃ᴬ[k] P₂) → (P₁ →ᴬ[k] P₂)) := by
intro e e' h
ext p
simp_rw [← coe_toContinuousAffineMap, h] | lemma | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | toContinuousAffineMap_injective | null |
toContinuousAffineMap_toAffineMap (e : P₁ ≃ᴬ[k] P₂) :
e.toContinuousAffineMap.toAffineMap = e.toAffineEquiv.toAffineMap :=
rfl | lemma | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | toContinuousAffineMap_toAffineMap | null |
toContinuousAffineMap_toContinuousMap (e : P₁ ≃ᴬ[k] P₂) :
e.toContinuousAffineMap.toContinuousMap = toContinuousMap e.toHomeomorph :=
rfl | lemma | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | toContinuousAffineMap_toContinuousMap | null |
refl : P₁ ≃ᴬ[k] P₁ where
toEquiv := Equiv.refl P₁
linear := LinearEquiv.refl k V₁
map_vadd' _ _ := rfl
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | refl | Identity map as a `ContinuousAffineEquiv`. |
coe_refl : ⇑(refl k P₁) = id :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | coe_refl | null |
refl_apply (x : P₁) : refl k P₁ x = x :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | refl_apply | null |
toAffineEquiv_refl : (refl k P₁).toAffineEquiv = AffineEquiv.refl k P₁ :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | toAffineEquiv_refl | null |
toEquiv_refl : (refl k P₁).toEquiv = Equiv.refl P₁ :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | toEquiv_refl | null |
@[symm]
symm (e : P₁ ≃ᴬ[k] P₂) : P₂ ≃ᴬ[k] P₁ where
toAffineEquiv := e.toAffineEquiv.symm
continuous_toFun := e.continuous_invFun
continuous_invFun := e.continuous_toFun
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm | Inverse of a continuous affine equivalence as a continuous affine equivalence. |
toAffineEquiv_symm (e : P₁ ≃ᴬ[k] P₂) : e.symm.toAffineEquiv = e.toAffineEquiv.symm :=
rfl
@[deprecated "use instead `toAffineEquiv_symm`, in the reverse direction" (since := "2025-06-08")] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | toAffineEquiv_symm | null |
symm_toAffineEquiv (e : P₁ ≃ᴬ[k] P₂) : e.toAffineEquiv.symm = e.symm.toAffineEquiv :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm_toAffineEquiv | null |
coe_symm_toAffineEquiv (e : P₁ ≃ᴬ[k] P₂) : ⇑e.toAffineEquiv.symm = e.symm := rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | coe_symm_toAffineEquiv | null |
toEquiv_symm (e : P₁ ≃ᴬ[k] P₂) : e.symm.toEquiv = e.toEquiv.symm := rfl
@[deprecated "use instead `symm_toEquiv`, in the reverse direction" (since := "2025-06-08")] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | toEquiv_symm | null |
symm_toEquiv (e : P₁ ≃ᴬ[k] P₂) : e.toEquiv.symm = e.symm.toEquiv := rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm_toEquiv | null |
coe_symm_toEquiv (e : P₁ ≃ᴬ[k] P₂) : ⇑e.toEquiv.symm = e.symm := rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | coe_symm_toEquiv | null |
apply_symm_apply (e : P₁ ≃ᴬ[k] P₂) (p : P₂) : e (e.symm p) = p :=
e.toEquiv.apply_symm_apply p
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | apply_symm_apply | null |
symm_apply_apply (e : P₁ ≃ᴬ[k] P₂) (p : P₁) : e.symm (e p) = p :=
e.toEquiv.symm_apply_apply p | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm_apply_apply | null |
apply_eq_iff_eq_symm_apply (e : P₁ ≃ᴬ[k] P₂) {p₁ p₂} : e p₁ = p₂ ↔ p₁ = e.symm p₂ :=
e.toEquiv.apply_eq_iff_eq_symm_apply | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | apply_eq_iff_eq_symm_apply | null |
apply_eq_iff_eq (e : P₁ ≃ᴬ[k] P₂) {p₁ p₂ : P₁} : e p₁ = e p₂ ↔ p₁ = p₂ :=
e.toEquiv.apply_eq_iff_eq
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | apply_eq_iff_eq | null |
symm_symm (e : P₁ ≃ᴬ[k] P₂) : e.symm.symm = e := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm_symm | null |
symm_bijective : Function.Bijective (symm : (P₁ ≃ᴬ[k] P₂) → _) :=
Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩ | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm_bijective | null |
symm_symm_apply (e : P₁ ≃ᴬ[k] P₂) (x : P₁) : e.symm.symm x = e x :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm_symm_apply | null |
symm_apply_eq (e : P₁ ≃ᴬ[k] P₂) {x y} : e.symm x = y ↔ x = e y :=
e.toAffineEquiv.symm_apply_eq | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm_apply_eq | null |
eq_symm_apply (e : P₁ ≃ᴬ[k] P₂) {x y} : y = e.symm x ↔ e y = x :=
e.toAffineEquiv.eq_symm_apply
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | eq_symm_apply | null |
image_symm (f : P₁ ≃ᴬ[k] P₂) (s : Set P₂) : f.symm '' s = f ⁻¹' s :=
f.symm.toEquiv.image_eq_preimage _
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | image_symm | null |
preimage_symm (f : P₁ ≃ᴬ[k] P₂) (s : Set P₁) : f.symm ⁻¹' s = f '' s :=
(f.symm.image_symm _).symm | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | preimage_symm | null |
protected bijective (e : P₁ ≃ᴬ[k] P₂) : Bijective e :=
e.toEquiv.bijective | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | bijective | null |
protected surjective (e : P₁ ≃ᴬ[k] P₂) : Surjective e :=
e.toEquiv.surjective | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | surjective | null |
protected injective (e : P₁ ≃ᴬ[k] P₂) : Injective e :=
e.toEquiv.injective | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | injective | null |
protected image_eq_preimage (e : P₁ ≃ᴬ[k] P₂) (s : Set P₁) : e '' s = e.symm ⁻¹' s :=
e.toEquiv.image_eq_preimage s | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | image_eq_preimage | null |
protected image_symm_eq_preimage (e : P₁ ≃ᴬ[k] P₂) (s : Set P₂) :
e.symm '' s = e ⁻¹' s := by
rw [e.symm.image_eq_preimage, e.symm_symm]
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | image_symm_eq_preimage | null |
image_preimage (e : P₁ ≃ᴬ[k] P₂) (s : Set P₂) : e '' (e ⁻¹' s) = s :=
e.surjective.image_preimage s
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | image_preimage | null |
preimage_image (e : P₁ ≃ᴬ[k] P₂) (s : Set P₁) : e ⁻¹' (e '' s) = s :=
e.injective.preimage_image s | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | preimage_image | null |
symm_image_image (e : P₁ ≃ᴬ[k] P₂) (s : Set P₁) : e.symm '' (e '' s) = s :=
e.toEquiv.symm_image_image s | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm_image_image | null |
image_symm_image (e : P₁ ≃ᴬ[k] P₂) (s : Set P₂) : e '' (e.symm '' s) = s :=
e.symm.symm_image_image s
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | image_symm_image | null |
refl_symm : (refl k P₁).symm = refl k P₁ :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | refl_symm | null |
symm_refl : (refl k P₁).symm = refl k P₁ :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm_refl | null |
@[trans]
trans (e : P₁ ≃ᴬ[k] P₂) (e' : P₂ ≃ᴬ[k] P₃) : P₁ ≃ᴬ[k] P₃ where
toAffineEquiv := e.toAffineEquiv.trans e'.toAffineEquiv
continuous_toFun := e'.continuous_toFun.comp (e.continuous_toFun)
continuous_invFun := e.continuous_invFun.comp (e'.continuous_invFun)
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | trans | Composition of two `ContinuousAffineEquiv`alences, applied left to right. |
coe_trans (e : P₁ ≃ᴬ[k] P₂) (e' : P₂ ≃ᴬ[k] P₃) : ⇑(e.trans e') = e' ∘ e :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | coe_trans | null |
trans_apply (e : P₁ ≃ᴬ[k] P₂) (e' : P₂ ≃ᴬ[k] P₃) (p : P₁) : e.trans e' p = e' (e p) :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | trans_apply | null |
trans_assoc (e₁ : P₁ ≃ᴬ[k] P₂) (e₂ : P₂ ≃ᴬ[k] P₃) (e₃ : P₃ ≃ᴬ[k] P₄) :
(e₁.trans e₂).trans e₃ = e₁.trans (e₂.trans e₃) :=
ext fun _ ↦ rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | trans_assoc | null |
trans_refl (e : P₁ ≃ᴬ[k] P₂) : e.trans (refl k P₂) = e :=
ext fun _ ↦ rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | trans_refl | null |
refl_trans (e : P₁ ≃ᴬ[k] P₂) : (refl k P₁).trans e = e :=
ext fun _ ↦ rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | refl_trans | null |
self_trans_symm (e : P₁ ≃ᴬ[k] P₂) : e.trans e.symm = refl k P₁ :=
ext e.symm_apply_apply
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | self_trans_symm | null |
symm_trans_self (e : P₁ ≃ᴬ[k] P₂) : e.symm.trans e = refl k P₂ :=
ext e.apply_symm_apply | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | symm_trans_self | null |
trans_toContinuousAffineMap (e : P₁ ≃ᴬ[k] P₂) (e' : P₂ ≃ᴬ[k] P₃) :
(e.trans e').toContinuousAffineMap = e'.toContinuousAffineMap.comp e.toContinuousAffineMap :=
rfl | lemma | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | trans_toContinuousAffineMap | null |
_root_.ContinuousLinearEquiv.toContinuousAffineEquiv (L : E ≃L[k] F) : E ≃ᴬ[k] F where
toAffineEquiv := L.toAffineEquiv
continuous_toFun := L.continuous_toFun
continuous_invFun := L.continuous_invFun
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | _root_.ContinuousLinearEquiv.toContinuousAffineEquiv | Reinterpret a continuous linear equivalence between modules
as a continuous affine equivalence. |
_root_.ContinuousLinearEquiv.coe_toContinuousAffineEquiv (e : E ≃L[k] F) :
⇑e.toContinuousAffineEquiv = e :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | _root_.ContinuousLinearEquiv.coe_toContinuousAffineEquiv | null |
_root_.ContinuousLinearEquiv.toContinuousAffineEquiv_toContinuousAffineMap (L : E ≃L[k] F) :
L.toContinuousAffineEquiv.toContinuousAffineMap =
L.toContinuousLinearMap.toContinuousAffineMap :=
rfl
variable (k P₁) in | lemma | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | _root_.ContinuousLinearEquiv.toContinuousAffineEquiv_toContinuousAffineMap | null |
constVAdd [ContinuousConstVAdd V₁ P₁] (v : V₁) : P₁ ≃ᴬ[k] P₁ where
toAffineEquiv := AffineEquiv.constVAdd k P₁ v
continuous_toFun := continuous_const_vadd v
continuous_invFun := continuous_const_vadd (-v) | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | constVAdd | The map `p ↦ v +ᵥ p` as a continuous affine automorphism of an affine space
on which addition is continuous. |
constVAdd_coe [ContinuousConstVAdd V₁ P₁] (v : V₁) :
(constVAdd k P₁ v).toAffineEquiv = .constVAdd k P₁ v := rfl | lemma | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | constVAdd_coe | null |
@[simps toAffineEquiv]
prodCongr : P₁ × P₃ ≃ᴬ[k] P₂ × P₄ where
__ := AffineEquiv.prodCongr e₁ e₂
continuous_toFun := by eta_expand; dsimp; fun_prop
continuous_invFun := by eta_expand; dsimp; fun_prop
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | prodCongr | Product of two continuous affine equivalences. The map comes from `Equiv.prodCongr` |
prodCongr_symm : (e₁.prodCongr e₂).symm = e₁.symm.prodCongr e₂.symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | prodCongr_symm | null |
prodCongr_apply (p : P₁ × P₃) : e₁.prodCongr e₂ p = (e₁ p.1, e₂ p.2) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | prodCongr_apply | null |
prodCongr_toContinuousAffineMap : (e₁.prodCongr e₂).toContinuousAffineMap =
e₁.toContinuousAffineMap.prodMap e₂.toContinuousAffineMap :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | prodCongr_toContinuousAffineMap | null |
@[simps! apply symm_apply toAffineEquiv]
prodComm : P₁ × P₂ ≃ᴬ[k] P₂ × P₁ where
__ := AffineEquiv.prodComm k P₁ P₂
continuous_toFun := continuous_swap
continuous_invFun := continuous_swap
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | prodComm | Product of affine spaces is commutative up to continuous affine isomorphism. |
prodComm_symm : (prodComm k P₁ P₂).symm = prodComm k P₂ P₁ :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | prodComm_symm | null |
@[simps! apply symm_apply toAffineEquiv]
prodAssoc : (P₁ × P₂) × P₃ ≃ᴬ[k] P₁ × (P₂ × P₃) where
__ := AffineEquiv.prodAssoc k P₁ P₂ P₃
continuous_toFun := by eta_expand; dsimp; fun_prop
continuous_invFun := by eta_expand; dsimp; fun_prop | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineEquiv",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.ContinuousAffineMap"
] | Mathlib/Topology/Algebra/ContinuousAffineEquiv.lean | prodAssoc | Product of affine spaces is associative up to continuous affine isomorphism. |
ContinuousAffineMap (R : Type*) {V W : Type*} (P Q : Type*) [Ring R] [AddCommGroup V]
[Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W]
[TopologicalSpace Q] [AddTorsor W Q] extends P →ᵃ[R] Q where
cont : Continuous toFun | structure | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | ContinuousAffineMap | A continuous map of affine spaces |
toContinuousMap (f : P →ᴬ[R] Q) : C(P, Q) :=
⟨f, f.cont⟩ | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | toContinuousMap | A continuous map of affine spaces -/
notation:25 P " →ᴬ[" R "] " Q => ContinuousAffineMap R P Q
namespace ContinuousAffineMap
variable {R V W P Q : Type*} [Ring R]
variable [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P]
variable [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q]
instance : Coe (P →ᴬ[R] Q) (P →ᵃ[R] Q) :=
⟨toAffineMap⟩
attribute [coe] ContinuousAffineMap.toAffineMap
theorem toAffineMap_injective {f g : P →ᴬ[R] Q} (h : (f : P →ᵃ[R] Q) = (g : P →ᵃ[R] Q)) :
f = g := by
cases f
cases g
congr
instance : FunLike (P →ᴬ[R] Q) P Q where
coe f := f.toAffineMap
coe_injective' _ _ h := toAffineMap_injective <| DFunLike.coe_injective h
instance : ContinuousMapClass (P →ᴬ[R] Q) P Q where
map_continuous := cont
theorem toFun_eq_coe (f : P →ᴬ[R] Q) : f.toFun = ⇑f := rfl
theorem coe_injective : @Function.Injective (P →ᴬ[R] Q) (P → Q) (⇑) :=
DFunLike.coe_injective
@[ext]
theorem ext {f g : P →ᴬ[R] Q} (h : ∀ x, f x = g x) : f = g :=
DFunLike.ext _ _ h
theorem congr_fun {f g : P →ᴬ[R] Q} (h : f = g) (x : P) : f x = g x :=
DFunLike.congr_fun h _
/-- Forgetting its algebraic properties, a continuous affine map is a continuous map. |
@[simp]
toContinuousMap_coe (f : P →ᴬ[R] Q) : f.toContinuousMap = ↑f := rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | toContinuousMap_coe | null |
coe_toAffineMap (f : P →ᴬ[R] Q) : ((f : P →ᵃ[R] Q) : P → Q) = f := rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_toAffineMap | null |
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