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 ⌀ |
|---|---|---|---|---|---|---|
coe_to_continuousMap (f : P →ᴬ[R] Q) : ((f : C(P, Q)) : P → Q) = f := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_to_continuousMap | null |
to_continuousMap_injective {f g : P →ᴬ[R] Q} (h : (f : C(P, Q)) = (g : C(P, Q))) :
f = g := by
ext a
exact ContinuousMap.congr_fun h a
@[norm_cast] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | to_continuousMap_injective | null |
coe_toAffineMap_mk (f : P →ᵃ[R] Q) (h) : ((⟨f, h⟩ : P →ᴬ[R] Q) : P →ᵃ[R] Q) = f := rfl
@[norm_cast] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_toAffineMap_mk | null |
coe_continuousMap_mk (f : P →ᵃ[R] Q) (h) : ((⟨f, h⟩ : P →ᴬ[R] Q) : C(P, Q)) = ⟨f, h⟩ := rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_continuousMap_mk | null |
coe_mk (f : P →ᵃ[R] Q) (h) : ((⟨f, h⟩ : P →ᴬ[R] Q) : P → Q) = f := rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_mk | null |
mk_coe (f : P →ᴬ[R] Q) (h) : (⟨(f : P →ᵃ[R] Q), h⟩ : P →ᴬ[R] Q) = f := by
ext
rfl
@[continuity] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | mk_coe | null |
protected continuous (f : P →ᴬ[R] Q) : Continuous f := f.2
variable (R P) | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | continuous | null |
const (q : Q) : P →ᴬ[R] Q :=
{ AffineMap.const R P q with cont := continuous_const }
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | const | The constant map as a continuous affine map |
coe_const (q : Q) : ⇑(const R P q) = Function.const P q := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_const | null |
id : P →ᴬ[R] P := { AffineMap.id R P with cont := continuous_id }
@[simp, norm_cast] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | id | The identity map as a continuous affine map |
coe_id : ⇑(id R P) = _root_.id := rfl
variable {R P} {W₂ Q₂ W₃ Q₃ : Type*}
variable [AddCommGroup W₂] [Module R W₂] [TopologicalSpace Q₂] [AddTorsor W₂ Q₂] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_id | null |
comp (f : Q →ᴬ[R] Q₂) (g : P →ᴬ[R] Q) : P →ᴬ[R] Q₂ :=
{ (f : Q →ᵃ[R] Q₂).comp (g : P →ᵃ[R] Q) with cont := f.cont.comp g.cont }
@[simp, norm_cast] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | comp | The composition of continuous affine maps as a continuous affine map |
coe_comp (f : Q →ᴬ[R] Q₂) (g : P →ᴬ[R] Q) : ⇑(f.comp g) = f ∘ g := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_comp | null |
comp_apply (f : Q →ᴬ[R] Q₂) (g : P →ᴬ[R] Q) (p : P) : f.comp g p = f (g p) := rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | comp_apply | null |
comp_id (f : P →ᴬ[R] Q) : f.comp (id R P) = f :=
ext fun _ => rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | comp_id | null |
id_comp (f : P →ᴬ[R] Q) : (id R Q).comp f = f :=
ext fun _ => rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | id_comp | null |
lineMap (p₀ p₁ : P) [TopologicalSpace R] [TopologicalSpace V]
[ContinuousSMul R V] [ContinuousVAdd V P] : R →ᴬ[R] P where
toAffineMap := AffineMap.lineMap p₀ p₁
cont := (continuous_id.smul continuous_const).vadd continuous_const
@[simp] lemma lineMap_toAffineMap (p₀ p₁ : P) [TopologicalSpace R] [TopologicalSpace V]
[ContinuousSMul R V] [ContinuousVAdd V P] :
(lineMap p₀ p₁).toAffineMap = AffineMap.lineMap (k := R) p₀ p₁ := rfl | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | lineMap | The continuous affine map sending `0` to `p₀` and `1` to `p₁` |
coe_lineMap_eq (p₀ p₁ : P) [TopologicalSpace R] [TopologicalSpace V]
[ContinuousSMul R V] [ContinuousVAdd V P] :
⇑(ContinuousAffineMap.lineMap p₀ p₁) = ⇑(AffineMap.lineMap (k := R) p₀ p₁) := rfl | lemma | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_lineMap_eq | null |
contLinear (f : P →ᴬ[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | contLinear | The linear map underlying a continuous affine map is continuous. |
coe_contLinear (f : P →ᴬ[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_contLinear | null |
coe_contLinear_eq_linear (f : P →ᴬ[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_contLinear_eq_linear | null |
coe_mk_contLinear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →ᴬ[R] Q).contLinear : V → W) = f.linear :=
rfl
@[deprecated (since := "2025-09-17")]
alias coe_mk_const_linear_eq_linear := coe_mk_contLinear_eq_linear | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_mk_contLinear_eq_linear | null |
coe_linear_eq_coe_contLinear (f : P →ᴬ[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_linear_eq_coe_contLinear | null |
comp_contLinear (f : P →ᴬ[R] Q) (g : Q →ᴬ[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | comp_contLinear | null |
map_vadd (f : P →ᴬ[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | map_vadd | null |
contLinear_map_vsub (f : P →ᴬ[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | contLinear_map_vsub | null |
const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | const_contLinear | null |
contLinear_eq_zero_iff_exists_const (f : P →ᴬ[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine ⟨fun h => ?_, fun h => ?_⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine ⟨fun h => ?_, fun h => ?_⟩ <;> ext
· rw [h]; rfl
· rw [← coe_toAffineMap, h, AffineMap.const_apply, coe_const, Function.const_apply]
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | contLinear_eq_zero_iff_exists_const | null |
@[norm_cast, simp]
coe_zero : ((0 : P →ᴬ[R] W) : P → W) = 0 := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_zero | null |
zero_apply (x : P) : (0 : P →ᴬ[R] W) x = 0 := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | zero_apply | null |
@[norm_cast, simp]
coe_smul (t : S) (f : P →ᴬ[R] W) : ⇑(t • f) = t • ⇑f := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_smul | null |
smul_apply (t : S) (f : P →ᴬ[R] W) (x : P) : (t • f) x = t • f x := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | smul_apply | null |
@[simp]
smul_contLinear (t : S) (f : P →ᴬ[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | smul_contLinear | null |
@[norm_cast, simp]
coe_add (f g : P →ᴬ[R] W) : ⇑(f + g) = f + g := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_add | null |
add_apply (f g : P →ᴬ[R] W) (x : P) : (f + g) x = f x + g x := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | add_apply | null |
@[norm_cast, simp]
coe_sub (f g : P →ᴬ[R] W) : ⇑(f - g) = f - g := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_sub | null |
sub_apply (f g : P →ᴬ[R] W) (x : P) : (f - g) x = f x - g x := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | sub_apply | null |
@[norm_cast, simp]
coe_neg (f : P →ᴬ[R] W) : ⇑(-f) = -f := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_neg | null |
neg_apply (f : P →ᴬ[R] W) (x : P) : (-f) x = -f x := rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | neg_apply | null |
@[simp]
zero_contLinear : (0 : P →ᴬ[R] W).contLinear = 0 :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | zero_contLinear | null |
add_contLinear (f g : P →ᴬ[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | add_contLinear | null |
sub_contLinear (f g : P →ᴬ[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | sub_contLinear | null |
neg_contLinear (f : P →ᴬ[R] W) : (-f).contLinear = -f.contLinear :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | neg_contLinear | null |
@[simps toAffineMap]
prod (f : P₁ →ᴬ[k] P₂) (g : P₁ →ᴬ[k] P₃) : P₁ →ᴬ[k] P₂ × P₃ where
__ := AffineMap.prod f g
cont := by eta_expand; dsimp; fun_prop | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | prod | The space of continuous affine maps from `P` to `Q` is an affine space over the space of
continuous affine maps from `P` to `W`. -/
instance : AddTorsor (P →ᴬ[R] W) (P →ᴬ[R] Q) where
vadd f g := { __ := f.toAffineMap +ᵥ g.toAffineMap, cont := f.cont.vadd g.cont }
zero_vadd _ := ext fun _ ↦ zero_vadd _ _
add_vadd _ _ _ := ext fun _ ↦ add_vadd _ _ _
vsub f g := { __ := f.toAffineMap -ᵥ g.toAffineMap, cont := f.cont.vsub g.cont }
vsub_vadd' _ _ := ext fun _ ↦ vsub_vadd _ _
vadd_vsub' _ _ := ext fun _ ↦ vadd_vsub _ _
@[simp] lemma vadd_apply (f : P →ᴬ[R] W) (g : P →ᴬ[R] Q) (p : P) : (f +ᵥ g) p = f p +ᵥ g p :=
rfl
@[simp] lemma vsub_apply (f g : P →ᴬ[R] Q) (p : P) : (f -ᵥ g) p = f p -ᵥ g p :=
rfl
@[simp] lemma vadd_toAffineMap (f : P →ᴬ[R] W) (g : P →ᴬ[R] Q) :
(f +ᵥ g).toAffineMap = f.toAffineMap +ᵥ g.toAffineMap :=
rfl
@[simp] lemma vsub_toAffineMap (f g : P →ᴬ[R] Q) :
(f -ᵥ g).toAffineMap = f.toAffineMap -ᵥ g.toAffineMap :=
rfl
variable [TopologicalSpace V] [IsTopologicalAddTorsor P]
@[simp] lemma vadd_contLinear (f : P →ᴬ[R] W) (g : P →ᴬ[R] Q) :
(f +ᵥ g).contLinear = f.contLinear + g.contLinear :=
rfl
@[simp] lemma vsub_contLinear (f g : P →ᴬ[R] Q) :
(f -ᵥ g).contLinear = f.contLinear - g.contLinear :=
rfl
end
section Prod
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]
/-- The product of two continuous affine maps is a continuous affine map. |
coe_prod (f : P₁ →ᴬ[k] P₂) (g : P₁ →ᴬ[k] P₃) : prod f g = Pi.prod f g :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_prod | null |
prod_apply (f : P₁ →ᴬ[k] P₂) (g : P₁ →ᴬ[k] P₃) (p : P₁) : prod f g p = (f p, g p) :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | prod_apply | null |
@[simps toAffineMap]
prodMap (f : P₁ →ᴬ[k] P₂) (g : P₃ →ᴬ[k] P₄) : P₁ × P₃ →ᴬ[k] P₂ × P₄ where
__ := AffineMap.prodMap f g
cont := by eta_expand; dsimp; fun_prop | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | prodMap | `Prod.map` of two continuous affine maps. |
coe_prodMap (f : P₁ →ᴬ[k] P₂) (g : P₃ →ᴬ[k] P₄) : ⇑(f.prodMap g) = Prod.map f g :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_prodMap | null |
prodMap_apply (f : P₁ →ᴬ[k] P₂) (g : P₃ →ᴬ[k] P₄) (x) : f.prodMap g x = (f x.1, g x.2) :=
rfl
variable
[TopologicalSpace V₁] [IsTopologicalAddTorsor P₁]
[TopologicalSpace V₂] [IsTopologicalAddTorsor P₂]
[TopologicalSpace V₃] [IsTopologicalAddTorsor P₃]
[TopologicalSpace V₄] [IsTopologicalAddTorsor P₄]
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | prodMap_apply | null |
prod_contLinear (f : P₁ →ᴬ[k] P₂) (g : P₁ →ᴬ[k] P₃) :
(f.prod g).contLinear = f.contLinear.prod g.contLinear :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | prod_contLinear | null |
prodMap_contLinear (f : P₁ →ᴬ[k] P₂) (g : P₃ →ᴬ[k] P₄) :
(f.prodMap g).contLinear = f.contLinear.prodMap g.contLinear :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | prodMap_contLinear | null |
toContinuousAffineMap (f : V →L[R] W) : V →ᴬ[R] W where
toFun := f
linear := f
map_vadd' := by simp
cont := f.cont
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | toContinuousAffineMap | A continuous linear map can be regarded as a continuous affine map. |
coe_toContinuousAffineMap (f : V →L[R] W) : ⇑f.toContinuousAffineMap = f := rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | coe_toContinuousAffineMap | null |
toContinuousAffineMap_map_zero (f : V →L[R] W) : f.toContinuousAffineMap 0 = 0 := by simp
variable [IsTopologicalAddGroup V] [IsTopologicalAddGroup W]
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | toContinuousAffineMap_map_zero | null |
toContinuousAffineMap_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f :=
rfl
@[deprecated (since := "2025-09-23")]
alias _root_.ContinuousAffineMap.to_affine_map_contLinear := toContinuousAffineMap_contLinear | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | toContinuousAffineMap_contLinear | null |
_root_.ContinuousAffineMap.decomp (f : V →ᴬ[R] W) :
(f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [ContinuousAffineMap.coe_mk_contLinear_eq_linear, ContinuousAffineMap.coe_mk, f.decomp,
Pi.add_apply, LinearMap.map_zero, zero_add, ← Function.const_def] | theorem | Topology | [
"Mathlib.LinearAlgebra.AffineSpace.AffineMap",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd",
"Mathlib.Topology.Algebra.Affine"
] | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | _root_.ContinuousAffineMap.decomp | null |
ContinuousAddMonoidHom (A B : Type*) [AddMonoid A] [AddMonoid B] [TopologicalSpace A]
[TopologicalSpace B] extends A →+ B, C(A, B) | structure | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | ContinuousAddMonoidHom | The type of continuous additive monoid homomorphisms from `A` to `B`.
When possible, instead of parametrizing results over `(f : ContinuousAddMonoidHom A B)`,
you should parametrize
over `(F : Type*) [FunLike F A B] [ContinuousMapClass F A B] [AddMonoidHomClass F A B] (f : F)`.
When you extend this structure,
make sure to extend `ContinuousMapClass` and/or `AddMonoidHomClass`, if needed. |
@[to_additive /-- The type of continuous additive monoid homomorphisms from `A` to `B`. -/]
ContinuousMonoidHom extends A →* B, C(A, B) | structure | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | ContinuousMonoidHom | The type of continuous monoid homomorphisms from `A` to `B`.
When possible, instead of parametrizing results over `(f : ContinuousMonoidHom A B)`,
you should parametrize
over `(F : Type*) [FunLike F A B] [ContinuousMapClass F A B] [MonoidHomClass F A B] (f : F)`.
When you extend this structure,
make sure to extend `ContinuousMapClass` and/or `MonoidHomClass`, if needed. |
@[to_additive (attr := coe) /-- Turn an element of a type `F` satisfying
`AddMonoidHomClass F A B` and `ContinuousMapClass F A B` into a`ContinuousAddMonoidHom`.
This is declared as the default coercion from `F` to `ContinuousAddMonoidHom A B`. -/]
toContinuousMonoidHom [MonoidHomClass F A B] [ContinuousMapClass F A B] (f : F) : A →ₜ* B :=
{ MonoidHomClass.toMonoidHom f with } | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | toContinuousMonoidHom | Reinterpret a `ContinuousMonoidHom` as a `MonoidHom`. -/
add_decl_doc ContinuousMonoidHom.toMonoidHom
/-- Reinterpret a `ContinuousAddMonoidHom` as an `AddMonoidHom`. -/
add_decl_doc ContinuousAddMonoidHom.toAddMonoidHom
/-- Reinterpret a `ContinuousMonoidHom` as a `ContinuousMap`. -/
add_decl_doc ContinuousMonoidHom.toContinuousMap
/-- Reinterpret a `ContinuousAddMonoidHom` as a `ContinuousMap`. -/
add_decl_doc ContinuousAddMonoidHom.toContinuousMap
namespace ContinuousMonoidHom
/-- The type of continuous monoid homomorphisms from `A` to `B`.-/
infixr:25 " →ₜ+ " => ContinuousAddMonoidHom
/-- The type of continuous monoid homomorphisms from `A` to `B`.-/
infixr:25 " →ₜ* " => ContinuousMonoidHom
variable {A B C D E}
@[to_additive]
instance instFunLike : FunLike (A →ₜ* B) A B where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g
congr
@[to_additive]
instance instMonoidHomClass : MonoidHomClass (A →ₜ* B) A B where
map_mul f := f.map_mul'
map_one f := f.map_one'
@[to_additive]
instance instContinuousMapClass : ContinuousMapClass (A →ₜ* B) A B where
map_continuous f := f.continuous_toFun
@[to_additive (attr := simp)]
lemma coe_toMonoidHom (f : A →ₜ* B) : f.toMonoidHom = f := rfl
@[to_additive (attr := simp)]
lemma coe_toContinuousMap (f : A →ₜ* B) : f.toContinuousMap = f := rfl
section
variable {F : Type*} [FunLike F A B]
/-- Turn an element of a type `F` satisfying `MonoidHomClass F A B` and `ContinuousMapClass F A B`
into a`ContinuousMonoidHom`. This is declared as the default coercion from `F` to
`(A →ₜ* B)`. |
@[to_additive (attr := simps!) /-- Composition of two continuous homomorphisms. -/]
comp (g : B →ₜ* C) (f : A →ₜ* B) : A →ₜ* C :=
⟨g.toMonoidHom.comp f.toMonoidHom, (map_continuous g).comp (map_continuous f)⟩
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | comp | Any type satisfying `MonoidHomClass` and `ContinuousMapClass` can be cast into
`ContinuousMonoidHom` via `ContinuousMonoidHom.toContinuousMonoidHom`. -/
@[to_additive /-- Any type satisfying `AddMonoidHomClass` and `ContinuousMapClass` can be cast into
`ContinuousAddMonoidHom` via `ContinuousAddMonoidHom.toContinuousAddMonoidHom`. -/]
instance [MonoidHomClass F A B] [ContinuousMapClass F A B] : CoeOut F (A →ₜ* B) :=
⟨ContinuousMonoidHom.toContinuousMonoidHom⟩
@[to_additive (attr := simp)]
lemma coe_coe [MonoidHomClass F A B] [ContinuousMapClass F A B] (f : F) :
⇑(f : A →ₜ* B) = f := rfl
@[to_additive (attr := simp, norm_cast)]
lemma toMonoidHom_toContinuousMonoidHom [MonoidHomClass F A B] [ContinuousMapClass F A B] (f : F) :
((f : A →ₜ* B) : A →* B) = f := rfl
@[to_additive (attr := simp, norm_cast)]
lemma toContinuousMap_toContinuousMonoidHom [MonoidHomClass F A B] [ContinuousMapClass F A B]
(f : F) : ((f : A →ₜ* B) : C(A, B)) = f := rfl
end
@[to_additive (attr := ext)]
theorem ext {f g : A →ₜ* B} (h : ∀ x, f x = g x) : f = g :=
DFunLike.ext _ _ h
@[to_additive]
theorem toContinuousMap_injective : Injective (toContinuousMap : _ → C(A, B)) := fun f g h =>
ext <| by convert DFunLike.ext_iff.1 h
/-- Composition of two continuous homomorphisms. |
coe_comp (g : ContinuousMonoidHom B C) (f : ContinuousMonoidHom A B) :
⇑(g.comp f) = ⇑g ∘ ⇑f := rfl | lemma | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | coe_comp | null |
@[to_additive (attr := simps!) prod
/-- Product of two continuous homomorphisms on the same space. -/]
prod (f : A →ₜ* B) (g : A →ₜ* C) : A →ₜ* (B × C) :=
⟨f.toMonoidHom.prod g.toMonoidHom, f.continuous_toFun.prodMk g.continuous_toFun⟩ | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | prod | Product of two continuous homomorphisms on the same space. |
@[to_additive (attr := simps!) prodMap
/-- Product of two continuous homomorphisms on different spaces. -/]
prodMap (f : A →ₜ* C) (g : B →ₜ* D) :
(A × B) →ₜ* (C × D) :=
⟨f.toMonoidHom.prodMap g.toMonoidHom, f.continuous_toFun.prodMap g.continuous_toFun⟩
variable (A B C D E) | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | prodMap | Product of two continuous homomorphisms on different spaces. |
@[to_additive (attr := simps!) /-- The identity continuous homomorphism. -/]
id : A →ₜ* A := ⟨.id A, continuous_id⟩
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | id | The trivial continuous homomorphism. -/
@[to_additive (attr := simps!) /-- The trivial continuous homomorphism. -/]
instance : One (A →ₜ* B) where
one := ⟨1, continuous_const⟩
@[to_additive (attr := simp)]
lemma coe_one : ⇑(1 : A →ₜ* B) = 1 :=
rfl
@[to_additive]
instance : Inhabited (A →ₜ* B) := ⟨1⟩
/-- The identity continuous homomorphism. |
coe_id : ⇑(ContinuousMonoidHom.id A) = _root_.id :=
rfl | lemma | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | coe_id | null |
@[to_additive (attr := simps!)
/-- The continuous homomorphism given by projection onto the first factor. -/]
fst : (A × B) →ₜ* A := ⟨MonoidHom.fst A B, continuous_fst⟩ | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | fst | The continuous homomorphism given by projection onto the first factor. |
@[to_additive (attr := simps!)
/-- The continuous homomorphism given by projection onto the second factor. -/]
snd : (A × B) →ₜ* B :=
⟨MonoidHom.snd A B, continuous_snd⟩ | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | snd | The continuous homomorphism given by projection onto the second factor. |
@[to_additive (attr := simps!)
/-- The continuous homomorphism given by inclusion of the first factor. -/]
inl : A →ₜ* (A × B) :=
prod (id A) 1 | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | inl | The continuous homomorphism given by inclusion of the first factor. |
@[to_additive (attr := simps!)
/-- The continuous homomorphism given by inclusion of the second factor. -/]
inr : B →ₜ* (A × B) :=
prod 1 (id B) | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | inr | The continuous homomorphism given by inclusion of the second factor. |
@[to_additive (attr := simps!) /-- The continuous homomorphism given by the diagonal embedding. -/]
diag : A →ₜ* (A × A) := prod (id A) (id A) | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | diag | The continuous homomorphism given by the diagonal embedding. |
@[to_additive (attr := simps!) /-- The continuous homomorphism given by swapping components. -/]
swap : (A × B) →ₜ* (B × A) := prod (snd A B) (fst A B) | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | swap | The continuous homomorphism given by swapping components. |
@[to_additive (attr := simps!) /-- The continuous homomorphism given by addition. -/]
mul : (E × E) →ₜ* E := ⟨mulMonoidHom, continuous_mul⟩
variable {A B C D E}
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | mul | The continuous homomorphism given by multiplication. |
@[to_additive (attr := simps!) /-- Coproduct of two continuous homomorphisms to the same space. -/]
coprod (f : ContinuousMonoidHom A E) (g : ContinuousMonoidHom B E) :
ContinuousMonoidHom (A × B) E :=
(mul E).comp (f.prodMap g) | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | coprod | Coproduct of two continuous homomorphisms to the same space. |
@[to_additive (attr := simps!) /-- The continuous homomorphism given by negation. -/]
inv : ContinuousMonoidHom E E :=
⟨invMonoidHom, continuous_inv⟩
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | inv | The continuous homomorphism given by inversion. |
@[to_additive /-- For `f : F` where `F` is a class of continuous additive monoid hom, this yields
an element `ContinuousAddMonoidHom A B`. -/]
ofClass (F : Type*) [FunLike F A B] [ContinuousMapClass F A B]
[MonoidHomClass F A B] (f : F) : (ContinuousMonoidHom A B) := toContinuousMonoidHom f | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | ofClass | For `f : F` where `F` is a class of continuous monoid hom, this yields an element
`ContinuousMonoidHom A B`. |
ContinuousAddEquiv [Add G] [Add H] extends G ≃+ H, G ≃ₜ H | structure | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | ContinuousAddEquiv | The structure of two-sided continuous isomorphisms between additive groups.
Note that both the map and its inverse have to be continuous. |
@[to_additive /-- The structure of two-sided continuous isomorphisms between additive groups.
Note that both the map and its inverse have to be continuous. -/]
ContinuousMulEquiv [Mul G] [Mul H] extends G ≃* H, G ≃ₜ H | structure | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | ContinuousMulEquiv | The structure of two-sided continuous isomorphisms between groups.
Note that both the map and its inverse have to be continuous. |
@[to_additive (attr := ext) /-- Two continuous additive isomorphisms agree if they are defined by
the same underlying function. -/]
ext {f g : M ≃ₜ* N} (h : ∀ x, f x = g x) : f = g :=
DFunLike.ext f g h
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | ext | The homeomorphism induced from a two-sided continuous isomorphism of groups. -/
add_decl_doc ContinuousMulEquiv.toHomeomorph
/-- The homeomorphism induced from a two-sided continuous isomorphism additive groups. -/
add_decl_doc ContinuousAddEquiv.toHomeomorph
@[inherit_doc]
infixl:25 " ≃ₜ* " => ContinuousMulEquiv
@[inherit_doc]
infixl:25 " ≃ₜ+ " => ContinuousAddEquiv
section
namespace ContinuousMulEquiv
variable {M N : Type*} [TopologicalSpace M] [TopologicalSpace N] [Mul M] [Mul N]
section coe
@[to_additive]
instance : EquivLike (M ≃ₜ* N) M N where
coe f := f.toFun
inv f := f.invFun
left_inv f := f.left_inv
right_inv f := f.right_inv
coe_injective' f g h₁ h₂ := by
cases f
cases g
congr
exact MulEquiv.ext_iff.mpr (congrFun h₁)
@[to_additive]
instance : MulEquivClass (M ≃ₜ* N) M N where
map_mul f := f.map_mul'
@[to_additive]
instance : HomeomorphClass (M ≃ₜ* N) M N where
map_continuous f := f.continuous_toFun
inv_continuous f := f.continuous_invFun
/-- Two continuous multiplicative isomorphisms agree if they are defined by the
same underlying function. |
coe_mk (f : M ≃* N) (hf1 hf2) : ⇑(mk f hf1 hf2) = f := rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | coe_mk | null |
toEquiv_eq_coe (f : M ≃ₜ* N) : f.toEquiv = f :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | toEquiv_eq_coe | null |
toMulEquiv_eq_coe (f : M ≃ₜ* N) : f.toMulEquiv = f :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | toMulEquiv_eq_coe | null |
toHomeomorph_eq_coe (f : M ≃ₜ* N) : f.toHomeomorph = f :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | toHomeomorph_eq_coe | null |
@[to_additive /-- Makes an continuous additive isomorphism from
a homeomorphism which preserves addition. -/]
mk' (f : M ≃ₜ N) (h : ∀ x y, f (x * y) = f x * f y) : M ≃ₜ* N :=
⟨⟨f.toEquiv,h⟩, f.continuous_toFun, f.continuous_invFun⟩
set_option linter.docPrime false in -- This is about `ContinuousMulEquiv.mk'`
@[simp] | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | mk' | Makes a continuous multiplicative isomorphism from
a homeomorphism which preserves multiplication. |
coe_mk' (f : M ≃ₜ N) (h : ∀ x y, f (x * y) = f x * f y) : ⇑(mk' f h) = f := rfl | lemma | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | coe_mk' | null |
@[to_additive]
protected bijective (e : M ≃ₜ* N) : Function.Bijective e :=
EquivLike.bijective e
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | bijective | null |
protected injective (e : M ≃ₜ* N) : Function.Injective e :=
EquivLike.injective e
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | injective | null |
protected surjective (e : M ≃ₜ* N) : Function.Surjective e :=
EquivLike.surjective e
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | surjective | null |
apply_eq_iff_eq (e : M ≃ₜ* N) {x y : M} : e x = e y ↔ x = y :=
e.injective.eq_iff | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | apply_eq_iff_eq | null |
@[to_additive (attr := refl) /-- The identity map is a continuous additive isomorphism. -/]
refl : M ≃ₜ* M :=
{ MulEquiv.refl _ with }
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | refl | The identity map is a continuous multiplicative isomorphism. |
@[to_additive (attr := simp, norm_cast)]
coe_refl : ↑(refl M) = id := rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | coe_refl | null |
refl_apply (m : M) : refl M m = m := rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | refl_apply | null |
@[to_additive (attr := symm) /-- The inverse of a ContinuousAddEquiv. -/]
symm (cme : M ≃ₜ* N) : N ≃ₜ* M :=
{ cme.toMulEquiv.symm with
continuous_toFun := cme.continuous_invFun
continuous_invFun := cme.continuous_toFun }
initialize_simps_projections ContinuousMulEquiv (toFun → apply, invFun → symm_apply)
initialize_simps_projections ContinuousAddEquiv (toFun → apply, invFun → symm_apply)
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | symm | The inverse of a ContinuousMulEquiv. |
invFun_eq_symm {f : M ≃ₜ* N} : f.invFun = f.symm := rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | invFun_eq_symm | null |
coe_toHomeomorph_symm (f : M ≃ₜ* N) : (f : M ≃ₜ N).symm = (f.symm : N ≃ₜ M) := rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | coe_toHomeomorph_symm | null |
equivLike_inv_eq_symm (f : M ≃ₜ* N) : EquivLike.inv f = f.symm := rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | equivLike_inv_eq_symm | null |
symm_symm (f : M ≃ₜ* N) : f.symm.symm = f := rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | symm_symm | null |
symm_bijective : Function.Bijective (symm : M ≃ₜ* N → _) :=
Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | symm_bijective | null |
@[to_additive (attr := simp)
/-- `e.symm` is a right inverse of `e`, written as `e (e.symm y) = y`. -/]
apply_symm_apply (e : M ≃ₜ* N) (y : N) : e (e.symm y) = y :=
e.toEquiv.apply_symm_apply y | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | apply_symm_apply | `e.symm` is a right inverse of `e`, written as `e (e.symm y) = y`. |
@[to_additive (attr := simp)
/-- `e.symm` is a left inverse of `e`, written as `e.symm (e y) = y`. -/]
symm_apply_apply (e : M ≃ₜ* N) (x : M) : e.symm (e x) = x :=
e.toEquiv.symm_apply_apply x
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | symm_apply_apply | `e.symm` is a left inverse of `e`, written as `e.symm (e y) = y`. |
symm_comp_self (e : M ≃ₜ* N) : e.symm ∘ e = id :=
funext e.symm_apply_apply
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Topology.Algebra.Group.Defs"
] | Mathlib/Topology/Algebra/ContinuousMonoidHom.lean | symm_comp_self | null |
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