Continuous Monoid Homs

This file defines the space of continuous homomorphisms between two topological groups.

Main definitions

• ContinuousMonoidHom A B: The continuous homomorphisms A →* B.
• ContinuousAddMonoidHom A B: The continuous additive homomorphisms A →+ B.

structure ContinuousAddMonoidHom (A : Type u_7) (B : Type u_8) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] extends AddMonoidHom : Type (max u_7 u_8)

The type of continuous additive monoid homomorphisms from A to B.

• toFun : A → B
• map_zero' : (↑self.toAddMonoidHom).toFun 0 = 0
• map_add' : ∀ (x y : A), (↑self.toAddMonoidHom).toFun (x + y) = (↑self.toAddMonoidHom).toFun x + (↑self.toAddMonoidHom).toFun y
• continuous_toFun : Continuous (↑self.toAddMonoidHom).toFun

  Proof of continuity of the Hom.

theorem ContinuousAddMonoidHom.continuous_toFun {A : Type u_7} {B : Type u_8} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (self : ContinuousAddMonoidHom A B) : Continuous (↑self.toAddMonoidHom).toFun

Proof of continuity of the Hom.

structure ContinuousMonoidHom (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] extends MonoidHom : Type (max u_2 u_3)

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*) [ContinuousMonoidHomClass F A B] (f : F).

When you extend this structure, make sure to extend ContinuousAddMonoidHomClass.

• toFun : A → B
• map_one' : (↑self.toMonoidHom).toFun 1 = 1
• map_mul' : ∀ (x y : A), (↑self.toMonoidHom).toFun (x * y) = (↑self.toMonoidHom).toFun x * (↑self.toMonoidHom).toFun y
• continuous_toFun : Continuous (↑self.toMonoidHom).toFun

  Proof of continuity of the Hom.

theorem ContinuousMonoidHom.continuous_toFun {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : ContinuousMonoidHom A B) : Continuous (↑self.toMonoidHom).toFun

Proof of continuity of the Hom.

class ContinuousAddMonoidHomClass (F : Type u_1) (A : outParam (Type u_7)) (B : outParam (Type u_8)) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] [FunLike F A B] extends AddMonoidHomClass : Prop

ContinuousAddMonoidHomClass F A B states that F is a type of continuous additive monoid homomorphisms.

You should also extend this typeclass when you extend ContinuousAddMonoidHom.

• map_add : ∀ (f : F) (x y : A), f (x + y) = f x + f y
• map_zero : ∀ (f : F), f 0 = 0
• map_continuous : ∀ (f : F), Continuous ⇑f

  Proof of the continuity of the map.

theorem ContinuousAddMonoidHomClass.map_continuous {F : Type u_1} {A : outParam (Type u_7)} {B : outParam (Type u_8)} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] [FunLike F A B] [self : ContinuousAddMonoidHomClass F A B] (f : F) : Continuous ⇑f

Proof of the continuity of the map.

class ContinuousMonoidHomClass (F : Type u_1) (A : outParam (Type u_7)) (B : outParam (Type u_8)) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [FunLike F A B] extends MonoidHomClass : Prop

ContinuousMonoidHomClass F A B states that F is a type of continuous monoid homomorphisms.

You should also extend this typeclass when you extend ContinuousMonoidHom.

• map_mul : ∀ (f : F) (x y : A), f (x * y) = f x * f y
• map_one : ∀ (f : F), f 1 = 1
• map_continuous : ∀ (f : F), Continuous ⇑f

  Proof of the continuity of the map.

theorem ContinuousMonoidHomClass.map_continuous {F : Type u_1} {A : outParam (Type u_7)} {B : outParam (Type u_8)} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [FunLike F A B] [self : ContinuousMonoidHomClass F A B] (f : F) : Continuous ⇑f

Proof of the continuity of the map.

@[instance 100]

instance ContinuousAddMonoidHomClass.toContinuousMapClass (F : Type u_1) (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] [FunLike F A B] [ContinuousAddMonoidHomClass F A B] : ContinuousMapClass F A B

Equations

• ⋯ = ⋯

@[instance 100]

instance ContinuousMonoidHomClass.toContinuousMapClass (F : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [FunLike F A B] [ContinuousMonoidHomClass F A B] : ContinuousMapClass F A B

Equations

• ⋯ = ⋯

instance ContinuousAddMonoidHom.funLike {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : FunLike (ContinuousAddMonoidHom A B) A B

Equations

• ContinuousAddMonoidHom.funLike = { coe := fun (f : ContinuousAddMonoidHom A B) => (↑f.toAddMonoidHom).toFun, coe_injective' := ⋯ }

theorem ContinuousAddMonoidHom.funLike.proof_1 {A : Type u_1} {B : Type u_2} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (f : ContinuousAddMonoidHom A B) (g : ContinuousAddMonoidHom A B) (h : (fun (f : ContinuousAddMonoidHom A B) => (↑f.toAddMonoidHom).toFun) f = (fun (f : ContinuousAddMonoidHom A B) => (↑f.toAddMonoidHom).toFun) g) : f = g

instance ContinuousMonoidHom.funLike {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : FunLike (ContinuousMonoidHom A B) A B

Equations

• ContinuousMonoidHom.funLike = { coe := fun (f : ContinuousMonoidHom A B) => (↑f.toMonoidHom).toFun, coe_injective' := ⋯ }

instance ContinuousAddMonoidHom.ContinuousAddMonoidHomClass {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousAddMonoidHomClass (ContinuousAddMonoidHom A B) A B

Equations

• ⋯ = ⋯

instance ContinuousMonoidHom.ContinuousMonoidHomClass {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousMonoidHomClass (ContinuousMonoidHom A B) A B

Equations

• ⋯ = ⋯

theorem ContinuousAddMonoidHom.ext {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] {f : ContinuousAddMonoidHom A B} {g : ContinuousAddMonoidHom A B} (h : ∀ (x : A), f x = g x) : f = g

theorem ContinuousMonoidHom.ext {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] {f : ContinuousMonoidHom A B} {g : ContinuousMonoidHom A B} (h : ∀ (x : A), f x = g x) : f = g

def ContinuousAddMonoidHom.toContinuousMap {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (f : ContinuousAddMonoidHom A B) : C(A, B)

Reinterpret a ContinuousAddMonoidHom as a ContinuousMap.

Equations

• f.toContinuousMap = { toFun := (↑f.toAddMonoidHom).toFun, continuous_toFun := ⋯ }

def ContinuousMonoidHom.toContinuousMap {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (f : ContinuousMonoidHom A B) : C(A, B)

Reinterpret a ContinuousMonoidHom as a ContinuousMap.

Equations

• f.toContinuousMap = { toFun := (↑f.toMonoidHom).toFun, continuous_toFun := ⋯ }

theorem ContinuousAddMonoidHom.toContinuousMap_injective {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : Function.Injective ContinuousAddMonoidHom.toContinuousMap

theorem ContinuousMonoidHom.toContinuousMap_injective {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Function.Injective ContinuousMonoidHom.toContinuousMap

def ContinuousAddMonoidHom.mk' {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (f : A →+ B) (hf : Continuous ⇑f) : ContinuousAddMonoidHom A B

Construct a ContinuousAddMonoidHom from a Continuous AddMonoidHom.

Equations

• ContinuousAddMonoidHom.mk' f hf = { toAddMonoidHom := f, continuous_toFun := hf }

def ContinuousMonoidHom.mk' {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (f : A →* B) (hf : Continuous ⇑f) : ContinuousMonoidHom A B

Construct a ContinuousMonoidHom from a Continuous MonoidHom.

Equations

• ContinuousMonoidHom.mk' f hf = { toMonoidHom := f, continuous_toFun := hf }

theorem ContinuousAddMonoidHom.comp.proof_1 {A : Type u_1} {B : Type u_3} {C : Type u_2} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : ContinuousAddMonoidHom B C) (f : ContinuousAddMonoidHom A B) : Continuous ((↑g.toAddMonoidHom).toFun ∘ ⇑f.toAddMonoidHom)

def ContinuousAddMonoidHom.comp {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : ContinuousAddMonoidHom B C) (f : ContinuousAddMonoidHom A B) : ContinuousAddMonoidHom A C

Composition of two continuous homomorphisms.

Equations

• g.comp f = ContinuousAddMonoidHom.mk' (g.comp f.toAddMonoidHom) ⋯

@[simp]

theorem ContinuousAddMonoidHom.comp_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : ContinuousAddMonoidHom B C) (f : ContinuousAddMonoidHom A B) : ∀ (a : A), (g.comp f) a = g.toAddMonoidHom (f.toAddMonoidHom a)

@[simp]

theorem ContinuousMonoidHom.comp_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : ContinuousMonoidHom B C) (f : ContinuousMonoidHom A B) : ∀ (a : A), (g.comp f) a = g.toMonoidHom (f.toMonoidHom a)

def ContinuousMonoidHom.comp {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : ContinuousMonoidHom B C) (f : ContinuousMonoidHom A B) : ContinuousMonoidHom A C

Composition of two continuous homomorphisms.

Equations

• g.comp f = ContinuousMonoidHom.mk' (g.comp f.toMonoidHom) ⋯

def ContinuousAddMonoidHom.sum {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousAddMonoidHom A B) (g : ContinuousAddMonoidHom A C) : ContinuousAddMonoidHom A (B × C)

Product of two continuous homomorphisms on the same space.

Equations

• f.sum g = ContinuousAddMonoidHom.mk' (f.prod g.toAddMonoidHom) ⋯

theorem ContinuousAddMonoidHom.sum.proof_1 {A : Type u_1} {B : Type u_3} {C : Type u_2} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousAddMonoidHom A B) (g : ContinuousAddMonoidHom A C) : Continuous fun (x : A) => ((↑f.toAddMonoidHom).toFun x, g.toAddMonoidHom x)

@[simp]

theorem ContinuousAddMonoidHom.sum_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousAddMonoidHom A B) (g : ContinuousAddMonoidHom A C) (i : A) : (f.sum g) i = (f.toAddMonoidHom i, g.toAddMonoidHom i)

@[simp]

theorem ContinuousMonoidHom.prod_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousMonoidHom A B) (g : ContinuousMonoidHom A C) (i : A) : (f.prod g) i = (f.toMonoidHom i, g.toMonoidHom i)

def ContinuousMonoidHom.prod {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousMonoidHom A B) (g : ContinuousMonoidHom A C) : ContinuousMonoidHom A (B × C)

Product of two continuous homomorphisms on the same space.

Equations

• f.prod g = ContinuousMonoidHom.mk' (f.prod g.toMonoidHom) ⋯

def ContinuousAddMonoidHom.sum_map {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [AddMonoid A] [AddMonoid B] [AddMonoid C] [AddMonoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : ContinuousAddMonoidHom A C) (g : ContinuousAddMonoidHom B D) : ContinuousAddMonoidHom (A × B) (C × D)

Product of two continuous homomorphisms on different spaces.

Equations

• f.sum_map g = ContinuousAddMonoidHom.mk' (f.prodMap g.toAddMonoidHom) ⋯

theorem ContinuousAddMonoidHom.sum_map.proof_1 {A : Type u_1} {B : Type u_2} {C : Type u_4} {D : Type u_3} [AddMonoid A] [AddMonoid B] [AddMonoid C] [AddMonoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : ContinuousAddMonoidHom A C) (g : ContinuousAddMonoidHom B D) : Continuous fun (p : A × B) => ((↑f.toAddMonoidHom).toFun p.1, (↑g.toAddMonoidHom).1 p.2)

@[simp]

theorem ContinuousAddMonoidHom.sum_map_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [AddMonoid A] [AddMonoid B] [AddMonoid C] [AddMonoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : ContinuousAddMonoidHom A C) (g : ContinuousAddMonoidHom B D) (i : A × B) : (f.sum_map g) i = (f.toAddMonoidHom i.1, g.toAddMonoidHom i.2)

@[simp]

theorem ContinuousMonoidHom.prod_map_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [Monoid A] [Monoid B] [Monoid C] [Monoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : ContinuousMonoidHom A C) (g : ContinuousMonoidHom B D) (i : A × B) : (f.prod_map g) i = (f.toMonoidHom i.1, g.toMonoidHom i.2)

def ContinuousMonoidHom.prod_map {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [Monoid A] [Monoid B] [Monoid C] [Monoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : ContinuousMonoidHom A C) (g : ContinuousMonoidHom B D) : ContinuousMonoidHom (A × B) (C × D)

Product of two continuous homomorphisms on different spaces.

Equations

• f.prod_map g = ContinuousMonoidHom.mk' (f.prodMap g.toMonoidHom) ⋯

def ContinuousAddMonoidHom.zero (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousAddMonoidHom A B

The trivial continuous homomorphism.

Equations

• ContinuousAddMonoidHom.zero A B = ContinuousAddMonoidHom.mk' 0 ⋯

theorem ContinuousAddMonoidHom.zero.proof_1 (A : Type u_1) (B : Type u_2) [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : Continuous fun (x : A) => AddZeroClass.toZero.1

@[simp]

theorem ContinuousMonoidHom.one_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ∀ (x : A), (ContinuousMonoidHom.one A B) x = 1

@[simp]

theorem ContinuousAddMonoidHom.zero_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : ∀ (x : A), (ContinuousAddMonoidHom.zero A B) x = 0

def ContinuousMonoidHom.one (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousMonoidHom A B

The trivial continuous homomorphism.

Equations

• ContinuousMonoidHom.one A B = ContinuousMonoidHom.mk' 1 ⋯

instance ContinuousAddMonoidHom.instInhabited (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : Inhabited (ContinuousAddMonoidHom A B)

Equations

• ContinuousAddMonoidHom.instInhabited A B = { default := ContinuousAddMonoidHom.zero A B }

instance ContinuousMonoidHom.instInhabited (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Inhabited (ContinuousMonoidHom A B)

Equations

• ContinuousMonoidHom.instInhabited A B = { default := ContinuousMonoidHom.one A B }

def ContinuousAddMonoidHom.id (A : Type u_2) [AddMonoid A] [TopologicalSpace A] : ContinuousAddMonoidHom A A

The identity continuous homomorphism.

Equations

• ContinuousAddMonoidHom.id A = ContinuousAddMonoidHom.mk' (AddMonoidHom.id A) ⋯

@[simp]

theorem ContinuousMonoidHom.id_toFun (A : Type u_2) [Monoid A] [TopologicalSpace A] (x : A) : (ContinuousMonoidHom.id A) x = x

@[simp]

theorem ContinuousAddMonoidHom.id_toFun (A : Type u_2) [AddMonoid A] [TopologicalSpace A] (x : A) : (ContinuousAddMonoidHom.id A) x = x

def ContinuousMonoidHom.id (A : Type u_2) [Monoid A] [TopologicalSpace A] : ContinuousMonoidHom A A

The identity continuous homomorphism.

Equations

• ContinuousMonoidHom.id A = ContinuousMonoidHom.mk' (MonoidHom.id A) ⋯

def ContinuousAddMonoidHom.fst (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousAddMonoidHom (A × B) A

The continuous homomorphism given by projection onto the first factor.

Equations

• ContinuousAddMonoidHom.fst A B = ContinuousAddMonoidHom.mk' (AddMonoidHom.fst A B) ⋯

@[simp]

theorem ContinuousAddMonoidHom.fst_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) : (ContinuousAddMonoidHom.fst A B) self = self.1

@[simp]

theorem ContinuousMonoidHom.fst_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) : (ContinuousMonoidHom.fst A B) self = self.1

def ContinuousMonoidHom.fst (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousMonoidHom (A × B) A

The continuous homomorphism given by projection onto the first factor.

Equations

• ContinuousMonoidHom.fst A B = ContinuousMonoidHom.mk' (MonoidHom.fst A B) ⋯

def ContinuousAddMonoidHom.snd (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousAddMonoidHom (A × B) B

The continuous homomorphism given by projection onto the second factor.

Equations

• ContinuousAddMonoidHom.snd A B = ContinuousAddMonoidHom.mk' (AddMonoidHom.snd A B) ⋯

@[simp]

theorem ContinuousAddMonoidHom.snd_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) : (ContinuousAddMonoidHom.snd A B) self = self.2

@[simp]

theorem ContinuousMonoidHom.snd_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) : (ContinuousMonoidHom.snd A B) self = self.2

def ContinuousMonoidHom.snd (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousMonoidHom (A × B) B

The continuous homomorphism given by projection onto the second factor.

Equations

• ContinuousMonoidHom.snd A B = ContinuousMonoidHom.mk' (MonoidHom.snd A B) ⋯

def ContinuousAddMonoidHom.inl (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousAddMonoidHom A (A × B)

The continuous homomorphism given by inclusion of the first factor.

Equations

• ContinuousAddMonoidHom.inl A B = (ContinuousAddMonoidHom.id A).sum (ContinuousAddMonoidHom.zero A B)

@[simp]

theorem ContinuousMonoidHom.inl_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A) : (ContinuousMonoidHom.inl A B) i = ((ContinuousMonoidHom.id A).toMonoidHom i, (ContinuousMonoidHom.one A B).toMonoidHom i)

@[simp]

theorem ContinuousAddMonoidHom.inl_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A) : (ContinuousAddMonoidHom.inl A B) i = ((ContinuousAddMonoidHom.id A).toAddMonoidHom i, (ContinuousAddMonoidHom.zero A B).toAddMonoidHom i)

def ContinuousMonoidHom.inl (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousMonoidHom A (A × B)

The continuous homomorphism given by inclusion of the first factor.

Equations

• ContinuousMonoidHom.inl A B = (ContinuousMonoidHom.id A).prod (ContinuousMonoidHom.one A B)

def ContinuousAddMonoidHom.inr (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousAddMonoidHom B (A × B)

The continuous homomorphism given by inclusion of the second factor.

Equations

• ContinuousAddMonoidHom.inr A B = (ContinuousAddMonoidHom.zero B A).sum (ContinuousAddMonoidHom.id B)

@[simp]

theorem ContinuousMonoidHom.inr_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : B) : (ContinuousMonoidHom.inr A B) i = ((ContinuousMonoidHom.one B A).toMonoidHom i, (ContinuousMonoidHom.id B).toMonoidHom i)

@[simp]

theorem ContinuousAddMonoidHom.inr_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (i : B) : (ContinuousAddMonoidHom.inr A B) i = ((ContinuousAddMonoidHom.zero B A).toAddMonoidHom i, (ContinuousAddMonoidHom.id B).toAddMonoidHom i)

def ContinuousMonoidHom.inr (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousMonoidHom B (A × B)

The continuous homomorphism given by inclusion of the second factor.

Equations

• ContinuousMonoidHom.inr A B = (ContinuousMonoidHom.one B A).prod (ContinuousMonoidHom.id B)

def ContinuousAddMonoidHom.diag (A : Type u_2) [AddMonoid A] [TopologicalSpace A] : ContinuousAddMonoidHom A (A × A)

The continuous homomorphism given by the diagonal embedding.

Equations

• ContinuousAddMonoidHom.diag A = (ContinuousAddMonoidHom.id A).sum (ContinuousAddMonoidHom.id A)

@[simp]

theorem ContinuousMonoidHom.diag_toFun (A : Type u_2) [Monoid A] [TopologicalSpace A] (i : A) : (ContinuousMonoidHom.diag A) i = ((ContinuousMonoidHom.id A).toMonoidHom i, (ContinuousMonoidHom.id A).toMonoidHom i)

@[simp]

theorem ContinuousAddMonoidHom.diag_toFun (A : Type u_2) [AddMonoid A] [TopologicalSpace A] (i : A) : (ContinuousAddMonoidHom.diag A) i = ((ContinuousAddMonoidHom.id A).toAddMonoidHom i, (ContinuousAddMonoidHom.id A).toAddMonoidHom i)

def ContinuousMonoidHom.diag (A : Type u_2) [Monoid A] [TopologicalSpace A] : ContinuousMonoidHom A (A × A)

The continuous homomorphism given by the diagonal embedding.

Equations

• ContinuousMonoidHom.diag A = (ContinuousMonoidHom.id A).prod (ContinuousMonoidHom.id A)

def ContinuousAddMonoidHom.swap (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousAddMonoidHom (A × B) (B × A)

The continuous homomorphism given by swapping components.

Equations

• ContinuousAddMonoidHom.swap A B = (ContinuousAddMonoidHom.snd A B).sum (ContinuousAddMonoidHom.fst A B)

@[simp]

theorem ContinuousAddMonoidHom.swap_toFun (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A × B) : (ContinuousAddMonoidHom.swap A B) i = ((ContinuousAddMonoidHom.snd A B).toAddMonoidHom i, (ContinuousAddMonoidHom.fst A B).toAddMonoidHom i)

@[simp]

theorem ContinuousMonoidHom.swap_toFun (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A × B) : (ContinuousMonoidHom.swap A B) i = ((ContinuousMonoidHom.snd A B).toMonoidHom i, (ContinuousMonoidHom.fst A B).toMonoidHom i)

def ContinuousMonoidHom.swap (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousMonoidHom (A × B) (B × A)

The continuous homomorphism given by swapping components.

Equations

• ContinuousMonoidHom.swap A B = (ContinuousMonoidHom.snd A B).prod (ContinuousMonoidHom.fst A B)

def ContinuousAddMonoidHom.add (E : Type u_6) [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] : ContinuousAddMonoidHom (E × E) E

The continuous homomorphism given by addition.

Equations

• ContinuousAddMonoidHom.add E = ContinuousAddMonoidHom.mk' addAddMonoidHom ⋯

theorem ContinuousAddMonoidHom.add.proof_1 (E : Type u_1) [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] : Continuous fun (p : E × E) => p.1 + p.2

@[simp]

theorem ContinuousAddMonoidHom.add_toFun (E : Type u_6) [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] : ∀ (a : E × E), (ContinuousAddMonoidHom.add E) a = a.1 + a.2

@[simp]

theorem ContinuousMonoidHom.mul_toFun (E : Type u_6) [CommGroup E] [TopologicalSpace E] [TopologicalGroup E] : ∀ (a : E × E), (ContinuousMonoidHom.mul E) a = a.1 * a.2

def ContinuousMonoidHom.mul (E : Type u_6) [CommGroup E] [TopologicalSpace E] [TopologicalGroup E] : ContinuousMonoidHom (E × E) E

The continuous homomorphism given by multiplication.

Equations

• ContinuousMonoidHom.mul E = ContinuousMonoidHom.mk' mulMonoidHom ⋯

theorem ContinuousAddMonoidHom.neg.proof_1 (E : Type u_1) [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] : Continuous fun (a : E) => -a

def ContinuousAddMonoidHom.neg (E : Type u_6) [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] : ContinuousAddMonoidHom E E

The continuous homomorphism given by negation.

Equations

• ContinuousAddMonoidHom.neg E = ContinuousAddMonoidHom.mk' negAddMonoidHom ⋯

@[simp]

theorem ContinuousAddMonoidHom.neg_toFun (E : Type u_6) [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] : ∀ (a : E), (ContinuousAddMonoidHom.neg E) a = -a

@[simp]

theorem ContinuousMonoidHom.inv_toFun (E : Type u_6) [CommGroup E] [TopologicalSpace E] [TopologicalGroup E] : ∀ (a : E), (ContinuousMonoidHom.inv E) a = a⁻¹

def ContinuousMonoidHom.inv (E : Type u_6) [CommGroup E] [TopologicalSpace E] [TopologicalGroup E] : ContinuousMonoidHom E E

The continuous homomorphism given by inversion.

Equations

• ContinuousMonoidHom.inv E = ContinuousMonoidHom.mk' invMonoidHom ⋯

def ContinuousAddMonoidHom.coprod {A : Type u_2} {B : Type u_3} {E : Type u_6} [AddMonoid A] [AddMonoid B] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) (g : ContinuousAddMonoidHom B E) : ContinuousAddMonoidHom (A × B) E

Coproduct of two continuous homomorphisms to the same space.

Equations

• f.coprod g = (ContinuousAddMonoidHom.add E).comp (f.sum_map g)

@[simp]

theorem ContinuousAddMonoidHom.coprod_toFun {A : Type u_2} {B : Type u_3} {E : Type u_6} [AddMonoid A] [AddMonoid B] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) (g : ContinuousAddMonoidHom B E) : ∀ (a : A × B), (f.coprod g) a = (ContinuousAddMonoidHom.add E).toAddMonoidHom ((f.sum_map g).toAddMonoidHom a)

@[simp]

theorem ContinuousMonoidHom.coprod_toFun {A : Type u_2} {B : Type u_3} {E : Type u_6} [Monoid A] [Monoid B] [CommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalGroup E] (f : ContinuousMonoidHom A E) (g : ContinuousMonoidHom B E) : ∀ (a : A × B), (f.coprod g) a = (ContinuousMonoidHom.mul E).toMonoidHom ((f.prod_map g).toMonoidHom a)

def ContinuousMonoidHom.coprod {A : Type u_2} {B : Type u_3} {E : Type u_6} [Monoid A] [Monoid B] [CommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalGroup E] (f : ContinuousMonoidHom A E) (g : ContinuousMonoidHom B E) : ContinuousMonoidHom (A × B) E

Coproduct of two continuous homomorphisms to the same space.

Equations

• f.coprod g = (ContinuousMonoidHom.mul E).comp (f.prod_map g)

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_3 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) : f + 0 = f

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_1 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) (g : ContinuousAddMonoidHom A E) (h : ContinuousAddMonoidHom A E) : f + g + h = f + (g + h)

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_6 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) (g : ContinuousAddMonoidHom A E) (h : ContinuousAddMonoidHom A E) : f + g + h = f + (g + h)

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_7 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) : 0 + f = f

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_4 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] : ∀ (x : ContinuousAddMonoidHom A E), nsmulRec 0 x = nsmulRec 0 x

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_5 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] : ∀ (n : ℕ) (x : ContinuousAddMonoidHom A E), nsmulRec (n + 1) x = nsmulRec (n + 1) x

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_11 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] : ∀ (n : ℕ) (a : ContinuousAddMonoidHom A E), zsmulRec (Int.ofNat n.succ) a = zsmulRec (Int.ofNat n.succ) a

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_12 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] : ∀ (n : ℕ) (a : ContinuousAddMonoidHom A E), zsmulRec (Int.negSucc n) a = zsmulRec (Int.negSucc n) a

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_8 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) : f + 0 = f

instance ContinuousAddMonoidHom.instAddCommGroup {A : Type u_2} {E : Type u_6} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] : AddCommGroup (ContinuousAddMonoidHom A E)

Equations

• ContinuousAddMonoidHom.instAddCommGroup = AddCommGroup.mk ⋯

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_13 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) : -f + f = 0

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_2 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) : 0 + f = f

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_14 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) (g : ContinuousAddMonoidHom A E) : f + g = g + f

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_9 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] : ∀ (a b : ContinuousAddMonoidHom A E), a - b = a - b

theorem ContinuousAddMonoidHom.instAddCommGroup.proof_10 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] : ∀ (a : ContinuousAddMonoidHom A E), zsmulRec 0 a = zsmulRec 0 a

instance ContinuousMonoidHom.instCommGroup {A : Type u_2} {E : Type u_6} [Monoid A] [CommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalGroup E] : CommGroup (ContinuousMonoidHom A E)

Equations

• ContinuousMonoidHom.instCommGroup = CommGroup.mk ⋯

instance ContinuousAddMonoidHom.instTopologicalSpace {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : TopologicalSpace (ContinuousAddMonoidHom A B)

Equations

• ContinuousAddMonoidHom.instTopologicalSpace = TopologicalSpace.induced ContinuousAddMonoidHom.toContinuousMap ContinuousMap.compactOpen

instance ContinuousMonoidHom.instTopologicalSpace {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : TopologicalSpace (ContinuousMonoidHom A B)

Equations

• ContinuousMonoidHom.instTopologicalSpace = TopologicalSpace.induced ContinuousMonoidHom.toContinuousMap ContinuousMap.compactOpen

theorem ContinuousAddMonoidHom.inducing_toContinuousMap (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : Inducing ContinuousAddMonoidHom.toContinuousMap

theorem ContinuousMonoidHom.inducing_toContinuousMap (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Inducing ContinuousMonoidHom.toContinuousMap

theorem ContinuousAddMonoidHom.embedding_toContinuousMap (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : Embedding ContinuousAddMonoidHom.toContinuousMap

theorem ContinuousMonoidHom.embedding_toContinuousMap (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Embedding ContinuousMonoidHom.toContinuousMap

theorem ContinuousAddMonoidHom.range_toContinuousMap (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : Set.range ContinuousAddMonoidHom.toContinuousMap = {f : C(A, B) | f 0 = 0 ∧ ∀ (x y : A), f (x + y) = f x + f y}

theorem ContinuousMonoidHom.range_toContinuousMap (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Set.range ContinuousMonoidHom.toContinuousMap = {f : C(A, B) | f 1 = 1 ∧ ∀ (x y : A), f (x * y) = f x * f y}

theorem ContinuousAddMonoidHom.closedEmbedding_toContinuousMap (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] [ContinuousAdd B] [T2Space B] : ClosedEmbedding ContinuousAddMonoidHom.toContinuousMap

theorem ContinuousMonoidHom.closedEmbedding_toContinuousMap (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [ContinuousMul B] [T2Space B] : ClosedEmbedding ContinuousMonoidHom.toContinuousMap

instance ContinuousAddMonoidHom.instT2Space {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] [T2Space B] : T2Space (ContinuousAddMonoidHom A B)

Equations

• ⋯ = ⋯

instance ContinuousMonoidHom.instT2Space {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [T2Space B] : T2Space (ContinuousMonoidHom A B)

Equations

• ⋯ = ⋯

instance ContinuousAddMonoidHom.instTopologicalAddGroup {A : Type u_2} {E : Type u_6} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] : TopologicalAddGroup (ContinuousAddMonoidHom A E)

Equations

• ⋯ = ⋯

instance ContinuousMonoidHom.instTopologicalGroup {A : Type u_2} {E : Type u_6} [Monoid A] [CommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalGroup E] : TopologicalGroup (ContinuousMonoidHom A E)

Equations

• ⋯ = ⋯

theorem ContinuousAddMonoidHom.continuous_of_continuous_uncurry {B : Type u_3} {C : Type u_4} [AddMonoid B] [AddMonoid C] [TopologicalSpace B] [TopologicalSpace C] {A : Type u_7} [TopologicalSpace A] (f : A → ContinuousAddMonoidHom B C) (h : Continuous (Function.uncurry fun (x : A) (y : B) => (f x) y)) : Continuous f

theorem ContinuousMonoidHom.continuous_of_continuous_uncurry {B : Type u_3} {C : Type u_4} [Monoid B] [Monoid C] [TopologicalSpace B] [TopologicalSpace C] {A : Type u_7} [TopologicalSpace A] (f : A → ContinuousMonoidHom B C) (h : Continuous (Function.uncurry fun (x : A) (y : B) => (f x) y)) : Continuous f

theorem ContinuousAddMonoidHom.continuous_comp {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [LocallyCompactSpace B] : Continuous fun (f : ContinuousAddMonoidHom A B × ContinuousAddMonoidHom B C) => f.2.comp f.1

theorem ContinuousMonoidHom.continuous_comp {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [LocallyCompactSpace B] : Continuous fun (f : ContinuousMonoidHom A B × ContinuousMonoidHom B C) => f.2.comp f.1

theorem ContinuousAddMonoidHom.continuous_comp_left {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousAddMonoidHom A B) : Continuous fun (g : ContinuousAddMonoidHom B C) => g.comp f

theorem ContinuousMonoidHom.continuous_comp_left {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousMonoidHom A B) : Continuous fun (g : ContinuousMonoidHom B C) => g.comp f

theorem ContinuousAddMonoidHom.continuous_comp_right {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousAddMonoidHom B C) : Continuous fun (g : ContinuousAddMonoidHom A B) => f.comp g

theorem ContinuousMonoidHom.continuous_comp_right {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousMonoidHom B C) : Continuous fun (g : ContinuousMonoidHom A B) => f.comp g

theorem ContinuousAddMonoidHom.compLeft.proof_1 {A : Type u_1} {B : Type u_3} (E : Type u_2) [AddMonoid A] [AddMonoid B] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A B) : (fun (g : ContinuousAddMonoidHom B E) => g.comp f) 0 = (fun (g : ContinuousAddMonoidHom B E) => g.comp f) 0

theorem ContinuousAddMonoidHom.compLeft.proof_3 {A : Type u_3} {B : Type u_1} (E : Type u_2) [AddMonoid A] [AddMonoid B] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] (f : ContinuousAddMonoidHom A B) : Continuous fun (g : ContinuousAddMonoidHom B E) => g.comp f

def ContinuousAddMonoidHom.compLeft {A : Type u_2} {B : Type u_3} (E : Type u_6) [AddMonoid A] [AddMonoid B] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A B) : ContinuousAddMonoidHom (ContinuousAddMonoidHom B E) (ContinuousAddMonoidHom A E)

ContinuousAddMonoidHom _ f is a functor.

Equations

• ContinuousAddMonoidHom.compLeft E f = { toFun := fun (g : ContinuousAddMonoidHom B E) => g.comp f, map_zero' := ⋯, map_add' := ⋯, continuous_toFun := ⋯ }

theorem ContinuousAddMonoidHom.compLeft.proof_2 {A : Type u_3} {B : Type u_1} (E : Type u_2) [AddMonoid A] [AddMonoid B] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A B) (_g : ContinuousAddMonoidHom B E) (_h : ContinuousAddMonoidHom B E) : { toFun := fun (g : ContinuousAddMonoidHom B E) => g.comp f, map_zero' := ⋯ }.toFun (_g + _h) = { toFun := fun (g : ContinuousAddMonoidHom B E) => g.comp f, map_zero' := ⋯ }.toFun (_g + _h)

def ContinuousMonoidHom.compLeft {A : Type u_2} {B : Type u_3} (E : Type u_6) [Monoid A] [Monoid B] [CommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalGroup E] (f : ContinuousMonoidHom A B) : ContinuousMonoidHom (ContinuousMonoidHom B E) (ContinuousMonoidHom A E)

ContinuousMonoidHom _ f is a functor.

Equations

• ContinuousMonoidHom.compLeft E f = { toFun := fun (g : ContinuousMonoidHom B E) => g.comp f, map_one' := ⋯, map_mul' := ⋯, continuous_toFun := ⋯ }

def ContinuousAddMonoidHom.compRight (A : Type u_2) {E : Type u_6} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] {B : Type u_7} [AddCommGroup B] [TopologicalSpace B] [TopologicalAddGroup B] (f : ContinuousAddMonoidHom B E) : ContinuousAddMonoidHom (ContinuousAddMonoidHom A B) (ContinuousAddMonoidHom A E)

ContinuousAddMonoidHom f _ is a functor.

Equations

• ContinuousAddMonoidHom.compRight A f = { toFun := fun (g : ContinuousAddMonoidHom A B) => f.comp g, map_zero' := ⋯, map_add' := ⋯, continuous_toFun := ⋯ }

theorem ContinuousAddMonoidHom.compRight.proof_1 (A : Type u_1) {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] {B : Type u_3} [AddCommGroup B] [TopologicalSpace B] [TopologicalAddGroup B] (f : ContinuousAddMonoidHom B E) : (fun (g : ContinuousAddMonoidHom A B) => f.comp g) 0 = 0

theorem ContinuousAddMonoidHom.compRight.proof_3 (A : Type u_1) {E : Type u_3} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] {B : Type u_2} [AddCommGroup B] [TopologicalSpace B] (f : ContinuousAddMonoidHom B E) : Continuous fun (g : ContinuousAddMonoidHom A B) => f.comp g

theorem ContinuousAddMonoidHom.compRight.proof_2 (A : Type u_1) {E : Type u_3} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] {B : Type u_2} [AddCommGroup B] [TopologicalSpace B] [TopologicalAddGroup B] (f : ContinuousAddMonoidHom B E) (g : ContinuousAddMonoidHom A B) (h : ContinuousAddMonoidHom A B) : { toFun := fun (g : ContinuousAddMonoidHom A B) => f.comp g, map_zero' := ⋯ }.toFun (g + h) = { toFun := fun (g : ContinuousAddMonoidHom A B) => f.comp g, map_zero' := ⋯ }.toFun g + { toFun := fun (g : ContinuousAddMonoidHom A B) => f.comp g, map_zero' := ⋯ }.toFun h

def ContinuousMonoidHom.compRight (A : Type u_2) {E : Type u_6} [Monoid A] [CommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalGroup E] {B : Type u_7} [CommGroup B] [TopologicalSpace B] [TopologicalGroup B] (f : ContinuousMonoidHom B E) : ContinuousMonoidHom (ContinuousMonoidHom A B) (ContinuousMonoidHom A E)

ContinuousMonoidHom f _ is a functor.

Equations

• ContinuousMonoidHom.compRight A f = { toFun := fun (g : ContinuousMonoidHom A B) => f.comp g, map_one' := ⋯, map_mul' := ⋯, continuous_toFun := ⋯ }

abbrev ContinuousAddMonoidHom.locallyCompactSpace_of_equicontinuousAt.match_2 {X : Type u_1} {Y : Type u_2} [AddGroup X] [AddCommGroup Y] (U : Set X) (W : Set Y) (f : X → Y) (motive : f ∈ (U.pi fun (x : X) => W) ∩ Set.range DFunLike.coe → Prop) : ∀ (x : f ∈ (U.pi fun (x : X) => W) ∩ Set.range DFunLike.coe), (∀ (hg : f ∈ U.pi fun (x : X) => W) (g : X →+ Y) (hf : ⇑g = f), motive ⋯) → motive x

Equations

• ⋯ = ⋯

theorem ContinuousAddMonoidHom.locallyCompactSpace_of_equicontinuousAt {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [AddGroup X] [TopologicalAddGroup X] [UniformSpace Y] [AddCommGroup Y] [UniformAddGroup Y] [T0Space Y] [CompactSpace Y] (U : Set X) (V : Set Y) (hU : IsCompact U) (hV : V ∈ nhds 0) (h : EquicontinuousAt (fun (f : ↑{f : X →+ Y | Set.MapsTo (⇑f) U V}) => ⇑↑f) 0) : LocallyCompactSpace (ContinuousAddMonoidHom X Y)

abbrev ContinuousAddMonoidHom.locallyCompactSpace_of_equicontinuousAt.match_1 {X : Type u_1} {Y : Type u_2} [AddGroup X] [AddCommGroup Y] (U : Set X) (W : Set Y) (f : X → Y) (motive : (∃ (x : X →+ Y), Set.MapsTo (⇑x) U W ∧ ⇑x = f) → Prop) : ∀ (x : ∃ (x : X →+ Y), Set.MapsTo (⇑x) U W ∧ ⇑x = f), (∀ (g : X →+ Y) (hg : Set.MapsTo (⇑g) U W) (hf : ⇑g = f), motive ⋯) → motive x

Equations

• ⋯ = ⋯

theorem ContinuousMonoidHom.locallyCompactSpace_of_equicontinuousAt {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [Group X] [TopologicalGroup X] [UniformSpace Y] [CommGroup Y] [UniformGroup Y] [T0Space Y] [CompactSpace Y] (U : Set X) (V : Set Y) (hU : IsCompact U) (hV : V ∈ nhds 1) (h : EquicontinuousAt (fun (f : ↑{f : X →* Y | Set.MapsTo (⇑f) U V}) => ⇑↑f) 1) : LocallyCompactSpace (ContinuousMonoidHom X Y)

theorem ContinuousAddMonoidHom.locallyCompactSpace_of_hasBasis {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [AddGroup X] [TopologicalAddGroup X] [UniformSpace Y] [AddCommGroup Y] [UniformAddGroup Y] [T0Space Y] [CompactSpace Y] [LocallyCompactSpace X] (V : ℕ → Set Y) (hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x + x ∈ V n → x ∈ V (n + 1)) (hVo : (nhds 0).HasBasis (fun (x : ℕ) => True) V) : LocallyCompactSpace (ContinuousAddMonoidHom X Y)

abbrev ContinuousAddMonoidHom.locallyCompactSpace_of_hasBasis.match_1 {X : Type u_1} {Y : Type u_2} [AddGroup X] [AddCommGroup Y] (V : ℕ → Set Y) (U : ℕ → Set X) (motive : ↑{f : X →+ Y | Set.MapsTo (⇑f) (U 0) (V 0)} → Prop) : ∀ (x : ↑{f : X →+ Y | Set.MapsTo (⇑f) (U 0) (V 0)}), (∀ (f : X →+ Y) (hf : f ∈ {f : X →+ Y | Set.MapsTo (⇑f) (U 0) (V 0)}), motive ⟨f, hf⟩) → motive x

Equations

• ⋯ = ⋯

theorem ContinuousMonoidHom.locallyCompactSpace_of_hasBasis {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [Group X] [TopologicalGroup X] [UniformSpace Y] [CommGroup Y] [UniformGroup Y] [T0Space Y] [CompactSpace Y] [LocallyCompactSpace X] (V : ℕ → Set Y) (hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)) (hVo : (nhds 1).HasBasis (fun (x : ℕ) => True) V) : LocallyCompactSpace (ContinuousMonoidHom X Y)