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)

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

  Proof of continuity of the Hom.

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

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)

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

  Proof of continuity of the Hom.

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.

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

• coe : F → A → B
• coe_injective' : Function.Injective FunLike.coe
• 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.

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.

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

• coe : F → A → B
• coe_injective' : Function.Injective FunLike.coe
• 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.

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

You should also extend this typeclass when you extend ContinuousMonoidHom.

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

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

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

theorem ContinuousAddMonoidHom.ContinuousMonoidHom.ContinuousAddMonoidHomClass.proof_3 {A : Type u_2} {B : Type u_1} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (f : ContinuousAddMonoidHom A B) : ZeroHom.toFun (↑f.toAddMonoidHom) 0 = 0

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

theorem ContinuousAddMonoidHom.ContinuousMonoidHom.ContinuousAddMonoidHomClass.proof_2 {A : Type u_2} {B : Type u_1} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (f : ContinuousAddMonoidHom A B) (x : A) (y : A) : ZeroHom.toFun (↑f.toAddMonoidHom) (x + y) = ZeroHom.toFun (↑f.toAddMonoidHom) x + ZeroHom.toFun (↑f.toAddMonoidHom) y

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

instance ContinuousAddMonoidHom.instCoeFunContinuousAddMonoidHomForAll {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] : CoeFun (ContinuousAddMonoidHom A B) fun x => A → B

Helper instance for when there's too many metavariables to apply FunLike.hasCoeToFun.

instance ContinuousMonoidHom.instCoeFunContinuousMonoidHomForAll {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : CoeFun (ContinuousMonoidHom A B) fun x => A → B

Helper instance for when there's too many metavariables to apply FunLike.hasCoeToFun directly.

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.

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.

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.

theorem ContinuousAddMonoidHom.mk'.proof_1 {A : Type u_2} {B : Type u_1} [AddMonoid A] [AddMonoid B] (f : A →+ B) (x : A) (y : A) : ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y

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.

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.

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

@[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), ↑(ContinuousAddMonoidHom.comp g 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), ↑(ContinuousMonoidHom.comp g 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.

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.

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

@[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) : ↑(ContinuousMonoidHom.prod f g) i = (↑f.toMonoidHom i, ↑g.toMonoidHom i)

@[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) : ↑(ContinuousAddMonoidHom.sum f g) i = (↑f.toAddMonoidHom i, ↑g.toAddMonoidHom 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.

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.

theorem ContinuousAddMonoidHom.sum_map.proof_1 {A : Type u_3} {B : Type u_4} {C : Type u_2} {D : Type u_1} [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 x => (ZeroHom.toFun (↑f.toAddMonoidHom) x.fst, (↑g.toAddMonoidHom).1 x.snd)

@[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) : ↑(ContinuousMonoidHom.prod_map f g) i = (↑f.toMonoidHom i.fst, ↑g.toMonoidHom i.snd)

@[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) : ↑(ContinuousAddMonoidHom.sum_map f g) i = (↑f.toAddMonoidHom i.fst, ↑g.toAddMonoidHom i.snd)

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.

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.

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

@[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

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.

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

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

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

The identity continuous homomorphism.

@[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.

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.

@[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.fst

@[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.fst

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.

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.

@[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.snd

@[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.snd

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.

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.

@[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.

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.

@[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.

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

The continuous homomorphism given by the diagonal embedding.

@[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.

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.

@[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)

@[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)

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.

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

The continuous homomorphism given by addition.

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

@[simp]

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

@[simp]

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

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

The continuous homomorphism given by multiplication.

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

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

The continuous homomorphism given by negation.

@[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.

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.

@[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), ↑(ContinuousAddMonoidHom.coprod f g) a = ↑(ContinuousAddMonoidHom.add E).toAddMonoidHom (↑(ContinuousAddMonoidHom.sum_map f 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), ↑(ContinuousMonoidHom.coprod f g) a = ↑(ContinuousMonoidHom.mul E).toMonoidHom (↑(ContinuousMonoidHom.prod_map f 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.

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

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

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

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

theorem ContinuousAddMonoidHom.instAddCommGroupContinuousAddMonoidHomToAddMonoidToSubNegAddMonoidToAddGroup.proof_12 {A : Type u_2} {E : Type u_1} [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.instAddCommGroupContinuousAddMonoidHomToAddMonoidToSubNegAddMonoidToAddGroup.proof_5 {A : Type u_2} {E : Type u_1} [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.instAddCommGroupContinuousAddMonoidHomToAddMonoidToSubNegAddMonoidToAddGroup.proof_1 {A : Type u_2} {E : Type u_1} [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.instAddCommGroupContinuousAddMonoidHomToAddMonoidToSubNegAddMonoidToAddGroup.proof_13 {A : Type u_2} {E : Type u_1} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) : -f + f = 0

theorem ContinuousAddMonoidHom.instAddCommGroupContinuousAddMonoidHomToAddMonoidToSubNegAddMonoidToAddGroup.proof_6 {A : Type u_2} {E : Type u_1} [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.instAddCommGroupContinuousAddMonoidHomToAddMonoidToSubNegAddMonoidToAddGroup.proof_8 {A : Type u_2} {E : Type u_1} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) : f + 0 = f

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

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

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

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

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

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

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

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

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.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

abbrev ContinuousAddMonoidHom.closedEmbedding_toContinuousMap.match_1 (A : Type u_1) (B : Type u_2) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (f : ContinuousAddMonoidHom A B) (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (motive : ContinuousAddMonoidHom.toContinuousMap f ∈ {f | ↑f x ∈ U} ∩ {f | ↑f y ∈ V} ∩ {f | ↑f (x + y) ∈ W} → Prop) : (x : ContinuousAddMonoidHom.toContinuousMap f ∈ {f | ↑f x ∈ U} ∩ {f | ↑f y ∈ V} ∩ {f | ↑f (x + y) ∈ W}) → ((hfU : ContinuousAddMonoidHom.toContinuousMap f ∈ {f | ↑f x ∈ U}) → (hfV : ContinuousAddMonoidHom.toContinuousMap f ∈ {f | ↑f y ∈ V}) → (hfW : ContinuousAddMonoidHom.toContinuousMap f ∈ {f | ↑f (x + y) ∈ W}) → motive (_ : ContinuousAddMonoidHom.toContinuousMap f ∈ {f | ↑f x ∈ U} ∩ {f | ↑f y ∈ V} ∧ ContinuousAddMonoidHom.toContinuousMap f ∈ {f | ↑f (x + y) ∈ W})) → motive x

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

theorem ContinuousAddMonoidHom.instT2SpaceContinuousAddMonoidHomInstTopologicalSpaceContinuousAddMonoidHom.proof_1 {A : Type u_1} {B : Type u_2} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] [T2Space B] : T2Space (ContinuousAddMonoidHom A B)

instance ContinuousAddMonoidHom.instT2SpaceContinuousAddMonoidHomInstTopologicalSpaceContinuousAddMonoidHom {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] [T2Space B] : T2Space (ContinuousAddMonoidHom A B)

instance ContinuousMonoidHom.instT2SpaceContinuousMonoidHomInstTopologicalSpaceContinuousMonoidHom {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [T2Space B] : T2Space (ContinuousMonoidHom A B)

theorem ContinuousAddMonoidHom.instTopologicalAddGroupContinuousAddMonoidHomToAddMonoidToSubNegAddMonoidToAddGroupInstTopologicalSpaceContinuousAddMonoidHomToAddGroupInstAddCommGroupContinuousAddMonoidHomToAddMonoidToSubNegAddMonoidToAddGroup.proof_1 {A : Type u_1} {E : Type u_2} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] : TopologicalAddGroup (ContinuousAddMonoidHom A E)

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

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

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 y => ↑(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 y => ↑(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.comp f.snd f.fst

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.comp f.snd f.fst

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.comp g 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.comp g 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.comp f 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.comp f g

theorem ContinuousAddMonoidHom.compLeft.proof_2 {A : Type u_3} {B : Type u_2} (E : Type u_1) [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) : ZeroHom.toFun { toFun := fun g => ContinuousAddMonoidHom.comp g f, map_zero' := (_ : (fun g => ContinuousAddMonoidHom.comp g f) 0 = (fun g => ContinuousAddMonoidHom.comp g f) 0) } (_g + _h) = ZeroHom.toFun { toFun := fun g => ContinuousAddMonoidHom.comp g f, map_zero' := (_ : (fun g => ContinuousAddMonoidHom.comp g f) 0 = (fun g => ContinuousAddMonoidHom.comp g f) 0) } (_g + _h)

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

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.comp g f) 0 = (fun g => ContinuousAddMonoidHom.comp g f) 0

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.

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.

theorem ContinuousAddMonoidHom.compRight.proof_2 (A : Type u_2) {E : Type u_3} [AddMonoid A] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalAddGroup E] {B : Type u_1} [AddCommGroup B] [TopologicalSpace B] [TopologicalAddGroup B] (f : ContinuousAddMonoidHom B E) (g : ContinuousAddMonoidHom A B) (h : ContinuousAddMonoidHom A B) : ZeroHom.toFun { toFun := fun g => ContinuousAddMonoidHom.comp f g, map_zero' := (_ : (fun g => ContinuousAddMonoidHom.comp f g) 0 = 0) } (g + h) = ZeroHom.toFun { toFun := fun g => ContinuousAddMonoidHom.comp f g, map_zero' := (_ : (fun g => ContinuousAddMonoidHom.comp f g) 0 = 0) } g + ZeroHom.toFun { toFun := fun g => ContinuousAddMonoidHom.comp f g, map_zero' := (_ : (fun g => ContinuousAddMonoidHom.comp f g) 0 = 0) } h

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

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.comp f g) 0 = 0

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.

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.

def PontryaginDual (A : Type u_2) [Monoid A] [TopologicalSpace A] : Type u_2

The Pontryagin dual of A is the group of continuous homomorphism A → circle.

instance instTopologicalSpacePontryaginDual (A : Type u_2) [Monoid A] [TopologicalSpace A] : TopologicalSpace (PontryaginDual A)

instance instT2SpacePontryaginDualInstTopologicalSpacePontryaginDual (A : Type u_2) [Monoid A] [TopologicalSpace A] : T2Space (PontryaginDual A)

noncomputable instance instCommGroupPontryaginDual (A : Type u_2) [Monoid A] [TopologicalSpace A] : CommGroup (PontryaginDual A)

instance instTopologicalGroupPontryaginDualInstTopologicalSpacePontryaginDualToGroupInstCommGroupPontryaginDual (A : Type u_2) [Monoid A] [TopologicalSpace A] : TopologicalGroup (PontryaginDual A)

noncomputable instance instInhabitedPontryaginDual (A : Type u_2) [Monoid A] [TopologicalSpace A] : Inhabited (PontryaginDual A)

noncomputable instance PontryaginDual.instContinuousMonoidHomClassPontryaginDualSubtypeComplexMemSubmonoidToMulOneClassToMulZeroOneClassToNonAssocSemiringInstSemiringComplexInstMembershipInstSetLikeSubmonoidCircleToMonoidToMonoidToMonoidWithZeroInstTopologicalSpaceSubtypeToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingInstNormedFieldComplex {A : Type u_2} [Monoid A] [TopologicalSpace A] : ContinuousMonoidHomClass (PontryaginDual A) A { x // x ∈ circle }

noncomputable def PontryaginDual.map {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (f : ContinuousMonoidHom A B) : ContinuousMonoidHom (PontryaginDual B) (PontryaginDual A)

PontryaginDual is a functor.

@[simp]

theorem PontryaginDual.map_apply {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (f : ContinuousMonoidHom A B) (x : PontryaginDual B) (y : A) : ↑(↑(PontryaginDual.map f) x) y = ↑x (↑f y)

@[simp]

theorem PontryaginDual.map_one {A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : PontryaginDual.map (ContinuousMonoidHom.one A B) = ContinuousMonoidHom.one (PontryaginDual B) (PontryaginDual A)

@[simp]

theorem PontryaginDual.map_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) : PontryaginDual.map (ContinuousMonoidHom.comp g f) = ContinuousMonoidHom.comp (PontryaginDual.map f) (PontryaginDual.map g)

@[simp]

theorem PontryaginDual.map_mul {A : Type u_2} {E : Type u_6} [Monoid A] [CommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalGroup E] (f : ContinuousMonoidHom A E) (g : ContinuousMonoidHom A E) : PontryaginDual.map (f * g) = PontryaginDual.map f * PontryaginDual.map g

noncomputable def PontryaginDual.mapHom (A : Type u_2) (E : Type u_6) [Monoid A] [CommGroup E] [TopologicalSpace A] [TopologicalSpace E] [TopologicalGroup E] [LocallyCompactSpace E] : ContinuousMonoidHom (ContinuousMonoidHom A E) (ContinuousMonoidHom (PontryaginDual E) (PontryaginDual A))

ContinuousMonoidHom.dual as a ContinuousMonoidHom.