# Documentation

Mathlib.Algebra.Hom.Equiv.Basic

# Multiplicative and additive equivs #

In this file we define two extensions of Equiv called AddEquiv and MulEquiv, which are datatypes representing isomorphisms of AddMonoids/AddGroups and Monoids/Groups.

## Notations #

• infix  ≃* :25 := MulEquiv≃* :25 := MulEquiv
• infix  ≃+ :25 := AddEquiv≃+ :25 := AddEquiv

The extended equivs all have coercions to functions, and the coercions are the canonical notation when treating the isomorphisms as maps.

## Tags #

def AddHom.inverse {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : AddHom M N) (g : NM) (h₁ : ) (h₂ : ) :

Equations
• AddHom.inverse f g h₁ h₂ = { toFun := g, map_add' := (_ : ∀ (x y : N), g (x + y) = g x + g y) }
def AddHom.inverse.proof_1 {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : AddHom M N) (g : NM) (h₁ : ) (h₂ : ) (x : N) (y : N) :
g (x + y) = g x + g y
Equations
• (_ : g (x + y) = g x + g y) = (_ : g (x + y) = g x + g y)
def MulHom.inverse {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M →ₙ* N) (g : NM) (h₁ : ) (h₂ : ) :

Makes a multiplicative inverse from a bijection which preserves multiplication.

Equations
• MulHom.inverse f g h₁ h₂ = { toFun := g, map_mul' := (_ : ∀ (x y : N), g (x * y) = g x * g y) }
def AddMonoidHom.inverse.proof_2 {A : Type u_1} {B : Type u_2} [inst : ] [inst : ] (f : A →+ B) (g : BA) (h₁ : ) (h₂ : ) (x : B) (y : B) :
Equations
• One or more equations did not get rendered due to their size.
def AddMonoidHom.inverse.proof_1 {A : Type u_1} {B : Type u_2} [inst : ] [inst : ] (f : A →+ B) (g : BA) (h₁ : ) :
g 0 = 0
Equations
• (_ : g 0 = 0) = (_ : g 0 = 0)
def AddMonoidHom.inverse {A : Type u_1} {B : Type u_2} [inst : ] [inst : ] (f : A →+ B) (g : BA) (h₁ : ) (h₂ : ) :
B →+ A

The inverse of a bijective AddMonoidHom is an AddMonoidHom.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem MonoidHom.inverse_apply {A : Type u_1} {B : Type u_2} [inst : ] [inst : ] (f : A →* B) (g : BA) (h₁ : ) (h₂ : ) :
∀ (a : B), ↑(MonoidHom.inverse f g h₁ h₂) a = g a
@[simp]
theorem AddMonoidHom.inverse_apply {A : Type u_1} {B : Type u_2} [inst : ] [inst : ] (f : A →+ B) (g : BA) (h₁ : ) (h₂ : ) :
∀ (a : B), ↑(AddMonoidHom.inverse f g h₁ h₂) a = g a
def MonoidHom.inverse {A : Type u_1} {B : Type u_2} [inst : ] [inst : ] (f : A →* B) (g : BA) (h₁ : ) (h₂ : ) :
B →* A

The inverse of a bijective MonoidHom is a MonoidHom.

Equations
• One or more equations did not get rendered due to their size.
structure AddEquiv (A : Type u_1) (B : Type u_2) [inst : Add A] [inst : Add B] extends :
Type (maxu_1u_2)

AddEquiv α β is the type of an equiv α ≃ β≃ β which preserves addition.

Instances For
class AddEquivClass (F : Type u_1) (A : outParam (Type u_2)) (B : outParam (Type u_3)) [inst : Add A] [inst : Add B] extends :
Type (max(maxu_1u_2)u_3)

map_add : ∀ (f : F) (a b : A), f (a + b) = f a + f b

AddEquivClass F A B states that F is a type of addition-preserving morphisms. You should extend this class when you extend AddEquiv.

Instances
abbrev AddEquiv.toAddHom {A : Type u_1} {B : Type u_2} [inst : Add A] [inst : Add B] (self : A ≃+ B) :

The AddHom underlying a AddEquiv.

Equations
structure MulEquiv (M : Type u_1) (N : Type u_2) [inst : Mul M] [inst : Mul N] extends :
Type (maxu_1u_2)

MulEquiv α β is the type of an equiv α ≃ β≃ β which preserves multiplication.

Instances For
abbrev MulEquiv.toMulHom {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (self : M ≃* N) :

The MulHom underlying a MulEquiv.

Equations
class MulEquivClass (F : Type u_1) (A : outParam (Type u_2)) (B : outParam (Type u_3)) [inst : Mul A] [inst : Mul B] extends :
Type (max(maxu_1u_2)u_3)
• Preserves multiplication.

map_mul : ∀ (f : F) (a b : A), f (a * b) = f a * f b

MulEquivClass F A B states that F is a type of multiplication-preserving morphisms. You should extend this class when you extend MulEquiv.

Instances

Notation for a MulEquiv.

Equations

Notation for an AddEquiv.

Equations
instance AddEquivClass.instAddHomClass (F : Type u_1) {M : Type u_2} {N : Type u_3} :
{x : Add M} → {x_1 : Add N} → [h : ] → AddHomClass F M N
Equations
def AddEquivClass.instAddHomClass.proof_1 (F : Type u_1) {M : Type u_2} {N : Type u_3} :
∀ {x : Add M} {x_1 : Add N} [h : ], Function.Injective FunLike.coe
Equations
instance MulEquivClass.instMulHomClass (F : Type u_1) {M : Type u_2} {N : Type u_3} :
{x : Mul M} → {x_1 : Mul N} → [h : ] → MulHomClass F M N
Equations
instance AddEquivClass.instAddMonoidHomClass (F : Type u_1) {M : Type u_2} {N : Type u_3} :
{x : } → {x_1 : } → [inst : ] →
Equations
def AddEquivClass.instAddMonoidHomClass.proof_2 (F : Type u_3) {M : Type u_2} {N : Type u_1} :
∀ {x : } {x_1 : } [inst : ] (f : F) (x_2 y : M), f (x_2 + y) = f x_2 + f y
Equations
• (_ : ∀ (f : F) (x y : M), f (x + y) = f x + f y) = (_ : ∀ (f : F) (x y : M), f (x + y) = f x + f y)
def AddEquivClass.instAddMonoidHomClass.proof_1 (F : Type u_1) {M : Type u_2} {N : Type u_3} :
∀ {x : } {x_1 : } [inst : ], Function.Injective FunLike.coe
Equations
def AddEquivClass.instAddMonoidHomClass.proof_3 (F : Type u_2) {M : Type u_3} {N : Type u_1} :
∀ {x : } {x_1 : } [inst : ] (e : F), e 0 = 0
Equations
• (_ : e 0 = 0) = (_ : e 0 = 0)
instance MulEquivClass.instMonoidHomClass (F : Type u_1) {M : Type u_2} {N : Type u_3} :
{x : } → {x_1 : } → [inst : ] →
Equations
instance MulEquivClass.toMonoidWithZeroHomClass (F : Type u_1) {α : Type u_2} {β : Type u_3} :
{x : } → {x_1 : } → [inst : ] →
Equations
@[simp]
theorem AddEquivClass.map_eq_zero_iff {F : Type u_3} {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] [inst : ] (h : F) {x : M} :
h x = 0 x = 0
@[simp]
theorem MulEquivClass.map_eq_one_iff {F : Type u_3} {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] [inst : ] (h : F) {x : M} :
h x = 1 x = 1
theorem AddEquivClass.map_ne_zero_iff {F : Type u_3} {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] [inst : ] (h : F) {x : M} :
h x 0 x 0
theorem MulEquivClass.map_ne_one_iff {F : Type u_3} {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] [inst : ] (h : F) {x : M} :
h x 1 x 1
def AddEquivClass.toAddEquiv {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Add α] [inst : Add β] [inst : ] (f : F) :
α ≃+ β

Turn an element of a type F satisfying AddEquivClass F α β into an actual AddEquiv. This is declared as the default coercion from F to α ≃+ β≃+ β.

Equations
• One or more equations did not get rendered due to their size.
def MulEquivClass.toMulEquiv {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Mul α] [inst : Mul β] [inst : ] (f : F) :
α ≃* β

Turn an element of a type F satisfying MulEquivClass F α β into an actual MulEquiv. This is declared as the default coercion from F to α ≃* β≃* β.

Equations
• One or more equations did not get rendered due to their size.
instance instCoeTCAddEquiv {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Add α] [inst : Add β] [inst : ] :
CoeTC F (α ≃+ β)

Any type satisfying AddEquivClass can be cast into AddEquiv via AddEquivClass.toAddEquiv.

Equations
instance instCoeTCMulEquiv {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Mul α] [inst : Mul β] [inst : ] :
CoeTC F (α ≃* β)

Any type satisfying MulEquivClass can be cast into MulEquiv via MulEquivClass.toMulEquiv.

Equations
• instCoeTCMulEquiv = { coe := MulEquivClass.toMulEquiv }
def AddEquiv.instAddEquivClassAddEquiv.proof_3 {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M ≃+ N) (g : M ≃+ N) (h₁ : (fun f => f.toEquiv.toFun) f = (fun f => f.toEquiv.toFun) g) (h₂ : (fun f => f.toEquiv.invFun) f = (fun f => f.toEquiv.invFun) g) :
f = g
Equations
• (_ : f = g) = (_ : f = g)
Equations
Function.LeftInverse f.toEquiv.invFun f.toEquiv.toFun
Equations
Function.RightInverse f.toEquiv.invFun f.toEquiv.toFun
Equations
instance MulEquiv.instMulEquivClassMulEquiv {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] :
Equations
@[simp]
theorem AddEquiv.toEquiv_eq_coe {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M ≃+ N) :
f.toEquiv = f
@[simp]
theorem MulEquiv.toEquiv_eq_coe {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M ≃* N) :
f.toEquiv = f
@[simp]
theorem AddEquiv.coe_toEquiv {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M ≃+ N) :
f = f
@[simp]
theorem MulEquiv.coe_toEquiv {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M ≃* N) :
f = f
@[simp]
theorem AddEquiv.coe_toAddHom {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] {f : M ≃+ N} :
↑() = f
@[simp]
theorem MulEquiv.coe_toMulHom {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] {f : M ≃* N} :
↑() = f
theorem AddEquiv.map_add {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M ≃+ N) (x : M) (y : M) :
f (x + y) = f x + f y

theorem MulEquiv.map_mul {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M ≃* N) (x : M) (y : M) :
f (x * y) = f x * f y

A multiplicative isomorphism preserves multiplication.

def AddEquiv.mk' {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M N) (h : ∀ (x y : M), f (x + y) = f x + f y) :
M ≃+ N

Equations
• = { toEquiv := f, map_add' := h }
def MulEquiv.mk' {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M N) (h : ∀ (x y : M), f (x * y) = f x * f y) :
M ≃* N

Makes a multiplicative isomorphism from a bijection which preserves multiplication.

Equations
• = { toEquiv := f, map_mul' := h }
theorem AddEquiv.bijective {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) :
theorem MulEquiv.bijective {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) :
theorem AddEquiv.injective {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) :
theorem MulEquiv.injective {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) :
theorem AddEquiv.surjective {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) :
theorem MulEquiv.surjective {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) :
M ≃+ M

The identity map is an additive isomorphism.

Equations
• One or more equations did not get rendered due to their size.
∀ (x x_1 : M), Equiv.toFun { toFun := ().toFun, invFun := ().invFun, left_inv := (_ : Function.LeftInverse ().invFun ().toFun), right_inv := (_ : Function.RightInverse ().invFun ().toFun) } (x + x_1) = Equiv.toFun { toFun := ().toFun, invFun := ().invFun, left_inv := (_ : Function.LeftInverse ().invFun ().toFun), right_inv := (_ : Function.RightInverse ().invFun ().toFun) } (x + x_1)
Equations
• One or more equations did not get rendered due to their size.
def MulEquiv.refl (M : Type u_1) [inst : Mul M] :
M ≃* M

The identity map is a multiplicative isomorphism.

Equations
• One or more equations did not get rendered due to their size.
Equations
instance MulEquiv.instInhabitedMulEquiv {M : Type u_1} [inst : Mul M] :
Equations
• MulEquiv.instInhabitedMulEquiv = { default := }
def AddEquiv.symm {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (h : M ≃+ N) :
N ≃+ M

The inverse of an isomorphism is an isomorphism.

Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.symm.proof_1 {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (h : M ≃+ N) (a : N) (b : N) :
↑(AddHom.inverse () ↑(Equiv.symm h.toEquiv) (_ : Function.LeftInverse h.toEquiv.invFun h.toEquiv.toFun) (_ : Function.RightInverse h.toEquiv.invFun h.toEquiv.toFun)) (a + b) = ↑(AddHom.inverse () ↑(Equiv.symm h.toEquiv) (_ : Function.LeftInverse h.toEquiv.invFun h.toEquiv.toFun) (_ : Function.RightInverse h.toEquiv.invFun h.toEquiv.toFun)) a + ↑(AddHom.inverse () ↑(Equiv.symm h.toEquiv) (_ : Function.LeftInverse h.toEquiv.invFun h.toEquiv.toFun) (_ : Function.RightInverse h.toEquiv.invFun h.toEquiv.toFun)) b
Equations
• One or more equations did not get rendered due to their size.
def MulEquiv.symm {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (h : M ≃* N) :
N ≃* M

The inverse of an isomorphism is an isomorphism.

Equations
• One or more equations did not get rendered due to their size.
theorem AddEquiv.invFun_eq_symm {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] {f : M ≃+ N} :
f.toEquiv.invFun = ↑()
theorem MulEquiv.invFun_eq_symm {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] {f : M ≃* N} :
f.toEquiv.invFun = ↑()
@[simp]
theorem AddEquiv.coe_toEquiv_symm {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M ≃+ N) :
↑() = ↑()
@[simp]
theorem MulEquiv.coe_toEquiv_symm {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M ≃* N) :
↑() = ↑()
@[simp]
theorem AddEquiv.equivLike_neg_eq_symm {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M ≃+ N) :
= ↑()
@[simp]
theorem MulEquiv.equivLike_inv_eq_symm {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M ≃* N) :
= ↑()
def AddEquiv.Simps.symm_apply {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) :
NM

See Note [custom simps projection]

Equations
def MulEquiv.Simps.symm_apply {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) :
NM

See Note [custom simps projection]

Equations
@[simp]
theorem AddEquiv.toEquiv_symm {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M ≃+ N) :
↑() =
@[simp]
theorem MulEquiv.toEquiv_symm {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M ≃* N) :
↑() =
@[simp]
theorem AddEquiv.coe_mk {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M N) (hf : ∀ (x y : M), f (x + y) = f x + f y) :
{ toEquiv := f, map_add' := hf } = f
@[simp]
theorem MulEquiv.coe_mk {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M N) (hf : ∀ (x y : M), f (x * y) = f x * f y) :
{ toEquiv := f, map_mul' := hf } = f
@[simp]
theorem AddEquiv.symm_symm {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M ≃+ N) :
@[simp]
theorem MulEquiv.symm_symm {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M ≃* N) :
theorem AddEquiv.symm_bijective {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] :
theorem MulEquiv.symm_bijective {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] :
Function.Bijective MulEquiv.symm
@[simp]
theorem AddEquiv.symm_mk {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : M N) (h : ∀ (x y : M), Equiv.toFun f (x + y) = + ) :
AddEquiv.symm { toEquiv := f, map_add' := h } = { toEquiv := , map_add' := (_ : ∀ (x y : N), Equiv.toFun (AddEquiv.symm { toEquiv := f, map_add' := h }).toEquiv (x + y) = Equiv.toFun (AddEquiv.symm { toEquiv := f, map_add' := h }).toEquiv x + Equiv.toFun (AddEquiv.symm { toEquiv := f, map_add' := h }).toEquiv y) }
@[simp]
theorem MulEquiv.symm_mk {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M N) (h : ∀ (x y : M), Equiv.toFun f (x * y) = * ) :
MulEquiv.symm { toEquiv := f, map_mul' := h } = { toEquiv := , map_mul' := (_ : ∀ (x y : N), Equiv.toFun (MulEquiv.symm { toEquiv := f, map_mul' := h }).toEquiv (x * y) = Equiv.toFun (MulEquiv.symm { toEquiv := f, map_mul' := h }).toEquiv x * Equiv.toFun (MulEquiv.symm { toEquiv := f, map_mul' := h }).toEquiv y) }
@[simp]
@[simp]
theorem MulEquiv.refl_symm {M : Type u_1} [inst : Mul M] :
def AddEquiv.trans.proof_1 {M : Type u_2} {N : Type u_3} {P : Type u_1} [inst : Add M] [inst : Add N] [inst : Add P] (h1 : M ≃+ N) (h2 : N ≃+ P) (x : M) (y : M) :
h2 (h1 (x + y)) = h2 (h1 x) + h2 (h1 y)
Equations
• (_ : h2 (h1 (x + y)) = h2 (h1 x) + h2 (h1 y)) = (_ : h2 (h1 (x + y)) = h2 (h1 x) + h2 (h1 y))
def AddEquiv.trans {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : Add M] [inst : Add N] [inst : Add P] (h1 : M ≃+ N) (h2 : N ≃+ P) :
M ≃+ P

Equations
• One or more equations did not get rendered due to their size.
def MulEquiv.trans {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : Mul M] [inst : Mul N] [inst : Mul P] (h1 : M ≃* N) (h2 : N ≃* P) :
M ≃* P

Transitivity of multiplication-preserving isomorphisms

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem AddEquiv.apply_symm_apply {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) (y : N) :
e (↑() y) = y

e.symm is a right inverse of e, written as e (e.symm y) = y.

@[simp]
theorem MulEquiv.apply_symm_apply {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) (y : N) :
e (↑() y) = y

e.symm is a right inverse of e, written as e (e.symm y) = y.

@[simp]
theorem AddEquiv.symm_apply_apply {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) (x : M) :
↑() (e x) = x

e.symm is a left inverse of e, written as e.symm (e y) = y.

@[simp]
theorem MulEquiv.symm_apply_apply {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) (x : M) :
↑() (e x) = x

e.symm is a left inverse of e, written as e.symm (e y) = y.

@[simp]
theorem AddEquiv.symm_comp_self {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) :
↑() e = id
@[simp]
theorem MulEquiv.symm_comp_self {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) :
↑() e = id
@[simp]
theorem AddEquiv.self_comp_symm {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) :
e ↑() = id
@[simp]
theorem MulEquiv.self_comp_symm {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) :
e ↑() = id
@[simp]
↑() = id
@[simp]
theorem MulEquiv.coe_refl {M : Type u_1} [inst : Mul M] :
↑() = id
@[simp]
theorem AddEquiv.refl_apply {M : Type u_1} [inst : Add M] (m : M) :
↑() m = m
@[simp]
theorem MulEquiv.refl_apply {M : Type u_1} [inst : Mul M] (m : M) :
↑() m = m
@[simp]
theorem AddEquiv.coe_trans {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : Add M] [inst : Add N] [inst : Add P] (e₁ : M ≃+ N) (e₂ : N ≃+ P) :
↑(AddEquiv.trans e₁ e₂) = e₂ e₁
@[simp]
theorem MulEquiv.coe_trans {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : Mul M] [inst : Mul N] [inst : Mul P] (e₁ : M ≃* N) (e₂ : N ≃* P) :
↑(MulEquiv.trans e₁ e₂) = e₂ e₁
@[simp]
theorem AddEquiv.trans_apply {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : Add M] [inst : Add N] [inst : Add P] (e₁ : M ≃+ N) (e₂ : N ≃+ P) (m : M) :
↑(AddEquiv.trans e₁ e₂) m = e₂ (e₁ m)
@[simp]
theorem MulEquiv.trans_apply {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : Mul M] [inst : Mul N] [inst : Mul P] (e₁ : M ≃* N) (e₂ : N ≃* P) (m : M) :
↑(MulEquiv.trans e₁ e₂) m = e₂ (e₁ m)
@[simp]
theorem AddEquiv.symm_trans_apply {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : Add M] [inst : Add N] [inst : Add P] (e₁ : M ≃+ N) (e₂ : N ≃+ P) (p : P) :
@[simp]
theorem MulEquiv.symm_trans_apply {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : Mul M] [inst : Mul N] [inst : Mul P] (e₁ : M ≃* N) (e₂ : N ≃* P) (p : P) :
↑(MulEquiv.symm (MulEquiv.trans e₁ e₂)) p = ↑() (↑() p)
theorem AddEquiv.apply_eq_iff_eq {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) {x : M} {y : M} :
e x = e y x = y
theorem MulEquiv.apply_eq_iff_eq {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) {x : M} {y : M} :
e x = e y x = y
theorem AddEquiv.apply_eq_iff_symm_apply {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) {x : M} {y : N} :
e x = y x = ↑() y
theorem MulEquiv.apply_eq_iff_symm_apply {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) {x : M} {y : N} :
e x = y x = ↑() y
theorem AddEquiv.symm_apply_eq {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) {x : N} {y : M} :
↑() x = y x = e y
theorem MulEquiv.symm_apply_eq {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) {x : N} {y : (fun x => M) x} :
↑() x = y x = e y
theorem AddEquiv.eq_symm_apply {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) {x : N} {y : M} :
y = ↑() x e y = x
theorem MulEquiv.eq_symm_apply {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) {x : N} {y : (fun x => M) x} :
y = ↑() x e y = x
theorem AddEquiv.eq_comp_symm {M : Type u_2} {N : Type u_3} [inst : Add M] [inst : Add N] {α : Type u_1} (e : M ≃+ N) (f : Nα) (g : Mα) :
f = g ↑() f e = g
theorem MulEquiv.eq_comp_symm {M : Type u_2} {N : Type u_3} [inst : Mul M] [inst : Mul N] {α : Type u_1} (e : M ≃* N) (f : Nα) (g : Mα) :
f = g ↑() f e = g
theorem AddEquiv.comp_symm_eq {M : Type u_2} {N : Type u_3} [inst : Add M] [inst : Add N] {α : Type u_1} (e : M ≃+ N) (f : Nα) (g : Mα) :
g ↑() = f g = f e
theorem MulEquiv.comp_symm_eq {M : Type u_2} {N : Type u_3} [inst : Mul M] [inst : Mul N] {α : Type u_1} (e : M ≃* N) (f : Nα) (g : Mα) :
g ↑() = f g = f e
theorem AddEquiv.eq_symm_comp {M : Type u_2} {N : Type u_3} [inst : Add M] [inst : Add N] {α : Type u_1} (e : M ≃+ N) (f : αM) (g : αN) :
f = ↑() g e f = g
theorem MulEquiv.eq_symm_comp {M : Type u_2} {N : Type u_3} [inst : Mul M] [inst : Mul N] {α : Type u_1} (e : M ≃* N) (f : αM) (g : αN) :
f = ↑() g e f = g
theorem AddEquiv.symm_comp_eq {M : Type u_2} {N : Type u_3} [inst : Add M] [inst : Add N] {α : Type u_1} (e : M ≃+ N) (f : αM) (g : αN) :
↑() g = f g = e f
theorem MulEquiv.symm_comp_eq {M : Type u_2} {N : Type u_3} [inst : Mul M] [inst : Mul N] {α : Type u_1} (e : M ≃* N) (f : αM) (g : αN) :
↑() g = f g = e f
@[simp]
theorem AddEquiv.symm_trans_self {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) :
@[simp]
theorem MulEquiv.symm_trans_self {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) :
@[simp]
theorem AddEquiv.self_trans_symm {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) :
@[simp]
theorem MulEquiv.self_trans_symm {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) :
theorem MulEquiv.coe_monoidHom_refl {M : Type u_1} [inst : ] :
↑() =
theorem AddEquiv.coe_addMonoidHom_trans {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : ] [inst : ] [inst : ] (e₁ : M ≃+ N) (e₂ : N ≃+ P) :
theorem MulEquiv.coe_monoidHom_trans {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : ] [inst : ] [inst : ] (e₁ : M ≃* N) (e₂ : N ≃* P) :
↑(MulEquiv.trans e₁ e₂) = MonoidHom.comp e₂ e₁
theorem AddEquiv.ext {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] {f : M ≃+ N} {g : M ≃+ N} (h : ∀ (x : M), f x = g x) :
f = g

Two additive isomorphisms agree if they are defined by the same underlying function.

theorem MulEquiv.ext {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] {f : M ≃* N} {g : M ≃* N} (h : ∀ (x : M), f x = g x) :
f = g

Two multiplicative isomorphisms agree if they are defined by the same underlying function.

theorem AddEquiv.ext_iff {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] {f : M ≃+ N} {g : M ≃+ N} :
f = g ∀ (x : M), f x = g x
theorem MulEquiv.ext_iff {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] {f : M ≃* N} {g : M ≃* N} :
f = g ∀ (x : M), f x = g x
@[simp]
theorem AddEquiv.mk_coe {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) (e' : NM) (h₁ : ) (h₂ : ) (h₃ : ∀ (x y : M), Equiv.toFun { toFun := e, invFun := e', left_inv := h₁, right_inv := h₂ } (x + y) = Equiv.toFun { toFun := e, invFun := e', left_inv := h₁, right_inv := h₂ } x + Equiv.toFun { toFun := e, invFun := e', left_inv := h₁, right_inv := h₂ } y) :
{ toEquiv := { toFun := e, invFun := e', left_inv := h₁, right_inv := h₂ }, map_add' := h₃ } = e
@[simp]
theorem MulEquiv.mk_coe {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) (e' : NM) (h₁ : ) (h₂ : ) (h₃ : ∀ (x y : M), Equiv.toFun { toFun := e, invFun := e', left_inv := h₁, right_inv := h₂ } (x * y) = Equiv.toFun { toFun := e, invFun := e', left_inv := h₁, right_inv := h₂ } x * Equiv.toFun { toFun := e, invFun := e', left_inv := h₁, right_inv := h₂ } y) :
{ toEquiv := { toFun := e, invFun := e', left_inv := h₁, right_inv := h₂ }, map_mul' := h₃ } = e
@[simp]
theorem AddEquiv.mk_coe' {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (e : M ≃+ N) (f : NM) (h₁ : Function.LeftInverse (e) f) (h₂ : ) (h₃ : ∀ (x y : N), Equiv.toFun { toFun := f, invFun := e, left_inv := h₁, right_inv := h₂ } (x + y) = Equiv.toFun { toFun := f, invFun := e, left_inv := h₁, right_inv := h₂ } x + Equiv.toFun { toFun := f, invFun := e, left_inv := h₁, right_inv := h₂ } y) :
{ toEquiv := { toFun := f, invFun := e, left_inv := h₁, right_inv := h₂ }, map_add' := h₃ } =
@[simp]
theorem MulEquiv.mk_coe' {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (e : M ≃* N) (f : NM) (h₁ : Function.LeftInverse (e) f) (h₂ : ) (h₃ : ∀ (x y : N), Equiv.toFun { toFun := f, invFun := e, left_inv := h₁, right_inv := h₂ } (x * y) = Equiv.toFun { toFun := f, invFun := e, left_inv := h₁, right_inv := h₂ } x * Equiv.toFun { toFun := f, invFun := e, left_inv := h₁, right_inv := h₂ } y) :
{ toEquiv := { toFun := f, invFun := e, left_inv := h₁, right_inv := h₂ }, map_mul' := h₃ } =
theorem AddEquiv.congr_arg {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] {f : M ≃+ N} {x : M} {x' : M} :
x = x'f x = f x'
theorem MulEquiv.congr_arg {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] {f : M ≃* N} {x : M} {x' : M} :
x = x'f x = f x'
theorem AddEquiv.congr_fun {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] {f : M ≃+ N} {g : M ≃+ N} (h : f = g) (x : M) :
f x = g x
theorem MulEquiv.congr_fun {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] {f : M ≃* N} {g : M ≃* N} (h : f = g) (x : M) :
f x = g x
def AddEquiv.addEquivOfUnique {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] [inst : Add M] [inst : Add N] :
M ≃+ N

The AddEquiv between two AddMonoids with a unique element.

Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.addEquivOfUnique.proof_1 {M : Type u_2} {N : Type u_1} [inst : ] [inst : ] [inst : Add M] [inst : Add N] :
∀ (x x_1 : M), Equiv.toFun { toFun := ().toFun, invFun := ().invFun, left_inv := (_ : Function.LeftInverse ().invFun ().toFun), right_inv := (_ : Function.RightInverse ().invFun ().toFun) } (x + x_1) = Equiv.toFun { toFun := ().toFun, invFun := ().invFun, left_inv := (_ : Function.LeftInverse ().invFun ().toFun), right_inv := (_ : Function.RightInverse ().invFun ().toFun) } x + Equiv.toFun { toFun := ().toFun, invFun := ().invFun, left_inv := (_ : Function.LeftInverse ().invFun ().toFun), right_inv := (_ : Function.RightInverse ().invFun ().toFun) } x_1
Equations
• One or more equations did not get rendered due to their size.
def MulEquiv.mulEquivOfUnique {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] [inst : Mul M] [inst : Mul N] :
M ≃* N

The MulEquiv between two monoids with a unique element.

Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.instUniqueAddEquiv.proof_1 {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] [inst : Add M] [inst : Add N] :
∀ (x : M ≃+ N), x = default
Equations
• (_ : x = default) = (_ : x = default)
instance AddEquiv.instUniqueAddEquiv {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] [inst : Add M] [inst : Add N] :
Unique (M ≃+ N)

There is a unique additive monoid homomorphism between two additive monoids with a unique element.

Equations
• AddEquiv.instUniqueAddEquiv = { toInhabited := { default := AddEquiv.addEquivOfUnique }, uniq := (_ : ∀ (x : M ≃+ N), x = default) }
instance MulEquiv.instUniqueMulEquiv {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] [inst : Mul M] [inst : Mul N] :
Unique (M ≃* N)

There is a unique monoid homomorphism between two monoids with a unique element.

Equations
• MulEquiv.instUniqueMulEquiv = { toInhabited := { default := MulEquiv.mulEquivOfUnique }, uniq := (_ : ∀ (x : M ≃* N), x = default) }

## Monoids #

theorem AddEquiv.map_zero {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (h : M ≃+ N) :
h 0 = 0

An additive isomorphism of additive monoids sends 0 to 0 (and is hence an additive monoid isomorphism).

theorem MulEquiv.map_one {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (h : M ≃* N) :
h 1 = 1

A multiplicative isomorphism of monoids sends 1 to 1 (and is hence a monoid isomorphism).

theorem AddEquiv.map_eq_zero_iff {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (h : M ≃+ N) {x : M} :
h x = 0 x = 0
theorem MulEquiv.map_eq_one_iff {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (h : M ≃* N) {x : M} :
h x = 1 x = 1
theorem AddEquiv.map_ne_zero_iff {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (h : M ≃+ N) {x : M} :
h x 0 x 0
theorem MulEquiv.map_ne_one_iff {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (h : M ≃* N) {x : M} :
h x 1 x 1
noncomputable def AddEquiv.ofBijective {M : Type u_1} {N : Type u_2} {F : Type u_3} [inst : Add M] [inst : Add N] [inst : AddHomClass F M N] (f : F) (hf : ) :
M ≃+ N

A bijective AddSemigroup homomorphism is an isomorphism

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem AddEquiv.ofBijective_apply {M : Type u_1} {N : Type u_2} {F : Type u_3} [inst : Add M] [inst : Add N] [inst : AddHomClass F M N] (f : F) (hf : ) (a : M) :
↑() a = f a
@[simp]
theorem MulEquiv.ofBijective_apply {M : Type u_1} {N : Type u_2} {F : Type u_3} [inst : Mul M] [inst : Mul N] [inst : MulHomClass F M N] (f : F) (hf : ) (a : M) :
↑() a = f a
noncomputable def MulEquiv.ofBijective {M : Type u_1} {N : Type u_2} {F : Type u_3} [inst : Mul M] [inst : Mul N] [inst : MulHomClass F M N] (f : F) (hf : ) :
M ≃* N

A bijective Semigroup homomorphism is an isomorphism

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem AddEquiv.ofBijective_apply_symm_apply {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] {n : N} (f : M →+ N) (hf : ) :
f (↑(Equiv.symm (Equiv.ofBijective (f) hf)) n) = n
@[simp]
theorem MulEquiv.ofBijective_apply_symm_apply {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] {n : N} (f : M →* N) (hf : ) :
f (↑(Equiv.symm (Equiv.ofBijective (f) hf)) n) = n
def AddEquiv.toAddMonoidHom {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (h : M ≃+ N) :
M →+ N

Extract the forward direction of an additive equivalence as an addition-preserving function.

Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.toAddMonoidHom.proof_1 {M : Type u_2} {N : Type u_1} [inst : ] [inst : ] (h : M ≃+ N) (x : M) (y : M) :
Equiv.toFun h.toEquiv (x + y) = Equiv.toFun h.toEquiv x + Equiv.toFun h.toEquiv y
Equations
• One or more equations did not get rendered due to their size.
def MulEquiv.toMonoidHom {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (h : M ≃* N) :
M →* N

Extract the forward direction of a multiplicative equivalence as a multiplication-preserving function.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem AddEquiv.coe_toAddMonoidHom {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (e : M ≃+ N) :
= e
@[simp]
theorem MulEquiv.coe_toMonoidHom {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (e : M ≃* N) :
↑() = e
theorem AddEquiv.toAddMonoidHom_injective {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] :
theorem MulEquiv.toMonoidHom_injective {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] :
Function.Injective MulEquiv.toMonoidHom
def AddEquiv.arrowCongr {M : Type u_1} {N : Type u_2} {P : Type u_3} {Q : Type u_4} [inst : Add P] [inst : Add Q] (f : M N) (g : P ≃+ Q) :
(MP) ≃+ (NQ)

An additive analogue of Equiv.arrowCongr, where the equivalence between the targets is additive.

Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.arrowCongr.proof_1 {M : Type u_1} {N : Type u_4} {P : Type u_2} {Q : Type u_3} [inst : Add P] [inst : Add Q] (f : M N) (g : P ≃+ Q) (h : MP) :
(fun k m => ↑() (k (f m))) ((fun h n => g (h (↑() n))) h) = h
Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.arrowCongr.proof_3 {M : Type u_3} {N : Type u_1} {P : Type u_4} {Q : Type u_2} [inst : Add P] [inst : Add Q] (f : M N) (g : P ≃+ Q) (h : MP) (k : MP) :
Equiv.toFun { toFun := fun h n => g (h (↑() n)), invFun := fun k m => ↑() (k (f m)), left_inv := (_ : ∀ (h : MP), (fun k m => ↑() (k (f m))) ((fun h n => g (h (↑() n))) h) = h), right_inv := (_ : ∀ (k : NQ), (fun h n => g (h (↑() n))) ((fun k m => ↑() (k (f m))) k) = k) } (h + k) = Equiv.toFun { toFun := fun h n => g (h (↑() n)), invFun := fun k m => ↑() (k (f m)), left_inv := (_ : ∀ (h : MP), (fun k m => ↑() (k (f m))) ((fun h n => g (h (↑() n))) h) = h), right_inv := (_ : ∀ (k : NQ), (fun h n => g (h (↑() n))) ((fun k m => ↑() (k (f m))) k) = k) } h + Equiv.toFun { toFun := fun h n => g (h (↑() n)), invFun := fun k m => ↑() (k (f m)), left_inv := (_ : ∀ (h : MP), (fun k m => ↑() (k (f m))) ((fun h n => g (h (↑() n))) h) = h), right_inv := (_ : ∀ (k : NQ), (fun h n => g (h (↑() n))) ((fun k m => ↑() (k (f m))) k) = k) } k
Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.arrowCongr.proof_2 {M : Type u_4} {N : Type u_1} {P : Type u_3} {Q : Type u_2} [inst : Add P] [inst : Add Q] (f : M N) (g : P ≃+ Q) (k : NQ) :
(fun h n => g (h (↑() n))) ((fun k m => ↑() (k (f m))) k) = k
Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem MulEquiv.arrowCongr_apply {M : Type u_1} {N : Type u_2} {P : Type u_3} {Q : Type u_4} [inst : Mul P] [inst : Mul Q] (f : M N) (g : P ≃* Q) (h : MP) (n : N) :
↑() h n = g (h (↑() n))
@[simp]
theorem AddEquiv.arrowCongr_apply {M : Type u_1} {N : Type u_2} {P : Type u_3} {Q : Type u_4} [inst : Add P] [inst : Add Q] (f : M N) (g : P ≃+ Q) (h : MP) (n : N) :
↑() h n = g (h (↑() n))
def MulEquiv.arrowCongr {M : Type u_1} {N : Type u_2} {P : Type u_3} {Q : Type u_4} [inst : Mul P] [inst : Mul Q] (f : M N) (g : P ≃* Q) :
(MP) ≃* (NQ)

A multiplicative analogue of Equiv.arrowCongr, where the equivalence between the targets is multiplicative.

Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.addMonoidHomCongr.proof_3 {M : Type u_1} {N : Type u_3} {P : Type u_2} {Q : Type u_4} [inst : ] [inst : ] [inst : ] [inst : ] (f : M ≃+ N) (g : P ≃+ Q) (h : M →+ P) (k : M →+ P) :
Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.addMonoidHomCongr {M : Type u_1} {N : Type u_2} {P : Type u_3} {Q : Type u_4} [inst : ] [inst : ] [inst : ] [inst : ] (f : M ≃+ N) (g : P ≃+ Q) :
(M →+ P) ≃+ (N →+ Q)

An additive analogue of Equiv.arrowCongr, for additive maps from an additive monoid to a commutative additive monoid.

Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.addMonoidHomCongr.proof_2 {M : Type u_3} {N : Type u_1} {P : Type u_4} {Q : Type u_2} [inst : ] [inst : ] [inst : ] [inst : ] (f : M ≃+ N) (g : P ≃+ Q) (k : N →+ Q) :
(fun h => ) ((fun k => ) k) = k
Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.addMonoidHomCongr.proof_1 {M : Type u_1} {N : Type u_3} {P : Type u_2} {Q : Type u_4} [inst : ] [inst : ] [inst : ] [inst : ] (f : M ≃+ N) (g : P ≃+ Q) (h : M →+ P) :
(fun k => ) ((fun h => ) h) = h
Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem MulEquiv.monoidHomCongr_apply {M : Type u_1} {N : Type u_2} {P : Type u_3} {Q : Type u_4} [inst : ] [inst : ] [inst : ] [inst : ] (f : M ≃* N) (g : P ≃* Q) (h : M →* P) :
↑() h =
@[simp]
theorem AddEquiv.addMonoidHomCongr_apply {M : Type u_1} {N : Type u_2} {P : Type u_3} {Q : Type u_4} [inst : ] [inst : ] [inst : ] [inst : ] (f : M ≃+ N) (g : P ≃+ Q) (h : M →+ P) :
def MulEquiv.monoidHomCongr {M : Type u_1} {N : Type u_2} {P : Type u_3} {Q : Type u_4} [inst : ] [inst : ] [inst : ] [inst : ] (f : M ≃* N) (g : P ≃* Q) :
(M →* P) ≃* (N →* Q)

A multiplicative analogue of Equiv.arrowCongr, for multiplicative maps from a monoid to a commutative monoid.

Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.piCongrRight {η : Type u_1} {Ms : ηType u_2} {Ns : ηType u_3} [inst : (j : η) → Add (Ms j)] [inst : (j : η) → Add (Ns j)] (es : (j : η) → Ms j ≃+ Ns j) :
((j : η) → Ms j) ≃+ ((j : η) → Ns j)

A family of additive equivalences Π j, (Ms j ≃+ Ns j)≃+ Ns j) generates an additive equivalence between Π j, Ms j and Π j, Ns j.

This is the AddEquiv version of Equiv.piCongrRight, and the dependent version of AddEquiv.arrowCongr.

Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.piCongrRight.proof_3 {η : Type u_1} {Ms : ηType u_3} {Ns : ηType u_2} [inst : (j : η) → Add (Ms j)] [inst : (j : η) → Add (Ns j)] (es : (j : η) → Ms j ≃+ Ns j) (x : (j : η) → Ms j) (y : (j : η) → Ms j) :
Equiv.toFun { toFun := fun x j => ↑(es j) (x j), invFun := fun x j => ↑(AddEquiv.symm (es j)) (x j), left_inv := (_ : Function.LeftInverse (Equiv.piCongrRight fun j => (es j).toEquiv).invFun (Equiv.piCongrRight fun j => (es j).toEquiv).toFun), right_inv := (_ : Function.RightInverse (Equiv.piCongrRight fun j => (es j).toEquiv).invFun (Equiv.piCongrRight fun j => (es j).toEquiv).toFun) } (x + y) = Equiv.toFun { toFun := fun x j => ↑(es j) (x j), invFun := fun x j => ↑(AddEquiv.symm (es j)) (x j), left_inv := (_ : Function.LeftInverse (Equiv.piCongrRight fun j => (es j).toEquiv).invFun (Equiv.piCongrRight fun j => (es j).toEquiv).toFun), right_inv := (_ : Function.RightInverse (Equiv.piCongrRight fun j => (es j).toEquiv).invFun (Equiv.piCongrRight fun j => (es j).toEquiv).toFun) } x + Equiv.toFun { toFun := fun x j => ↑(es j) (x j), invFun := fun x j => ↑(AddEquiv.symm (es j)) (x j), left_inv := (_ : Function.LeftInverse (Equiv.piCongrRight fun j => (es j).toEquiv).invFun (Equiv.piCongrRight fun j => (es j).toEquiv).toFun), right_inv := (_ : Function.RightInverse (Equiv.piCongrRight fun j => (es j).toEquiv).invFun (Equiv.piCongrRight fun j => (es j).toEquiv).toFun) } y
Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.piCongrRight.proof_2 {η : Type u_1} {Ms : ηType u_2} {Ns : ηType u_3} [inst : (j : η) → Add (Ms j)] [inst : (j : η) → Add (Ns j)] (es : (j : η) → Ms j ≃+ Ns j) :
Function.RightInverse (Equiv.piCongrRight fun j => (es j).toEquiv).invFun (Equiv.piCongrRight fun j => (es j).toEquiv).toFun
Equations
• One or more equations did not get rendered due to their size.
def AddEquiv.piCongrRight.proof_1 {η : Type u_1} {Ms : ηType u_2} {Ns : ηType u_3} [inst : (j : η) → Add (Ms j)] [inst : (j : η) → Add (Ns j)] (es : (j : η) → Ms j ≃+ Ns j) :
Function.LeftInverse (Equiv.piCongrRight fun j => (es j).toEquiv).invFun (Equiv.piCongrRight fun j => (es j).toEquiv).toFun
Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem AddEquiv.piCongrRight_apply {η : Type u_1} {Ms : ηType u_2} {Ns : ηType u_3} [inst : (j : η) → Add (Ms j)] [inst : (j : η) → Add (Ns j)] (es : (j : η) → Ms j ≃+ Ns j) (x : (j : η) → Ms j) (j : η) :
↑() x j = ↑(es j) (x j)
@[simp]
theorem MulEquiv.piCongrRight_apply {η : Type u_1} {Ms : ηType u_2} {Ns : ηType u_3} [inst : (j : η) → Mul (Ms j)] [inst : (j : η) → Mul (Ns j)] (es : (j : η) → Ms j ≃* Ns j) (x : (j : η) → Ms j) (j : η) :
↑() x j = ↑(es j) (x j)
def MulEquiv.piCongrRight {η : Type u_1} {Ms : ηType u_2} {Ns : ηType u_3} [inst : (j : η) → Mul (Ms j)] [inst : (j : η) → Mul (Ns j)] (es : (j : η) → Ms j ≃* Ns j) :
((j : η) → Ms j) ≃* ((j : η) → Ns j)

A family of multiplicative equivalences Π j, (Ms j ≃* Ns j)≃* Ns j) generates a multiplicative equivalence between Π j, Ms j and Π j, Ns j.

This is the MulEquiv version of Equiv.piCongrRight, and the dependent version of MulEquiv.arrowCongr.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem AddEquiv.piCongrRight_refl {η : Type u_1} {Ms : ηType u_2} [inst : (j : η) → Add (Ms j)] :
@[simp]
theorem MulEquiv.piCongrRight_refl {η : Type u_1} {Ms : ηType u_2} [inst : (j : η) → Mul (Ms j)] :
(MulEquiv.piCongrRight fun j => MulEquiv.refl (Ms j)) = MulEquiv.refl ((j : η) → Ms j)
@[simp]
theorem AddEquiv.piCongrRight_symm {η : Type u_1} {Ms : ηType u_2} {Ns : ηType u_3} [inst : (j : η) → Add (Ms j)] [inst : (j : η) → Add (Ns j)] (es : (j : η) → Ms j ≃+ Ns j) :
@[simp]
theorem MulEquiv.piCongrRight_symm {η : Type u_1} {Ms : ηType u_2} {Ns : ηType u_3} [inst : (j : η) → Mul (Ms j)] [inst : (j : η) → Mul (Ns j)] (es : (j : η) → Ms j ≃* Ns j) :
@[simp]
theorem AddEquiv.piCongrRight_trans {η : Type u_1} {Ms : ηType u_2} {Ns : ηType u_3} {Ps : ηType u_4} [inst : (j : η) → Add (Ms j)] [inst : (j : η) → Add (Ns j)] [inst : (j : η) → Add (Ps j)] (es : (j : η) → Ms j ≃+ Ns j) (fs : (j : η) → Ns j ≃+ Ps j) :
@[simp]
theorem MulEquiv.piCongrRight_trans {η : Type u_1} {Ms : ηType u_2} {Ns : ηType u_3} {Ps : ηType u_4} [inst : (j : η) → Mul (Ms j)] [inst : (j : η) → Mul (Ns j)] [inst : (j : η) → Mul (Ps j)] (es : (j : η) → Ms j ≃* Ns j) (fs : (j : η) → Ns j ≃* Ps j) :
= MulEquiv.piCongrRight fun i => MulEquiv.trans (es i) (fs i)
def AddEquiv.piSubsingleton.proof_1 {ι : Type u_1} (M : ιType u_2) [inst : ] (i : ι) :
Function.LeftInverse ().invFun ().toFun
Equations
def AddEquiv.piSubsingleton.proof_3 {ι : Type u_2} (M : ιType u_1) [inst : (j : ι) → Add (M j)] (i : ι) :
∀ (x x_1 : (j : ι) → M j), (((i : ι) → M i) + ((i : ι) → M i)) ((i : ι) → M i) instHAdd x x_1 i = x i + x_1 i
Equations
def AddEquiv.piSubsingleton.proof_2 {ι : Type u_1} (M : ιType u_2) [inst : ] (i : ι) :
Function.RightInverse ().invFun ().toFun
Equations
def AddEquiv.piSubsingleton {ι : Type u_1} (M : ιType u_2) [inst : (j : ι) → Add (M j)] [inst : ] (i : ι) :
((j : ι) → M j) ≃+ M i

A family indexed by a nonempty subsingleton type is equivalent to the element at the single index.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem MulEquiv.piSubsingleton_apply {ι : Type u_1} (M : ιType u_2) [inst : (j : ι) → Mul (M j)] [inst : ] (i : ι) (f : (x : ι) → M x) :
↑() f = f i
@[simp]
theorem AddEquiv.piSubsingleton_symm_apply {ι : Type u_1} (M : ιType u_2) [inst : (j : ι) → Add (M j)] [inst : ] (i : ι) (x : M i) (b : ι) :
↑() x b = cast (_ : M i = M b) x
@[simp]
theorem MulEquiv.piSubsingleton_symm_apply {ι : Type u_1} (M : ιType u_2) [inst : (j : ι) → Mul (M j)] [inst : ] (i : ι) (x : M i) (b : ι) :
↑() x b = cast (_ : M i = M b) x
@[simp]
theorem AddEquiv.piSubsingleton_apply {ι : Type u_1} (M : ιType u_2) [inst : (j : ι) → Add (M j)] [inst : ] (i : ι) (f : (x : ι) → M x) :
↑() f = f i
def MulEquiv.piSubsingleton {ι : Type u_1} (M : ιType u_2) [inst : (j : ι) → Mul (M j)] [inst : ] (i : ι) :
((j : ι) → M j) ≃* M i

A family indexed by a nonempty subsingleton type is equivalent to the element at the single index.

Equations
• One or more equations did not get rendered due to their size.

# Groups #

theorem AddEquiv.map_neg {G : Type u_1} {H : Type u_2} [inst : ] [inst : ] (h : G ≃+ H) (x : G) :
h (-x) = -h x

theorem MulEquiv.map_inv {G : Type u_1} {H : Type u_2} [inst : ] [inst : ] (h : G ≃* H) (x : G) :
h x⁻¹ = (h x)⁻¹

A multiplicative equivalence of groups preserves inversion.

theorem AddEquiv.map_sub {G : Type u_1} {H : Type u_2} [inst : ] [inst : ] (h : G ≃+ H) (x : G) (y : G) :
h (x - y) = h x - h y

theorem MulEquiv.map_div {G : Type u_1} {H : Type u_2} [inst : ] [inst : ] (h : G ≃* H) (x : G) (y : G) :
h (x / y) = h x / h y

A multiplicative equivalence of groups preserves division.

def AddHom.toAddEquiv {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : AddHom M N) (g : AddHom N M) (h₁ : = ) (h₂ : = ) :
M ≃+ N

Given a pair of additive homomorphisms f, g such that g.comp f = id and f.comp g = id, returns an additive equivalence with toFun = f and invFun = g. This constructor is useful if the underlying type(s) have specialized ext lemmas for additive homomorphisms.

Equations
• One or more equations did not get rendered due to their size.
def AddHom.toAddEquiv.proof_2 {M : Type u_2} {N : Type u_1} [inst : Add M] [inst : Add N] (f : AddHom M N) (g : AddHom N M) (h₂ : = ) (x : N) :
↑() x = ↑() x
Equations
• (_ : ∀ (x : N), ↑() x = ↑() x) = (_ : ∀ (x : N), ↑() x = ↑() x)
def AddHom.toAddEquiv.proof_1 {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : AddHom M N) (g : AddHom N M) (h₁ : = ) (x : M) :
↑() x = ↑() x
Equations
• (_ : ∀ (x : M), ↑() x = ↑() x) = (_ : ∀ (x : M), ↑() x = ↑() x)
def MulHom.toMulEquiv {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M →ₙ* N) (g : N →ₙ* M) (h₁ : = ) (h₂ : = ) :
M ≃* N

Given a pair of multiplicative homomorphisms f, g such that g.comp f = id and f.comp g = id, returns an multiplicative equivalence with toFun = f and invFun = g. This constructor is useful if the underlying type(s) have specialized ext lemmas for multiplicative homomorphisms.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem AddHom.toAddEquiv_apply {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : AddHom M N) (g : AddHom N M) (h₁ : = ) (h₂ : = ) :
@[simp]
theorem MulHom.toMulEquiv_apply {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M →ₙ* N) (g : N →ₙ* M) (h₁ : = ) (h₂ : = ) :
↑(MulHom.toMulEquiv f g h₁ h₂) = f
@[simp]
theorem AddHom.toAddEquiv_symm_apply {M : Type u_1} {N : Type u_2} [inst : Add M] [inst : Add N] (f : AddHom M N) (g : AddHom N M) (h₁ : = ) (h₂ : = ) :
@[simp]
theorem MulHom.toMulEquiv_symm_apply {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst : Mul N] (f : M →ₙ* N) (g : N →ₙ* M) (h₁ : = ) (h₂ : = ) :
↑(MulEquiv.symm (MulHom.toMulEquiv f g h₁ h₂)) = g
def AddMonoidHom.toAddEquiv.proof_1 {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (f : M →+ N) (g : N →+ M) (h₁ : ) (x : M) :
↑() x = ↑() x
Equations
• (_ : ∀ (x : M), ↑() x = ↑() x) = (_ : ∀ (x : M), ↑() x = ↑() x)
def AddMonoidHom.toAddEquiv.proof_2 {M : Type u_2} {N : Type u_1} [inst : ] [inst : ] (f : M →+ N) (g : N →+ M) (h₂ : ) (x : N) :
↑() x = ↑() x
Equations
• (_ : ∀ (x : N), ↑() x = ↑() x) = (_ : ∀ (x : N), ↑() x = ↑() x)
def AddMonoidHom.toAddEquiv {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (f : M →+ N) (g : N →+ M) (h₁ : ) (h₂ : ) :
M ≃+ N

Given a pair of additive monoid homomorphisms f, g such that g.comp f = id and f.comp g = id, returns an additive equivalence with toFun = f and invFun = g. This constructor is useful if the underlying type(s) have specialized ext lemmas for additive monoid homomorphisms.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem MonoidHom.toMulEquiv_symm_apply {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (f : M →* N) (g : N →* M) (h₁ : ) (h₂ : ) :
↑(MulEquiv.symm (MonoidHom.toMulEquiv f g h₁ h₂)) = g
@[simp]
theorem AddMonoidHom.toAddEquiv_symm_apply {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (f : M →+ N) (g : N →+ M) (h₁ : ) (h₂ : ) :
@[simp]
theorem AddMonoidHom.toAddEquiv_apply {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (f : M →+ N) (g : N →+ M) (h₁ : ) (h₂ : ) :
@[simp]
theorem MonoidHom.toMulEquiv_apply {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (f : M →* N) (g : N →* M) (h₁ : ) (h₂ : ) :
↑(MonoidHom.toMulEquiv f g h₁ h₂) = f
def MonoidHom.toMulEquiv {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (f : M →* N) (g : N →* M) (h₁ : ) (h₂ : ) :
M ≃* N

Given a pair of monoid homomorphisms f, g such that g.comp f = id and f.comp g = id, returns an multiplicative equivalence with toFun = f and invFun = g. This constructor is useful if the underlying type(s) have specialized ext lemmas for monoid homomorphisms.

Equations
• One or more equations did not get rendered due to their size.
def Equiv.neg (G : Type u_1) [inst : ] :

Negation on an AddGroup is a permutation of the underlying type.

Equations
@[simp]
theorem Equiv.neg_apply (G : Type u_1) [inst : ] :
↑() = Neg.neg
@[simp]
theorem Equiv.inv_apply (G : Type u_1) [inst : ] :
↑() = Inv.inv
def Equiv.inv (G : Type u_1) [inst : ] :

Inversion on a Group or GroupWithZero is a permutation of the underlying type.

Equations
@[simp]
theorem Equiv.neg_symm {G : Type u_1} [inst : ] :
@[simp]
theorem Equiv.inv_symm {G : Type u_1} [inst : ] :