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.

Main definitions

• ≃* (MulEquiv), ≃+ (AddEquiv): bundled equivalences that preserve multiplication/addition (and are therefore monoid and group isomorphisms).
• MulEquivClass, AddEquivClass: classes for types containing bundled equivalences that preserve multiplication/addition.

Notations

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

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

Tags

Equiv, MulEquiv, AddEquiv

@[simp]

theorem EmbeddingLike.map_eq_one_iff {F : Type u_1} {M : Type u_4} {N : Type u_5} [One M] [One N] [FunLike F M N] [EmbeddingLike F M N] [OneHomClass F M N] {f : F} {x : M} : f x = 1 ↔ x = 1

@[simp]

theorem EmbeddingLike.map_eq_zero_iff {F : Type u_1} {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] [FunLike F M N] [EmbeddingLike F M N] [ZeroHomClass F M N] {f : F} {x : M} : f x = 0 ↔ x = 0

theorem EmbeddingLike.map_ne_one_iff {F : Type u_1} {M : Type u_4} {N : Type u_5} [One M] [One N] [FunLike F M N] [EmbeddingLike F M N] [OneHomClass F M N] {f : F} {x : M} : f x ≠ 1 ↔ x ≠ 1

theorem EmbeddingLike.map_ne_zero_iff {F : Type u_1} {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] [FunLike F M N] [EmbeddingLike F M N] [ZeroHomClass F M N] {f : F} {x : M} : f x ≠ 0 ↔ x ≠ 0

structure AddEquiv (A : Type u_9) (B : Type u_10) [Add A] [Add B] extends A ≃ B, A →ₙ+ B : Type (max u_10 u_9)

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

• toFun : A → B
• invFun : B → A
• left_inv : Function.LeftInverse self.invFun self.toFun
• right_inv : Function.RightInverse self.invFun self.toFun
• map_add' (x y : A) : self.toFun (x + y) = self.toFun x + self.toFun y

class AddEquivClass (F : Type u_9) (A : outParam (Type u_10)) (B : outParam (Type u_11)) [Add A] [Add B] [EquivLike F A B] : Prop

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

• map_add (f : F) (a b : A) : f (a + b) = f a + f b

  Preserves addition.

structure MulEquiv (M : Type u_9) (N : Type u_10) [Mul M] [Mul N] extends M ≃ N, M →ₙ* N : Type (max u_10 u_9)

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

• toFun : M → N
• invFun : N → M
• left_inv : Function.LeftInverse self.invFun self.toFun
• right_inv : Function.RightInverse self.invFun self.toFun
• map_mul' (x y : M) : self.toFun (x * y) = self.toFun x * self.toFun y

def «term_≃*_» : Lean.TrailingParserDescr

Notation for a MulEquiv.

def «term_≃+_» : Lean.TrailingParserDescr

Notation for an AddEquiv.

theorem MulEquiv.toEquiv_injective {α : Type u_9} {β : Type u_10} [Mul α] [Mul β] : Function.Injective toEquiv

theorem AddEquiv.toEquiv_injective {α : Type u_9} {β : Type u_10} [Add α] [Add β] : Function.Injective toEquiv

class MulEquivClass (F : Type u_9) (A : outParam (Type u_10)) (B : outParam (Type u_11)) [Mul A] [Mul B] [EquivLike F A B] : Prop

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

• map_mul (f : F) (a b : A) : f (a * b) = f a * f b

  Preserves multiplication.

@[deprecated EmbeddingLike.map_eq_one_iff (since := "2024-11-10")]

theorem MulEquivClass.map_eq_one_iff {F : Type u_1} {M : Type u_4} {N : Type u_5} [One M] [One N] [FunLike F M N] [EmbeddingLike F M N] [OneHomClass F M N] {f : F} {x : M} : f x = 1 ↔ x = 1

Alias of EmbeddingLike.map_eq_one_iff.

@[deprecated EmbeddingLike.map_eq_zero_iff (since := "2024-11-10")]

theorem AddEquivClass.map_eq_zero_iff {F : Type u_1} {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] [FunLike F M N] [EmbeddingLike F M N] [ZeroHomClass F M N] {f : F} {x : M} : f x = 0 ↔ x = 0

@[deprecated EmbeddingLike.map_ne_one_iff (since := "2024-11-10")]

theorem MulEquivClass.map_ne_one_iff {F : Type u_1} {M : Type u_4} {N : Type u_5} [One M] [One N] [FunLike F M N] [EmbeddingLike F M N] [OneHomClass F M N] {f : F} {x : M} : f x ≠ 1 ↔ x ≠ 1

Alias of EmbeddingLike.map_ne_one_iff.

@[deprecated EmbeddingLike.map_ne_zero_iff (since := "2024-11-10")]

theorem AddEquivClass.map_ne_zero_iff {F : Type u_1} {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] [FunLike F M N] [EmbeddingLike F M N] [ZeroHomClass F M N] {f : F} {x : M} : f x ≠ 0 ↔ x ≠ 0

@[instance 100]

instance MulEquivClass.instMulHomClass {M : Type u_4} {N : Type u_5} (F : Type u_9) [Mul M] [Mul N] [EquivLike F M N] [h : MulEquivClass F M N] : MulHomClass F M N

@[instance 100]

instance AddEquivClass.instAddHomClass {M : Type u_4} {N : Type u_5} (F : Type u_9) [Add M] [Add N] [EquivLike F M N] [h : AddEquivClass F M N] : AddHomClass F M N

@[instance 100]

instance MulEquivClass.instMonoidHomClass (F : Type u_1) {M : Type u_4} {N : Type u_5} [EquivLike F M N] [MulOneClass M] [MulOneClass N] [MulEquivClass F M N] : MonoidHomClass F M N

@[instance 100]

instance AddEquivClass.instAddMonoidHomClass (F : Type u_1) {M : Type u_4} {N : Type u_5} [EquivLike F M N] [AddZeroClass M] [AddZeroClass N] [AddEquivClass F M N] : AddMonoidHomClass F M N

def MulEquivClass.toMulEquiv {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [Mul α] [Mul β] [MulEquivClass F α β] (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.

def AddEquivClass.toAddEquiv {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [Add α] [Add β] [AddEquivClass F α β] (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.

instance instCoeTCMulEquivOfMulEquivClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [Mul α] [Mul β] [MulEquivClass F α β] : CoeTC F (α ≃* β)

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

instance instCoeTCAddEquivOfAddEquivClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [Add α] [Add β] [AddEquivClass F α β] : CoeTC F (α ≃+ β)

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

instance MulEquiv.instEquivLike {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] : EquivLike (M ≃* N) M N

instance AddEquiv.instEquivLike {M : Type u_4} {N : Type u_5} [Add M] [Add N] : EquivLike (M ≃+ N) M N

instance MulEquiv.instCoeFunForall {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] : CoeFun (M ≃* N) fun (x : M ≃* N) => M → N

instance AddEquiv.instCoeFunForall {M : Type u_4} {N : Type u_5} [Add M] [Add N] : CoeFun (M ≃+ N) fun (x : M ≃+ N) => M → N

instance MulEquiv.instMulEquivClass {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] : MulEquivClass (M ≃* N) M N

instance AddEquiv.instAddEquivClass {M : Type u_4} {N : Type u_5} [Add M] [Add N] : AddEquivClass (M ≃+ N) M N

theorem MulEquiv.ext {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {f 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 {M : Type u_4} {N : Type u_5} [Add M] [Add N] {f 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 AddEquiv.ext_iff {M : Type u_4} {N : Type u_5} [Add M] [Add N] {f g : M ≃+ N} : f = g ↔ ∀ (x : M), f x = g x

theorem MulEquiv.ext_iff {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {f g : M ≃* N} : f = g ↔ ∀ (x : M), f x = g x

theorem MulEquiv.congr_arg {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {f : M ≃* N} {x x' : M} : x = x' → f x = f x'

theorem AddEquiv.congr_arg {M : Type u_4} {N : Type u_5} [Add M] [Add N] {f : M ≃+ N} {x x' : M} : x = x' → f x = f x'

theorem MulEquiv.congr_fun {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {f g : M ≃* N} (h : f = g) (x : M) : f x = g x

theorem AddEquiv.congr_fun {M : Type u_4} {N : Type u_5} [Add M] [Add N] {f g : M ≃+ N} (h : f = g) (x : M) : f x = g x

@[simp]

theorem MulEquiv.coe_mk {M : Type u_4} {N : Type u_5} [Mul M] [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.coe_mk {M : Type u_4} {N : Type u_5} [Add M] [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.mk_coe {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) (e' : N → M) (h₁ : Function.LeftInverse e' ⇑e) (h₂ : Function.RightInverse e' ⇑e) (h₃ : ∀ (x y : M), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) : { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃ } = e

@[simp]

theorem AddEquiv.mk_coe {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) (e' : N → M) (h₁ : Function.LeftInverse e' ⇑e) (h₂ : Function.RightInverse e' ⇑e) (h₃ : ∀ (x y : M), { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂ }.toFun y) : { toFun := ⇑e, invFun := e', left_inv := h₁, right_inv := h₂, map_add' := h₃ } = e

@[simp]

theorem MulEquiv.toEquiv_eq_coe {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) : f.toEquiv = ↑f

@[simp]

theorem AddEquiv.toEquiv_eq_coe {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) : f.toEquiv = ↑f

@[simp]

theorem MulEquiv.toMulHom_eq_coe {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) : f.toMulHom = ↑f

The simp-normal form to turn something into a MulHom is via MulHomClass.toMulHom.

@[simp]

theorem AddEquiv.toAddHom_eq_coe {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) : f.toAddHom = ↑f

theorem MulEquiv.toFun_eq_coe {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) : f.toFun = ⇑f

theorem AddEquiv.toFun_eq_coe {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) : f.toFun = ⇑f

@[simp]

theorem MulEquiv.coe_toEquiv {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) : ⇑↑f = ⇑f

simp-normal form of toFun_eq_coe.

@[simp]

theorem AddEquiv.coe_toEquiv {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) : ⇑↑f = ⇑f

@[simp]

theorem MulEquiv.coe_toMulHom {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {f : M ≃* N} : ⇑f.toMulHom = ⇑f

@[simp]

theorem AddEquiv.coe_toAddHom {M : Type u_4} {N : Type u_5} [Add M] [Add N] {f : M ≃+ N} : ⇑f.toAddHom = ⇑f

def MulEquiv.mk' {M : Type u_4} {N : Type u_5} [Mul M] [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.

def AddEquiv.mk' {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃ N) (h : ∀ (x y : M), f (x + y) = f x + f y) : M ≃+ N

Makes an additive isomorphism from a bijection which preserves addition.

theorem MulEquiv.map_mul {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) (x y : M) : f (x * y) = f x * f y

A multiplicative isomorphism preserves multiplication.

theorem AddEquiv.map_add {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) (x y : M) : f (x + y) = f x + f y

An additive isomorphism preserves addition.

theorem MulEquiv.bijective {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) : Function.Bijective ⇑e

theorem AddEquiv.bijective {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) : Function.Bijective ⇑e

theorem MulEquiv.injective {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) : Function.Injective ⇑e

theorem AddEquiv.injective {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) : Function.Injective ⇑e

theorem MulEquiv.surjective {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) : Function.Surjective ⇑e

theorem AddEquiv.surjective {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) : Function.Surjective ⇑e

theorem MulEquiv.apply_eq_iff_eq {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) {x y : M} : e x = e y ↔ x = y

theorem AddEquiv.apply_eq_iff_eq {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) {x y : M} : e x = e y ↔ x = y

def MulEquiv.refl (M : Type u_9) [Mul M] : M ≃* M

The identity map is a multiplicative isomorphism.

def AddEquiv.refl (M : Type u_9) [Add M] : M ≃+ M

The identity map is an additive isomorphism.

instance MulEquiv.instInhabited {M : Type u_4} [Mul M] : Inhabited (M ≃* M)

instance AddEquiv.instInhabited {M : Type u_4} [Add M] : Inhabited (M ≃+ M)

@[simp]

theorem MulEquiv.coe_refl {M : Type u_4} [Mul M] : ⇑(refl M) = id

@[simp]

theorem AddEquiv.coe_refl {M : Type u_4} [Add M] : ⇑(refl M) = id

@[simp]

theorem MulEquiv.refl_apply {M : Type u_4} [Mul M] (m : M) : (refl M) m = m

@[simp]

theorem AddEquiv.refl_apply {M : Type u_4} [Add M] (m : M) : (refl M) m = m

theorem MulEquiv.symm_map_mul {M : Type u_9} {N : Type u_10} [Mul M] [Mul N] (h : M ≃* N) (x y : N) : h.symm (x * y) = h.symm x * h.symm y

An alias for h.symm.map_mul. Introduced to fix the issue in https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/!4.234183.20.60simps.60.20maximum.20recursion.20depth

theorem AddEquiv.symm_map_add {M : Type u_9} {N : Type u_10} [Add M] [Add N] (h : M ≃+ N) (x y : N) : h.symm (x + y) = h.symm x + h.symm y

def MulEquiv.symm {M : Type u_9} {N : Type u_10} [Mul M] [Mul N] (h : M ≃* N) : N ≃* M

The inverse of an isomorphism is an isomorphism.

def AddEquiv.symm {M : Type u_9} {N : Type u_10} [Add M] [Add N] (h : M ≃+ N) : N ≃+ M

The inverse of an isomorphism is an isomorphism.

theorem MulEquiv.invFun_eq_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {f : M ≃* N} : f.invFun = ⇑f.symm

theorem AddEquiv.invFun_eq_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] {f : M ≃+ N} : f.invFun = ⇑f.symm

@[simp]

theorem MulEquiv.coe_toEquiv_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) : ⇑(↑f).symm = ⇑f.symm

simp-normal form of invFun_eq_symm.

@[simp]

theorem AddEquiv.coe_toEquiv_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) : ⇑(↑f).symm = ⇑f.symm

@[simp]

theorem MulEquiv.equivLike_inv_eq_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) : EquivLike.inv f = ⇑f.symm

@[simp]

theorem AddEquiv.equivLike_neg_eq_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) : EquivLike.inv f = ⇑f.symm

@[simp]

theorem MulEquiv.toEquiv_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) : ↑f.symm = (↑f).symm

@[simp]

theorem AddEquiv.toEquiv_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) : ↑f.symm = (↑f).symm

@[simp]

theorem MulEquiv.symm_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) : f.symm.symm = f

@[simp]

theorem AddEquiv.symm_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) : f.symm.symm = f

theorem MulEquiv.symm_bijective {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] : Function.Bijective symm

theorem AddEquiv.symm_bijective {M : Type u_4} {N : Type u_5} [Add M] [Add N] : Function.Bijective symm

@[simp]

theorem MulEquiv.mk_coe' {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) (f : N → M) (h₁ : Function.LeftInverse (⇑e) f) (h₂ : Function.RightInverse (⇑e) f) (h₃ : ∀ (x y : N), { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun (x * y) = { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun x * { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun y) : { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂, map_mul' := h₃ } = e.symm

@[simp]

theorem AddEquiv.mk_coe' {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) (f : N → M) (h₁ : Function.LeftInverse (⇑e) f) (h₂ : Function.RightInverse (⇑e) f) (h₃ : ∀ (x y : N), { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun (x + y) = { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun x + { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂ }.toFun y) : { toFun := f, invFun := ⇑e, left_inv := h₁, right_inv := h₂, map_add' := h₃ } = e.symm

@[simp]

theorem MulEquiv.symm_mk {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃ N) (h : ∀ (x y : M), f.toFun (x * y) = f.toFun x * f.toFun y) : { toEquiv := f, map_mul' := h }.symm = { toEquiv := f.symm, map_mul' := ⋯ }

@[simp]

theorem AddEquiv.symm_mk {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃ N) (h : ∀ (x y : M), f.toFun (x + y) = f.toFun x + f.toFun y) : { toEquiv := f, map_add' := h }.symm = { toEquiv := f.symm, map_add' := ⋯ }

@[simp]

theorem MulEquiv.refl_symm {M : Type u_4} [Mul M] : (refl M).symm = refl M

@[simp]

theorem AddEquiv.refl_symm {M : Type u_4} [Add M] : (refl M).symm = refl M

@[simp]

theorem MulEquiv.apply_symm_apply {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) (y : N) : e (e.symm y) = y

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

@[simp]

theorem AddEquiv.apply_symm_apply {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) (y : N) : e (e.symm y) = y

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

@[simp]

theorem MulEquiv.symm_apply_apply {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) (x : M) : e.symm (e x) = x

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

@[simp]

theorem AddEquiv.symm_apply_apply {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) (x : M) : e.symm (e x) = x

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

@[simp]

theorem MulEquiv.symm_comp_self {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) : ⇑e.symm ∘ ⇑e = id

@[simp]

theorem AddEquiv.symm_comp_self {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) : ⇑e.symm ∘ ⇑e = id

@[simp]

theorem MulEquiv.self_comp_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) : ⇑e ∘ ⇑e.symm = id

@[simp]

theorem AddEquiv.self_comp_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) : ⇑e ∘ ⇑e.symm = id

theorem MulEquiv.apply_eq_iff_symm_apply {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) {x : M} {y : N} : e x = y ↔ x = e.symm y

theorem AddEquiv.apply_eq_iff_symm_apply {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) {x : M} {y : N} : e x = y ↔ x = e.symm y

theorem MulEquiv.symm_apply_eq {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) {x : N} {y : M} : e.symm x = y ↔ x = e y

theorem AddEquiv.symm_apply_eq {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) {x : N} {y : M} : e.symm x = y ↔ x = e y

theorem MulEquiv.eq_symm_apply {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) {x : N} {y : M} : y = e.symm x ↔ e y = x

theorem AddEquiv.eq_symm_apply {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) {x : N} {y : M} : y = e.symm x ↔ e y = x

theorem MulEquiv.eq_comp_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {α : Type u_9} (e : M ≃* N) (f : N → α) (g : M → α) : f = g ∘ ⇑e.symm ↔ f ∘ ⇑e = g

theorem AddEquiv.eq_comp_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] {α : Type u_9} (e : M ≃+ N) (f : N → α) (g : M → α) : f = g ∘ ⇑e.symm ↔ f ∘ ⇑e = g

theorem MulEquiv.comp_symm_eq {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {α : Type u_9} (e : M ≃* N) (f : N → α) (g : M → α) : g ∘ ⇑e.symm = f ↔ g = f ∘ ⇑e

theorem AddEquiv.comp_symm_eq {M : Type u_4} {N : Type u_5} [Add M] [Add N] {α : Type u_9} (e : M ≃+ N) (f : N → α) (g : M → α) : g ∘ ⇑e.symm = f ↔ g = f ∘ ⇑e

theorem MulEquiv.eq_symm_comp {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {α : Type u_9} (e : M ≃* N) (f : α → M) (g : α → N) : f = ⇑e.symm ∘ g ↔ ⇑e ∘ f = g

theorem AddEquiv.eq_symm_comp {M : Type u_4} {N : Type u_5} [Add M] [Add N] {α : Type u_9} (e : M ≃+ N) (f : α → M) (g : α → N) : f = ⇑e.symm ∘ g ↔ ⇑e ∘ f = g

theorem MulEquiv.symm_comp_eq {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {α : Type u_9} (e : M ≃* N) (f : α → M) (g : α → N) : ⇑e.symm ∘ g = f ↔ g = ⇑e ∘ f

theorem AddEquiv.symm_comp_eq {M : Type u_4} {N : Type u_5} [Add M] [Add N] {α : Type u_9} (e : M ≃+ N) (f : α → M) (g : α → N) : ⇑e.symm ∘ g = f ↔ g = ⇑e ∘ f

@[simp]

theorem MulEquivClass.apply_coe_symm_apply {α : Type u_9} {β : Type u_10} [Mul α] [Mul β] {F : Type u_11} [EquivLike F α β] [MulEquivClass F α β] (e : F) (x : β) : e ((↑e).symm x) = x

@[simp]

theorem AddEquivClass.apply_coe_symm_apply {α : Type u_9} {β : Type u_10} [Add α] [Add β] {F : Type u_11} [EquivLike F α β] [AddEquivClass F α β] (e : F) (x : β) : e ((↑e).symm x) = x

@[simp]

theorem MulEquivClass.coe_symm_apply_apply {α : Type u_9} {β : Type u_10} [Mul α] [Mul β] {F : Type u_11} [EquivLike F α β] [MulEquivClass F α β] (e : F) (x : α) : (↑e).symm (e x) = x

@[simp]

theorem AddEquivClass.coe_symm_apply_apply {α : Type u_9} {β : Type u_10} [Add α] [Add β] {F : Type u_11} [EquivLike F α β] [AddEquivClass F α β] (e : F) (x : α) : (↑e).symm (e x) = x

def MulEquiv.Simps.symm_apply {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) : N → M

See Note [custom simps projection]

def AddEquiv.Simps.symm_apply {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) : N → M

See Note [custom simps projection]

def MulEquiv.trans {M : Type u_4} {N : Type u_5} {P : Type u_6} [Mul M] [Mul N] [Mul P] (h1 : M ≃* N) (h2 : N ≃* P) : M ≃* P

Transitivity of multiplication-preserving isomorphisms

def AddEquiv.trans {M : Type u_4} {N : Type u_5} {P : Type u_6} [Add M] [Add N] [Add P] (h1 : M ≃+ N) (h2 : N ≃+ P) : M ≃+ P

Transitivity of addition-preserving isomorphisms

@[simp]

theorem MulEquiv.coe_trans {M : Type u_4} {N : Type u_5} {P : Type u_6} [Mul M] [Mul N] [Mul P] (e₁ : M ≃* N) (e₂ : N ≃* P) : ⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁

@[simp]

theorem AddEquiv.coe_trans {M : Type u_4} {N : Type u_5} {P : Type u_6} [Add M] [Add N] [Add P] (e₁ : M ≃+ N) (e₂ : N ≃+ P) : ⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁

@[simp]

theorem MulEquiv.trans_apply {M : Type u_4} {N : Type u_5} {P : Type u_6} [Mul M] [Mul N] [Mul P] (e₁ : M ≃* N) (e₂ : N ≃* P) (m : M) : (e₁.trans e₂) m = e₂ (e₁ m)

@[simp]

theorem AddEquiv.trans_apply {M : Type u_4} {N : Type u_5} {P : Type u_6} [Add M] [Add N] [Add P] (e₁ : M ≃+ N) (e₂ : N ≃+ P) (m : M) : (e₁.trans e₂) m = e₂ (e₁ m)

@[simp]

theorem MulEquiv.symm_trans_apply {M : Type u_4} {N : Type u_5} {P : Type u_6} [Mul M] [Mul N] [Mul P] (e₁ : M ≃* N) (e₂ : N ≃* P) (p : P) : (e₁.trans e₂).symm p = e₁.symm (e₂.symm p)

@[simp]

theorem AddEquiv.symm_trans_apply {M : Type u_4} {N : Type u_5} {P : Type u_6} [Add M] [Add N] [Add P] (e₁ : M ≃+ N) (e₂ : N ≃+ P) (p : P) : (e₁.trans e₂).symm p = e₁.symm (e₂.symm p)

@[simp]

theorem MulEquiv.symm_trans_self {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) : e.symm.trans e = refl N

@[simp]

theorem AddEquiv.symm_trans_self {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) : e.symm.trans e = refl N

@[simp]

theorem MulEquiv.self_trans_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) : e.trans e.symm = refl M

@[simp]

theorem AddEquiv.self_trans_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) : e.trans e.symm = refl M

Monoids

@[simp]

theorem MulEquiv.coe_monoidHom_refl {M : Type u_4} [MulOneClass M] : ↑(refl M) = MonoidHom.id M

@[simp]

theorem AddEquiv.coe_addMonoidHom_refl {M : Type u_4} [AddZeroClass M] : ↑(refl M) = AddMonoidHom.id M

@[simp]

theorem MulEquiv.coe_monoidHom_trans {M : Type u_4} {N : Type u_5} {P : Type u_6} [MulOneClass M] [MulOneClass N] [MulOneClass P] (e₁ : M ≃* N) (e₂ : N ≃* P) : ↑(e₁.trans e₂) = (↑e₂).comp ↑e₁

@[simp]

theorem AddEquiv.coe_addMonoidHom_trans {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] (e₁ : M ≃+ N) (e₂ : N ≃+ P) : ↑(e₁.trans e₂) = (↑e₂).comp ↑e₁

@[simp]

theorem MulEquiv.coe_monoidHom_comp_coe_monoidHom_symm {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (e : M ≃* N) : (↑e).comp ↑e.symm = MonoidHom.id N

@[simp]

theorem AddEquiv.coe_addMonoidHom_comp_coe_addMonoidHom_symm {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (e : M ≃+ N) : (↑e).comp ↑e.symm = AddMonoidHom.id N

@[simp]

theorem MulEquiv.coe_monoidHom_symm_comp_coe_monoidHom {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (e : M ≃* N) : (↑e.symm).comp ↑e = MonoidHom.id M

@[simp]

theorem AddEquiv.coe_addMonoidHom_symm_comp_coe_addMonoidHom {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (e : M ≃+ N) : (↑e.symm).comp ↑e = AddMonoidHom.id M

theorem MulEquiv.comp_left_injective {M : Type u_4} {N : Type u_5} {P : Type u_6} [MulOneClass M] [MulOneClass N] [MulOneClass P] (e : M ≃* N) : Function.Injective fun (f : N →* P) => f.comp ↑e

theorem AddEquiv.comp_left_injective {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] (e : M ≃+ N) : Function.Injective fun (f : N →+ P) => f.comp ↑e

theorem MulEquiv.comp_right_injective {M : Type u_4} {N : Type u_5} {P : Type u_6} [MulOneClass M] [MulOneClass N] [MulOneClass P] (e : M ≃* N) : Function.Injective fun (f : P →* M) => (↑e).comp f

theorem AddEquiv.comp_right_injective {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] (e : M ≃+ N) : Function.Injective fun (f : P →+ M) => (↑e).comp f

theorem MulEquiv.map_one {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (h : M ≃* N) : h 1 = 1

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

theorem AddEquiv.map_zero {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (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_eq_one_iff {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (h : M ≃* N) {x : M} : h x = 1 ↔ x = 1

theorem AddEquiv.map_eq_zero_iff {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (h : M ≃+ N) {x : M} : h x = 0 ↔ x = 0

theorem MulEquiv.map_ne_one_iff {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (h : M ≃* N) {x : M} : h x ≠ 1 ↔ x ≠ 1

theorem AddEquiv.map_ne_zero_iff {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (h : M ≃+ N) {x : M} : h x ≠ 0 ↔ x ≠ 0

noncomputable def MulEquiv.ofBijective {M : Type u_9} {N : Type u_10} {F : Type u_11} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N] (f : F) (hf : Function.Bijective ⇑f) : M ≃* N

A bijective Semigroup homomorphism is an isomorphism

noncomputable def AddEquiv.ofBijective {M : Type u_9} {N : Type u_10} {F : Type u_11} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] (f : F) (hf : Function.Bijective ⇑f) : M ≃+ N

A bijective AddSemigroup homomorphism is an isomorphism

@[simp]

theorem AddEquiv.ofBijective_apply {M : Type u_9} {N : Type u_10} {F : Type u_11} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] (f : F) (hf : Function.Bijective ⇑f) (a : M) : (ofBijective f hf) a = f a

@[simp]

theorem MulEquiv.ofBijective_apply {M : Type u_9} {N : Type u_10} {F : Type u_11} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N] (f : F) (hf : Function.Bijective ⇑f) (a : M) : (ofBijective f hf) a = f a

@[simp]

theorem MulEquiv.ofBijective_apply_symm_apply {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] {n : N} (f : M →* N) (hf : Function.Bijective ⇑f) : f ((ofBijective f hf).symm n) = n

@[simp]

theorem AddEquiv.ofBijective_apply_symm_apply {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] {n : N} (f : M →+ N) (hf : Function.Bijective ⇑f) : f ((ofBijective f hf).symm n) = n

def MulEquiv.toMonoidHom {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (h : M ≃* N) : M →* N

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

def AddEquiv.toAddMonoidHom {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (h : M ≃+ N) : M →+ N

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

@[simp]

theorem MulEquiv.coe_toMonoidHom {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (e : M ≃* N) : ⇑e.toMonoidHom = ⇑e

@[simp]

theorem AddEquiv.coe_toAddMonoidHom {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (e : M ≃+ N) : ⇑e.toAddMonoidHom = ⇑e

@[simp]

theorem MulEquiv.toMonoidHom_eq_coe {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M ≃* N) : f.toMonoidHom = ↑f

@[simp]

theorem AddEquiv.toAddMonoidHom_eq_coe {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M ≃+ N) : f.toAddMonoidHom = ↑f

theorem MulEquiv.toMonoidHom_injective {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] : Function.Injective toMonoidHom

theorem AddEquiv.toAddMonoidHom_injective {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] : Function.Injective toAddMonoidHom

Groups

theorem MulEquiv.map_inv {G : Type u_7} {H : Type u_8} [Group G] [DivisionMonoid H] (h : G ≃* H) (x : G) : h x⁻¹ = (h x)⁻¹

A multiplicative equivalence of groups preserves inversion.

theorem AddEquiv.map_neg {G : Type u_7} {H : Type u_8} [AddGroup G] [SubtractionMonoid H] (h : G ≃+ H) (x : G) : h (-x) = -h x

An additive equivalence of additive groups preserves negation.

theorem MulEquiv.map_div {G : Type u_7} {H : Type u_8} [Group G] [DivisionMonoid H] (h : G ≃* H) (x y : G) : h (x / y) = h x / h y

A multiplicative equivalence of groups preserves division.

theorem AddEquiv.map_sub {G : Type u_7} {H : Type u_8} [AddGroup G] [SubtractionMonoid H] (h : G ≃+ H) (x y : G) : h (x - y) = h x - h y

An additive equivalence of additive groups preserves subtractions.

def MulHom.toMulEquiv {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M →ₙ* N) (g : N →ₙ* M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : M ≃* N

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

def AddHom.toAddEquiv {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M →ₙ+ N) (g : N →ₙ+ M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : 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.

@[simp]

theorem AddHom.toAddEquiv_symm_apply {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M →ₙ+ N) (g : N →ₙ+ M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : ⇑(f.toAddEquiv g h₁ h₂).symm = ⇑g

@[simp]

theorem MulHom.toMulEquiv_symm_apply {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M →ₙ* N) (g : N →ₙ* M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : ⇑(f.toMulEquiv g h₁ h₂).symm = ⇑g

@[simp]

theorem AddHom.toAddEquiv_apply {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M →ₙ+ N) (g : N →ₙ+ M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : ⇑(f.toAddEquiv g h₁ h₂) = ⇑f

@[simp]

theorem MulHom.toMulEquiv_apply {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M →ₙ* N) (g : N →ₙ* M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : ⇑(f.toMulEquiv g h₁ h₂) = ⇑f

def MonoidHom.toMulEquiv {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) (g : N →* M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : M ≃* N

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

def AddMonoidHom.toAddEquiv {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (g : N →+ M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : 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.

@[simp]

theorem MonoidHom.toMulEquiv_apply {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) (g : N →* M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : ⇑(f.toMulEquiv g h₁ h₂) = ⇑f

@[simp]

theorem AddMonoidHom.toAddEquiv_apply {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (g : N →+ M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : ⇑(f.toAddEquiv g h₁ h₂) = ⇑f

@[simp]

theorem MonoidHom.toMulEquiv_symm_apply {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) (g : N →* M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : ⇑(f.toMulEquiv g h₁ h₂).symm = ⇑g

@[simp]

theorem AddMonoidHom.toAddEquiv_symm_apply {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (g : N →+ M) (h₁ : g.comp f = id M) (h₂ : f.comp g = id N) : ⇑(f.toAddEquiv g h₁ h₂).symm = ⇑g