Documentation

Mathlib.Algebra.Group.Equiv.Defs

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 #

Notations #

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

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@[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
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@[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
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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
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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
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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.

Instances For
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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.

Instances
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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.

Instances For
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def «term_≃*_» :
Lean.TrailingParserDescr

Notation for a MulEquiv.

Equations
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def «term_≃+_» :
Lean.TrailingParserDescr

Notation for an AddEquiv.

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

Instances
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@[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.

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@[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
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@[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.

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@[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
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@[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
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@[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
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@[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
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@[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
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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.

Equations
  • ↑f = { toEquiv := ↑f, map_mul' := ⋯ }
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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.

Equations
  • ↑f = { toEquiv := ↑f, map_add' := ⋯ }
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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.

Equations
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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.

Equations
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instance MulEquiv.instEquivLike {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] :
EquivLike (M ≃* N) M N
Equations
  • MulEquiv.instEquivLike = { coe := fun (f : M ≃* N) => f.toFun, inv := fun (f : M ≃* N) => f.invFun, left_inv := ⋯, right_inv := ⋯, coe_injective' := ⋯ }
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instance AddEquiv.instEquivLike {M : Type u_4} {N : Type u_5} [Add M] [Add N] :
EquivLike (M ≃+ N) M N
Equations
  • AddEquiv.instEquivLike = { coe := fun (f : M ≃+ N) => f.toFun, inv := fun (f : M ≃+ N) => f.invFun, left_inv := ⋯, right_inv := ⋯, coe_injective' := ⋯ }
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instance MulEquiv.instCoeFunForall {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] :
CoeFun (M ≃* N) fun (x : M ≃* N) => M → N
Equations
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instance AddEquiv.instCoeFunForall {M : Type u_4} {N : Type u_5} [Add M] [Add N] :
CoeFun (M ≃+ N) fun (x : M ≃+ N) => M → N
Equations
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instance MulEquiv.instMulEquivClass {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] :
MulEquivClass (M ≃* N) M N
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instance AddEquiv.instAddEquivClass {M : Type u_4} {N : Type u_5} [Add M] [Add N] :
AddEquivClass (M ≃+ N) M N
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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.

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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.

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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'
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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'
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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
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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
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@[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
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@[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
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@[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
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@[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
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@[simp]
theorem MulEquiv.toEquiv_eq_coe {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) :
f.toEquiv = ↑f
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@[simp]
theorem AddEquiv.toEquiv_eq_coe {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) :
f.toEquiv = ↑f
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@[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.

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@[simp]
theorem AddEquiv.toAddHom_eq_coe {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) :
f.toAddHom = ↑f
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theorem MulEquiv.toFun_eq_coe {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) :
f.toFun = ⇑f
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theorem AddEquiv.toFun_eq_coe {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) :
f.toFun = ⇑f
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@[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.

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@[simp]
theorem AddEquiv.coe_toEquiv {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) :
⇑↑f = ⇑f
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@[simp]
theorem MulEquiv.coe_toMulHom {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {f : M ≃* N} :
⇑f.toMulHom = ⇑f
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@[simp]
theorem AddEquiv.coe_toAddHom {M : Type u_4} {N : Type u_5} [Add M] [Add N] {f : M ≃+ N} :
⇑f.toAddHom = ⇑f
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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.

Equations
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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.

Equations
source
@[deprecated map_mul (since := "2024-08-08")]
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.

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@[deprecated map_add (since := "2024-08-08")]
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.

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theorem MulEquiv.bijective {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) :
Function.Bijective ⇑e
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theorem AddEquiv.bijective {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) :
Function.Bijective ⇑e
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theorem MulEquiv.injective {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) :
Function.Injective ⇑e
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theorem AddEquiv.injective {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) :
Function.Injective ⇑e
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theorem MulEquiv.surjective {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (e : M ≃* N) :
Function.Surjective ⇑e
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theorem AddEquiv.surjective {M : Type u_4} {N : Type u_5} [Add M] [Add N] (e : M ≃+ N) :
Function.Surjective ⇑e
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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
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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
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def MulEquiv.refl (M : Type u_9) [Mul M] :
M ≃* M

The identity map is a multiplicative isomorphism.

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

The identity map is an additive isomorphism.

Equations
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instance MulEquiv.instInhabited {M : Type u_4} [Mul M] :
Inhabited (M ≃* M)
Equations
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instance AddEquiv.instInhabited {M : Type u_4} [Add M] :
Inhabited (M ≃+ M)
Equations
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@[simp]
theorem MulEquiv.coe_refl {M : Type u_4} [Mul M] :
⇑(refl M) = id
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@[simp]
theorem AddEquiv.coe_refl {M : Type u_4} [Add M] :
⇑(refl M) = id
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@[simp]
theorem MulEquiv.refl_apply {M : Type u_4} [Mul M] (m : M) :
(refl M) m = m
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@[simp]
theorem AddEquiv.refl_apply {M : Type u_4} [Add M] (m : M) :
(refl M) m = m
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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

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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
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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.

Equations
  • h.symm = { toEquiv := h.symm, map_mul' := ⋯ }
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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.

Equations
  • h.symm = { toEquiv := h.symm, map_add' := ⋯ }
source
theorem MulEquiv.invFun_eq_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] {f : M ≃* N} :
f.invFun = ⇑f.symm
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theorem AddEquiv.invFun_eq_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] {f : M ≃+ N} :
f.invFun = ⇑f.symm
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@[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.

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@[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
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@[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
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@[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
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@[simp]
theorem MulEquiv.toEquiv_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) :
↑f.symm = (↑f).symm
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@[simp]
theorem AddEquiv.toEquiv_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) :
↑f.symm = (↑f).symm
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@[simp]
theorem MulEquiv.symm_symm {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M ≃* N) :
f.symm.symm = f
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@[simp]
theorem AddEquiv.symm_symm {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M ≃+ N) :
f.symm.symm = f
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theorem MulEquiv.symm_bijective {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] :
Function.Bijective symm
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theorem AddEquiv.symm_bijective {M : Type u_4} {N : Type u_5} [Add M] [Add N] :
Function.Bijective symm
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@[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
source
@[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
source
@[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' := ⋯ }
source
@[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' := ⋯ }
source
@[simp]
theorem MulEquiv.refl_symm {M : Type u_4} [Mul M] :
(refl M).symm = refl M
source
@[simp]
theorem AddEquiv.refl_symm {M : Type u_4} [Add M] :
(refl M).symm = refl M
source
@[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.

source
@[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.

source
@[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.

source
@[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.

source
@[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
source
@[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
source
@[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
source
@[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
source
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
source
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
source
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
source
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
source
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
source
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
source
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
source
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
source
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
source
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
source
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
source
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
source
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
source
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
source
@[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
source
@[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
source
@[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
source
@[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
source
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]

Equations
source
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]

Equations
source
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

Equations
  • h1.trans h2 = { toEquiv := h1.trans h2.toEquiv, map_mul' := ⋯ }
source
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

Equations
  • h1.trans h2 = { toEquiv := h1.trans h2.toEquiv, map_add' := ⋯ }
source
@[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
source
@[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
source
@[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)
source
@[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)
source
@[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)
source
@[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)
source
@[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
source
@[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
source
@[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
source
@[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 #

source
theorem MulEquiv.coe_monoidHom_refl {M : Type u_4} [MulOneClass M] :
↑(refl M) = MonoidHom.id M
source
theorem AddEquiv.coe_addMonoidHom_refl {M : Type u_4} [AddZeroClass M] :
↑(refl M) = AddMonoidHom.id M
source
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
source
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
source
@[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
source
@[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
source
@[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
source
@[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
source
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
source
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
source
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
source
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
source
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).

source
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).

source
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
source
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
source
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
source
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
source
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

Equations
source
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

Equations
source
@[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
source
@[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
source
@[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
source
@[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
source
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.

Equations
  • h.toMonoidHom = { toFun := h.toFun, map_one' := ⋯, map_mul' := ⋯ }
source
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.

Equations
  • h.toAddMonoidHom = { toFun := h.toFun, map_zero' := ⋯, map_add' := ⋯ }
source
@[simp]
theorem MulEquiv.coe_toMonoidHom {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (e : M ≃* N) :
⇑e.toMonoidHom = ⇑e
source
@[simp]
theorem AddEquiv.coe_toAddMonoidHom {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (e : M ≃+ N) :
⇑e.toAddMonoidHom = ⇑e
source
@[simp]
theorem MulEquiv.toMonoidHom_eq_coe {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M ≃* N) :
f.toMonoidHom = ↑f
source
@[simp]
theorem AddEquiv.toAddMonoidHom_eq_coe {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M ≃+ N) :
f.toAddMonoidHom = ↑f
source
theorem MulEquiv.toMonoidHom_injective {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] :
Function.Injective toMonoidHom
source
theorem AddEquiv.toAddMonoidHom_injective {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] :
Function.Injective toAddMonoidHom

Groups #

source
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.

source
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.

source
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.

source
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.

source
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.

Equations
  • f.toMulEquiv g h₁ h₂ = { toFun := ⇑f, invFun := ⇑g, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ }
source
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.

Equations
  • f.toAddEquiv g h₁ h₂ = { toFun := ⇑f, invFun := ⇑g, left_inv := ⋯, right_inv := ⋯, map_add' := ⋯ }
source
@[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
source
@[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
source
@[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
source
@[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
source
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.

Equations
  • f.toMulEquiv g h₁ h₂ = { toFun := ⇑f, invFun := ⇑g, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ }
source
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.

Equations
  • f.toAddEquiv g h₁ h₂ = { toFun := ⇑f, invFun := ⇑g, left_inv := ⋯, right_inv := ⋯, map_add' := ⋯ }
source
@[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
source
@[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
source
@[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
source
@[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