# Documentation

Mathlib.Deprecated.Group

# Unbundled monoid and group homomorphisms #

This file is deprecated, and is no longer imported by anything in mathlib other than other deprecated files, and test files. You should not need to import it.

This file defines predicates for unbundled monoid and group homomorphisms. Instead of using this file, please use MonoidHom, defined in Algebra.Hom.Group, with notation →*→*, for morphisms between monoids or groups. For example use φ : G →* H→* H to represent a group homomorphism between multiplicative groups, and ψ : A →+ B→+ B to represent a group homomorphism between additive groups.

## Main Definitions #

IsMonoidHom (deprecated), IsGroupHom (deprecated)

## Tags #

IsGroupHom, IsMonoidHom

structure IsAddHom {α : Type u_1} {β : Type u_2} [inst : Add α] [inst : Add β] (f : αβ) :
• The proposition that f preserves addition.

map_add : ∀ (x y : α), f (x + y) = f x + f y

Predicate for maps which preserve an addition.

Instances For
structure IsMulHom {α : Type u_1} {β : Type u_2} [inst : Mul α] [inst : Mul β] (f : αβ) :
• The proposition that f preserves multiplication.

map_mul : ∀ (x y : α), f (x * y) = f x * f y

Predicate for maps which preserve a multiplication.

Instances For

theorem IsMulHom.id {α : Type u} [inst : Mul α] :

The identity map preserves multiplication.

theorem IsAddHom.comp {α : Type u} {β : Type v} [inst : Add α] [inst : Add β] {γ : Type u_1} [inst : Add γ] {f : αβ} {g : βγ} (hf : ) (hg : ) :

theorem IsMulHom.comp {α : Type u} {β : Type v} [inst : Mul α] [inst : Mul β] {γ : Type u_1} [inst : Mul γ] {f : αβ} {g : βγ} (hf : ) (hg : ) :

The composition of maps which preserve multiplication, also preserves multiplication.

theorem IsAddHom.add {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {f : αβ} {g : αβ} (hf : ) (hg : ) :
IsAddHom fun a => f a + g a

A sum of maps which preserves addition, preserves addition when the target is commutative.

theorem IsMulHom.mul {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {f : αβ} {g : αβ} (hf : ) (hg : ) :
IsMulHom fun a => f a * g a

A product of maps which preserve multiplication, preserves multiplication when the target is commutative.

theorem IsAddHom.neg {α : Type u_1} {β : Type u_2} [inst : Add α] [inst : ] {f : αβ} (hf : ) :
IsAddHom fun a => -f a

The negation of a map which preserves addition, preserves addition when the target is commutative.

theorem IsMulHom.inv {α : Type u_1} {β : Type u_2} [inst : Mul α] [inst : ] {f : αβ} (hf : ) :
IsMulHom fun a => (f a)⁻¹

The inverse of a map which preserves multiplication, preserves multiplication when the target is commutative.

structure IsAddMonoidHom {α : Type u} {β : Type v} [inst : ] [inst : ] (f : αβ) extends :
• The proposition that f preserves the additive identity.

map_zero : f 0 = 0

Predicate for additive monoid homomorphisms (deprecated -- use the bundled MonoidHom version).

Instances For
structure IsMonoidHom {α : Type u} {β : Type v} [inst : ] [inst : ] (f : αβ) extends :
• The proposition that f preserves the multiplicative identity.

map_one : f 1 = 1

Predicate for monoid homomorphisms (deprecated -- use the bundled MonoidHom version).

Instances For
def AddMonoidHom.of {M : Type u_1} {N : Type u_2} {mM : } {mN : } {f : MN} (h : ) :
M →+ N

Interpret a map f : M → N→ N as a homomorphism M →+ N→+ N.

Equations
• = { toZeroHom := { toFun := f, map_zero' := (_ : f 0 = 0) }, map_add' := (_ : ∀ (x y : M), f (x + y) = f x + f y) }
def AddMonoidHom.of.proof_1 {M : Type u_2} {N : Type u_1} {mM : } {mN : } {f : MN} (h : ) (x : M) (y : M) :
f (x + y) = f x + f y
Equations
• (_ : ∀ (x y : M), f (x + y) = f x + f y) = (_ : ∀ (x y : M), f (x + y) = f x + f y)
def MonoidHom.of {M : Type u_1} {N : Type u_2} {mM : } {mN : } {f : MN} (h : ) :
M →* N

Interpret a map f : M → N→ N as a homomorphism M →* N→* N.

Equations
• = { toOneHom := { toFun := f, map_one' := (_ : f 1 = 1) }, map_mul' := (_ : ∀ (x y : M), f (x * y) = f x * f y) }
@[simp]
theorem AddMonoidHom.coe_of {M : Type u_1} {N : Type u_2} {mM : } {mN : } {f : MN} (hf : ) :
↑() = f
@[simp]
theorem MonoidHom.coe_of {M : Type u_1} {N : Type u_2} {mM : } {mN : } {f : MN} (hf : ) :
↑() = f
theorem AddMonoidHom.isAddMonoidHom_coe {M : Type u_1} {N : Type u_2} {mM : } {mN : } (f : M →+ N) :
theorem MonoidHom.isMonoidHom_coe {M : Type u_1} {N : Type u_2} {mM : } {mN : } (f : M →* N) :
theorem AddEquiv.isAddHom {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (h : M ≃+ N) :

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

A multiplicative isomorphism preserves multiplication (deprecated).

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

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

A multiplicative bijection between two monoids is a monoid hom (deprecated -- use MulEquiv.toMonoidHom).

theorem IsAddMonoidHom.map_add' {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) (x : α) (y : α) :
f (x + y) = f x + f y

theorem IsMonoidHom.map_mul' {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) (x : α) (y : α) :
f (x * y) = f x * f y

A monoid homomorphism preserves multiplication.

theorem IsAddMonoidHom.neg {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {f : αβ} (hf : ) :
IsAddMonoidHom fun a => -f a

The negation of a map which preserves addition, preserves addition when the target is commutative.

theorem IsMonoidHom.inv {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {f : αβ} (hf : ) :
IsMonoidHom fun a => (f a)⁻¹

The inverse of a map which preserves multiplication, preserves multiplication when the target is commutative.

theorem IsAddHom.to_isAddMonoidHom {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :

theorem IsMulHom.to_isMonoidHom {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :

A map to a group preserving multiplication is a monoid homomorphism.

theorem IsAddMonoidHom.id {α : Type u} [inst : ] :

The identity map is an additive monoid homomorphism.

theorem IsMonoidHom.id {α : Type u} [inst : ] :

The identity map is a monoid homomorphism.

theorem IsAddMonoidHom.comp {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) {γ : Type u_1} [inst : ] {g : βγ} (hg : ) :

The composite of two additive monoid homomorphisms is an additive monoid homomorphism.

theorem IsMonoidHom.comp {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) {γ : Type u_1} [inst : ] {g : βγ} (hg : ) :

The composite of two monoid homomorphisms is a monoid homomorphism.

theorem IsAddMonoidHom.isAddMonoidHom_mul_left {γ : Type u_1} [inst : ] (x : γ) :
IsAddMonoidHom fun y => x * y

Left multiplication in a ring is an additive monoid morphism.

theorem IsAddMonoidHom.isAddMonoidHom_mul_right {γ : Type u_1} [inst : ] (x : γ) :
IsAddMonoidHom fun y => y * x

Right multiplication in a ring is an additive monoid morphism.

structure IsAddGroupHom {α : Type u} {β : Type v} [inst : ] [inst : ] (f : αβ) extends :

Predicate for additive group homomorphism (deprecated -- use bundled MonoidHom).

Instances For
structure IsGroupHom {α : Type u} {β : Type v} [inst : ] [inst : ] (f : αβ) extends :

Predicate for group homomorphisms (deprecated -- use bundled MonoidHom).

Instances For
∀ {x : } {x_1 : } (f : G →+ H),
theorem MonoidHom.isGroupHom {G : Type u_1} {H : Type u_2} :
∀ {x : } {x_1 : } (f : G →* H),
∀ {x : } {x_1 : } (h : G ≃+ H),
theorem MulEquiv.isGroupHom {G : Type u_1} {H : Type u_2} :
∀ {x : } {x_1 : } (h : G ≃* H),
theorem IsAddGroupHom.mk' {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ∀ (x y : α), f (x + y) = f x + f y) :

Construct IsAddGroupHom from its only hypothesis.

theorem IsGroupHom.mk' {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ∀ (x y : α), f (x * y) = f x * f y) :

Construct IsGroupHom from its only hypothesis.

theorem IsGroupHom.map_mul' {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) (x : α) (y : α) :
f (x * y) = f x * f y
theorem IsAddGroupHom.to_isAddMonoidHom {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :

theorem IsGroupHom.to_isMonoidHom {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :

A group homomorphism is a monoid homomorphism.

theorem IsAddGroupHom.map_zero {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :
f 0 = 0

An additive group homomorphism sends 0 to 0.

theorem IsGroupHom.map_one {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :
f 1 = 1

A group homomorphism sends 1 to 1.

theorem IsAddGroupHom.map_neg {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) (a : α) :
f (-a) = -f a

An additive group homomorphism sends negations to negations.

theorem IsGroupHom.map_inv {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) (a : α) :
f a⁻¹ = (f a)⁻¹

A group homomorphism sends inverses to inverses.

theorem IsAddGroupHom.map_sub {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) (a : α) (b : α) :
f (a - b) = f a - f b
theorem IsGroupHom.map_div {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) (a : α) (b : α) :
f (a / b) = f a / f b
theorem IsAddGroupHom.id {α : Type u} [inst : ] :

The identity is an additive group homomorphism.

theorem IsGroupHom.id {α : Type u} [inst : ] :

The identity is a group homomorphism.

theorem IsAddGroupHom.comp {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) {γ : Type u_1} [inst : ] {g : βγ} (hg : ) :

The composition of two additive group homomorphisms is an additive group homomorphism.

theorem IsGroupHom.comp {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) {γ : Type u_1} [inst : ] {g : βγ} (hg : ) :

The composition of two group homomorphisms is a group homomorphism.

theorem IsAddGroupHom.injective_iff {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :
∀ (a : α), f a = 0a = 0

An additive group homomorphism is injective if its kernel is trivial.

theorem IsGroupHom.injective_iff {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :
∀ (a : α), f a = 1a = 1

A group homomorphism is injective iff its kernel is trivial.

theorem IsAddGroupHom.add {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {f : αβ} {g : αβ} (hf : ) (hg : ) :
IsAddGroupHom fun a => f a + g a

The sum of two additive group homomorphisms is an additive group homomorphism if the target is commutative.

theorem IsGroupHom.mul {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {f : αβ} {g : αβ} (hf : ) (hg : ) :
IsGroupHom fun a => f a * g a

The product of group homomorphisms is a group homomorphism if the target is commutative.

theorem IsAddGroupHom.neg {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {f : αβ} (hf : ) :
IsAddGroupHom fun a => -f a

The negation of an additive group homomorphism is an additive group homomorphism if the target is commutative.

theorem IsGroupHom.inv {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {f : αβ} (hf : ) :
IsGroupHom fun a => (f a)⁻¹

The inverse of a group homomorphism is a group homomorphism if the target is commutative.

These instances look redundant, because Deprecated.Ring provides IsRingHom for a →+*→+*. Nevertheless these are harmless, and helpful for stripping out dependencies on Deprecated.Ring.

theorem RingHom.to_isMonoidHom {R : Type u_1} {S : Type u_2} [inst : ] [inst : ] (f : R →+* S) :
theorem RingHom.to_isAddMonoidHom {R : Type u_1} {S : Type u_2} [inst : ] [inst : ] (f : R →+* S) :
theorem RingHom.to_isAddGroupHom {R : Type u_1} {S : Type u_2} [inst : Ring R] [inst : Ring S] (f : R →+* S) :
theorem Neg.isAddGroupHom {α : Type u} [inst : ] :

Negation is an AddGroup homomorphism if the AddGroup is commutative.

theorem Inv.isGroupHom {α : Type u} [inst : ] :
IsGroupHom Inv.inv

Inversion is a group homomorphism if the group is commutative.

theorem IsAddGroupHom.sub {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {f : αβ} {g : αβ} (hf : ) (hg : ) :
IsAddGroupHom fun a => f a - g a

The difference of two additive group homomorphisms is an additive group homomorphism if the target is commutative.

def Units.map' {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] {f : MN} (hf : ) :

The group homomorphism on units induced by a multiplicative morphism.

Equations
@[simp]
theorem Units.coe_map' {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] {f : MN} (hf : ) (x : Mˣ) :
↑(↑() x) = f x
theorem Units.coe_isMonoidHom {M : Type u_1} [inst : ] :
IsMonoidHom fun x => x
theorem IsUnit.map' {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] {f : MN} (hf : ) {x : M} (h : ) :
IsUnit (f x)
theorem Additive.isAddHom {α : Type u} {β : Type v} [inst : Mul α] [inst : Mul β] {f : αβ} (hf : ) :
theorem Multiplicative.isMulHom {α : Type u} {β : Type v} [inst : Add α] [inst : Add β] {f : αβ} (hf : ) :
theorem Additive.isAddMonoidHom {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :
theorem Multiplicative.isMonoidHom {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :
theorem Additive.isAddGroupHom {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :
theorem Multiplicative.isGroupHom {α : Type u} {β : Type v} [inst : ] [inst : ] {f : αβ} (hf : ) :