Monoid and group homomorphisms

This file defines the bundled structures for monoid and group homomorphisms. Namely, we define MonoidHom (resp., AddMonoidHom) to be bundled homomorphisms between multiplicative (resp., additive) monoids or groups.

We also define coercion to a function, and usual operations: composition, identity homomorphism, pointwise multiplication and pointwise inversion.

This file also defines the lesser-used (and notation-less) homomorphism types which are used as building blocks for other homomorphisms:

• ZeroHom
• OneHom
• AddHom
• MulHom

Notations

• →+: Bundled AddMonoid homs. Also use for AddGroup homs.
• →*: Bundled Monoid homs. Also use for Group homs.
• →ₙ+: Bundled AddSemigroup homs.
• →ₙ*: Bundled Semigroup homs.

Implementation notes

There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion.

There is no GroupHom -- the idea is that MonoidHom is used. The constructor for MonoidHom needs a proof of map_one as well as map_mul; a separate constructor MonoidHom.mk' will construct group homs (i.e. monoid homs between groups) given only a proof that multiplication is preserved,

Implicit {} brackets are often used instead of type class [] brackets. This is done when the instances can be inferred because they are implicit arguments to the type MonoidHom. When they can be inferred from the type it is faster to use this method than to use type class inference.

Historically this file also included definitions of unbundled homomorphism classes; they were deprecated and moved to Deprecated/Group.

Tags

MonoidHom, AddMonoidHom

structure ZeroHom (M : Type u_10) (N : Type u_11) [Zero M] [Zero N] : Type (max u_10 u_11)

ZeroHom M N is the type of functions M → N that preserve zero.

When possible, instead of parametrizing results over (f : ZeroHom M N), you should parametrize over (F : Type*) [ZeroHomClass F M N] (f : F).

When you extend this structure, make sure to also extend ZeroHomClass.

• toFun : M → N

  The underlying function

• map_zero' : self.toFun 0 = 0

  The proposition that the function preserves 0

class ZeroHomClass (F : Type u_10) (M : outParam (Type u_11)) (N : outParam (Type u_12)) [Zero M] [Zero N] [FunLike F M N] : Prop

ZeroHomClass F M N states that F is a type of zero-preserving homomorphisms.

You should extend this typeclass when you extend ZeroHom.

• map_zero : ∀ (f : F), f 0 = 0

  The proposition that the function preserves 0

structure AddHom (M : Type u_10) (N : Type u_11) [Add M] [Add N] : Type (max u_10 u_11)

M →ₙ+ N is the type of functions M → N that preserve addition. The ₙ in the notation stands for "non-unital" because it is intended to match the notation for NonUnitalAlgHom and NonUnitalRingHom, so a AddHom is a non-unital additive monoid hom.

When possible, instead of parametrizing results over (f : AddHom M N), you should parametrize over (F : Type*) [AddHomClass F M N] (f : F).

When you extend this structure, make sure to extend AddHomClass.

• toFun : M → N

  The underlying function

• map_add' : ∀ (x y : M), self.toFun (x + y) = self.toFun x + self.toFun y

  The proposition that the function preserves addition

def «term_→ₙ+_» : Lean.TrailingParserDescr

M →ₙ+ N denotes the type of addition-preserving maps from M to N.

Equations

• «term_→ₙ+_» = Lean.ParserDescr.trailingNode `«term_→ₙ+_» 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →ₙ+ ") (Lean.ParserDescr.cat `term 25))

class AddHomClass (F : Type u_10) (M : outParam (Type u_11)) (N : outParam (Type u_12)) [Add M] [Add N] [FunLike F M N] : Prop

AddHomClass F M N states that F is a type of addition-preserving homomorphisms. You should declare an instance of this typeclass when you extend AddHom.

• map_add : ∀ (f : F) (x y : M), f (x + y) = f x + f y

  The proposition that the function preserves addition

structure AddMonoidHom (M : Type u_10) (N : Type u_11) [AddZeroClass M] [AddZeroClass N] extends ZeroHom M N, M →ₙ+ N : Type (max u_10 u_11)

M →+ N is the type of functions M → N that preserve the AddZeroClass structure.

AddMonoidHom is also used for group homomorphisms.

When possible, instead of parametrizing results over (f : M →+ N), you should parametrize over (F : Type*) [AddMonoidHomClass F M N] (f : F).

When you extend this structure, make sure to extend AddMonoidHomClass.

• toFun : M → N
• map_zero' : (↑self).toFun 0 = 0
• map_add' : ∀ (x y : M), (↑self).toFun (x + y) = (↑self).toFun x + (↑self).toFun y

def «term_→+_» : Lean.TrailingParserDescr

M →+ N denotes the type of additive monoid homomorphisms from M to N.

Equations

• «term_→+_» = Lean.ParserDescr.trailingNode `«term_→+_» 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →+ ") (Lean.ParserDescr.cat `term 25))

class AddMonoidHomClass (F : Type u_10) (M : outParam (Type u_11)) (N : outParam (Type u_12)) [AddZeroClass M] [AddZeroClass N] [FunLike F M N] extends AddHomClass F M N, ZeroHomClass F M N : Prop

AddMonoidHomClass F M N states that F is a type of AddZeroClass-preserving homomorphisms.

You should also extend this typeclass when you extend AddMonoidHom.

• map_add : ∀ (f : F) (x y : M), f (x + y) = f x + f y
• map_zero : ∀ (f : F), f 0 = 0

structure OneHom (M : Type u_10) (N : Type u_11) [One M] [One N] : Type (max u_10 u_11)

OneHom M N is the type of functions M → N that preserve one.

When possible, instead of parametrizing results over (f : OneHom M N), you should parametrize over (F : Type*) [OneHomClass F M N] (f : F).

When you extend this structure, make sure to also extend OneHomClass.

• toFun : M → N

  The underlying function

• map_one' : self.toFun 1 = 1

  The proposition that the function preserves 1

class OneHomClass (F : Type u_10) (M : outParam (Type u_11)) (N : outParam (Type u_12)) [One M] [One N] [FunLike F M N] : Prop

OneHomClass F M N states that F is a type of one-preserving homomorphisms. You should extend this typeclass when you extend OneHom.

• map_one : ∀ (f : F), f 1 = 1

  The proposition that the function preserves 1

instance OneHom.funLike {M : Type u_4} {N : Type u_5} [One M] [One N] : FunLike (OneHom M N) M N

Equations

• OneHom.funLike = { coe := OneHom.toFun, coe_injective' := ⋯ }

instance ZeroHom.funLike {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] : FunLike (ZeroHom M N) M N

Equations

• ZeroHom.funLike = { coe := ZeroHom.toFun, coe_injective' := ⋯ }

theorem OneHom.oneHomClass {M : Type u_4} {N : Type u_5} [One M] [One N] : OneHomClass (OneHom M N) M N

theorem ZeroHom.zeroHomClass {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] : ZeroHomClass (ZeroHom M N) M N

@[simp]

theorem map_one {M : Type u_4} {N : Type u_5} {F : Type u_9} [One M] [One N] [FunLike F M N] [OneHomClass F M N] (f : F) : f 1 = 1

See note [low priority simp lemmas]

@[simp]

theorem map_zero {M : Type u_4} {N : Type u_5} {F : Type u_9} [Zero M] [Zero N] [FunLike F M N] [ZeroHomClass F M N] (f : F) : f 0 = 0

theorem map_comp_one {ι : Type u_1} {M : Type u_4} {N : Type u_5} {F : Type u_9} [One M] [One N] [FunLike F M N] [OneHomClass F M N] (f : F) : ⇑f ∘ 1 = 1

theorem map_comp_zero {ι : Type u_1} {M : Type u_4} {N : Type u_5} {F : Type u_9} [Zero M] [Zero N] [FunLike F M N] [ZeroHomClass F M N] (f : F) : ⇑f ∘ 0 = 0

theorem Subsingleton.of_oneHomClass {M : Type u_4} {N : Type u_5} {F : Type u_9} [One M] [One N] [FunLike F M N] [Subsingleton M] [OneHomClass F M N] : Subsingleton F

In principle this could be an instance, but in practice it causes performance issues.

theorem Subsingleton.of_zeroHomClass {M : Type u_4} {N : Type u_5} {F : Type u_9} [Zero M] [Zero N] [FunLike F M N] [Subsingleton M] [ZeroHomClass F M N] : Subsingleton F

theorem instSubsingletonOneHom {M : Type u_4} {N : Type u_5} [One M] [One N] [Subsingleton M] : Subsingleton (OneHom M N)

theorem instSubsingletonZeroHom {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] [Subsingleton M] : Subsingleton (ZeroHom M N)

theorem map_eq_one_iff {M : Type u_4} {N : Type u_5} {F : Type u_9} [One M] [One N] [FunLike F M N] [OneHomClass F M N] (f : F) (hf : Function.Injective ⇑f) {x : M} : f x = 1 ↔ x = 1

theorem map_eq_zero_iff {M : Type u_4} {N : Type u_5} {F : Type u_9} [Zero M] [Zero N] [FunLike F M N] [ZeroHomClass F M N] (f : F) (hf : Function.Injective ⇑f) {x : M} : f x = 0 ↔ x = 0

theorem map_ne_one_iff {R : Type u_10} {S : Type u_11} {F : Type u_12} [One R] [One S] [FunLike F R S] [OneHomClass F R S] (f : F) (hf : Function.Injective ⇑f) {x : R} : f x ≠ 1 ↔ x ≠ 1

theorem map_ne_zero_iff {R : Type u_10} {S : Type u_11} {F : Type u_12} [Zero R] [Zero S] [FunLike F R S] [ZeroHomClass F R S] (f : F) (hf : Function.Injective ⇑f) {x : R} : f x ≠ 0 ↔ x ≠ 0

theorem ne_one_of_map {R : Type u_10} {S : Type u_11} {F : Type u_12} [One R] [One S] [FunLike F R S] [OneHomClass F R S] {f : F} {x : R} (hx : f x ≠ 1) : x ≠ 1

theorem ne_zero_of_map {R : Type u_10} {S : Type u_11} {F : Type u_12} [Zero R] [Zero S] [FunLike F R S] [ZeroHomClass F R S] {f : F} {x : R} (hx : f x ≠ 0) : x ≠ 0

def OneHomClass.toOneHom {M : Type u_4} {N : Type u_5} {F : Type u_9} [One M] [One N] [FunLike F M N] [OneHomClass F M N] (f : F) : OneHom M N

Turn an element of a type F satisfying OneHomClass F M N into an actual OneHom. This is declared as the default coercion from F to OneHom M N.

Equations

• ↑f = { toFun := ⇑f, map_one' := ⋯ }

def ZeroHomClass.toZeroHom {M : Type u_4} {N : Type u_5} {F : Type u_9} [Zero M] [Zero N] [FunLike F M N] [ZeroHomClass F M N] (f : F) : ZeroHom M N

Turn an element of a type F satisfying ZeroHomClass F M N into an actual ZeroHom. This is declared as the default coercion from F to ZeroHom M N.

Equations

• ↑f = { toFun := ⇑f, map_zero' := ⋯ }

instance instCoeTCOneHomOfOneHomClass {M : Type u_4} {N : Type u_5} {F : Type u_9} [One M] [One N] [FunLike F M N] [OneHomClass F M N] : CoeTC F (OneHom M N)

Any type satisfying OneHomClass can be cast into OneHom via OneHomClass.toOneHom.

Equations

• instCoeTCOneHomOfOneHomClass = { coe := OneHomClass.toOneHom }

instance instCoeTCZeroHomOfZeroHomClass {M : Type u_4} {N : Type u_5} {F : Type u_9} [Zero M] [Zero N] [FunLike F M N] [ZeroHomClass F M N] : CoeTC F (ZeroHom M N)

Any type satisfying ZeroHomClass can be cast into ZeroHom via ZeroHomClass.toZeroHom.

Equations

• instCoeTCZeroHomOfZeroHomClass = { coe := ZeroHomClass.toZeroHom }

@[simp]

theorem OneHom.coe_coe {M : Type u_4} {N : Type u_5} {F : Type u_9} [One M] [One N] [FunLike F M N] [OneHomClass F M N] (f : F) : ⇑↑f = ⇑f

@[simp]

theorem ZeroHom.coe_coe {M : Type u_4} {N : Type u_5} {F : Type u_9} [Zero M] [Zero N] [FunLike F M N] [ZeroHomClass F M N] (f : F) : ⇑↑f = ⇑f

structure MulHom (M : Type u_10) (N : Type u_11) [Mul M] [Mul N] : Type (max u_10 u_11)

M →ₙ* N is the type of functions M → N that preserve multiplication. The ₙ in the notation stands for "non-unital" because it is intended to match the notation for NonUnitalAlgHom and NonUnitalRingHom, so a MulHom is a non-unital monoid hom.

When possible, instead of parametrizing results over (f : M →ₙ* N), you should parametrize over (F : Type*) [MulHomClass F M N] (f : F). When you extend this structure, make sure to extend MulHomClass.

• toFun : M → N

  The underlying function

• map_mul' : ∀ (x y : M), self.toFun (x * y) = self.toFun x * self.toFun y

  The proposition that the function preserves multiplication

def «term_→ₙ*_» : Lean.TrailingParserDescr

M →ₙ* N denotes the type of multiplication-preserving maps from M to N.

Equations

• «term_→ₙ*_» = Lean.ParserDescr.trailingNode `«term_→ₙ*_» 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →ₙ* ") (Lean.ParserDescr.cat `term 25))

class MulHomClass (F : Type u_10) (M : outParam (Type u_11)) (N : outParam (Type u_12)) [Mul M] [Mul N] [FunLike F M N] : Prop

MulHomClass F M N states that F is a type of multiplication-preserving homomorphisms.

You should declare an instance of this typeclass when you extend MulHom.

• map_mul : ∀ (f : F) (x y : M), f (x * y) = f x * f y

  The proposition that the function preserves multiplication

instance MulHom.funLike {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] : FunLike (M →ₙ* N) M N

Equations

• MulHom.funLike = { coe := MulHom.toFun, coe_injective' := ⋯ }

instance AddHom.funLike {M : Type u_4} {N : Type u_5} [Add M] [Add N] : FunLike (M →ₙ+ N) M N

Equations

• AddHom.funLike = { coe := AddHom.toFun, coe_injective' := ⋯ }

theorem MulHom.mulHomClass {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] : MulHomClass (M →ₙ* N) M N

MulHom is a type of multiplication-preserving homomorphisms

theorem AddHom.addHomClass {M : Type u_4} {N : Type u_5} [Add M] [Add N] : AddHomClass (M →ₙ+ N) M N

AddHom is a type of addition-preserving homomorphisms

@[simp]

theorem map_mul {M : Type u_4} {N : Type u_5} {F : Type u_9} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N] (f : F) (x y : M) : f (x * y) = f x * f y

See note [low priority simp lemmas]

@[simp]

theorem map_add {M : Type u_4} {N : Type u_5} {F : Type u_9} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] (f : F) (x y : M) : f (x + y) = f x + f y

@[simp]

theorem map_comp_mul {ι : Type u_1} {M : Type u_4} {N : Type u_5} {F : Type u_9} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N] (f : F) (g h : ι → M) : ⇑f ∘ (g * h) = ⇑f ∘ g * ⇑f ∘ h

@[simp]

theorem map_comp_add {ι : Type u_1} {M : Type u_4} {N : Type u_5} {F : Type u_9} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] (f : F) (g h : ι → M) : ⇑f ∘ (g + h) = ⇑f ∘ g + ⇑f ∘ h

def MulHomClass.toMulHom {M : Type u_4} {N : Type u_5} {F : Type u_9} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N] (f : F) : M →ₙ* N

Turn an element of a type F satisfying MulHomClass F M N into an actual MulHom. This is declared as the default coercion from F to M →ₙ* N.

Equations

• ↑f = { toFun := ⇑f, map_mul' := ⋯ }

def AddHomClass.toAddHom {M : Type u_4} {N : Type u_5} {F : Type u_9} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] (f : F) : M →ₙ+ N

Turn an element of a type F satisfying AddHomClass F M N into an actual AddHom. This is declared as the default coercion from F to M →ₙ+ N.

Equations

• ↑f = { toFun := ⇑f, map_add' := ⋯ }

instance instCoeTCMulHomOfMulHomClass {M : Type u_4} {N : Type u_5} {F : Type u_9} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N] : CoeTC F (M →ₙ* N)

Any type satisfying MulHomClass can be cast into MulHom via MulHomClass.toMulHom.

Equations

• instCoeTCMulHomOfMulHomClass = { coe := MulHomClass.toMulHom }

instance instCoeTCAddHomOfAddHomClass {M : Type u_4} {N : Type u_5} {F : Type u_9} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] : CoeTC F (M →ₙ+ N)

Any type satisfying AddHomClass can be cast into AddHom via AddHomClass.toAddHom.

Equations

• instCoeTCAddHomOfAddHomClass = { coe := AddHomClass.toAddHom }

@[simp]

theorem MulHom.coe_coe {M : Type u_4} {N : Type u_5} {F : Type u_9} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N] (f : F) : ⇑↑f = ⇑f

@[simp]

theorem AddHom.coe_coe {M : Type u_4} {N : Type u_5} {F : Type u_9} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] (f : F) : ⇑↑f = ⇑f

structure MonoidHom (M : Type u_10) (N : Type u_11) [MulOneClass M] [MulOneClass N] extends OneHom M N, M →ₙ* N : Type (max u_10 u_11)

M →* N is the type of functions M → N that preserve the Monoid structure. MonoidHom is also used for group homomorphisms.

When possible, instead of parametrizing results over (f : M →* N), you should parametrize over (F : Type*) [MonoidHomClass F M N] (f : F).

When you extend this structure, make sure to extend MonoidHomClass.

• toFun : M → N
• map_one' : (↑self).toFun 1 = 1
• map_mul' : ∀ (x y : M), (↑self).toFun (x * y) = (↑self).toFun x * (↑self).toFun y

def «term_→*_» : Lean.TrailingParserDescr

M →* N denotes the type of monoid homomorphisms from M to N.

Equations

• «term_→*_» = Lean.ParserDescr.trailingNode `«term_→*_» 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →* ") (Lean.ParserDescr.cat `term 25))

class MonoidHomClass (F : Type u_10) (M : outParam (Type u_11)) (N : outParam (Type u_12)) [MulOneClass M] [MulOneClass N] [FunLike F M N] extends MulHomClass F M N, OneHomClass F M N : Prop

MonoidHomClass F M N states that F is a type of Monoid-preserving homomorphisms. You should also extend this typeclass when you extend MonoidHom.

• map_mul : ∀ (f : F) (x y : M), f (x * y) = f x * f y
• map_one : ∀ (f : F), f 1 = 1

instance MonoidHom.instFunLike {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] : FunLike (M →* N) M N

Equations

• MonoidHom.instFunLike = { coe := fun (f : M →* N) => (↑f).toFun, coe_injective' := ⋯ }

instance AddMonoidHom.instFunLike {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] : FunLike (M →+ N) M N

Equations

• AddMonoidHom.instFunLike = { coe := fun (f : M →+ N) => (↑f).toFun, coe_injective' := ⋯ }

theorem MonoidHom.instMonoidHomClass {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] : MonoidHomClass (M →* N) M N

theorem AddMonoidHom.instAddMonoidHomClass {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] : AddMonoidHomClass (M →+ N) M N

theorem instSubsingletonMonoidHom {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] [Subsingleton M] : Subsingleton (M →* N)

theorem instSubsingletonAddMonoidHom {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] [Subsingleton M] : Subsingleton (M →+ N)

def MonoidHomClass.toMonoidHom {M : Type u_4} {N : Type u_5} {F : Type u_9} [MulOneClass M] [MulOneClass N] [FunLike F M N] [MonoidHomClass F M N] (f : F) : M →* N

Turn an element of a type F satisfying MonoidHomClass F M N into an actual MonoidHom. This is declared as the default coercion from F to M →* N.

Equations

• ↑f = { toFun := (↑f).toFun, map_one' := ⋯, map_mul' := ⋯ }

def AddMonoidHomClass.toAddMonoidHom {M : Type u_4} {N : Type u_5} {F : Type u_9} [AddZeroClass M] [AddZeroClass N] [FunLike F M N] [AddMonoidHomClass F M N] (f : F) : M →+ N

Turn an element of a type F satisfying AddMonoidHomClass F M N into an actual MonoidHom. This is declared as the default coercion from F to M →+ N.

Equations

• ↑f = { toFun := (↑f).toFun, map_zero' := ⋯, map_add' := ⋯ }

instance instCoeTCMonoidHomOfMonoidHomClass {M : Type u_4} {N : Type u_5} {F : Type u_9} [MulOneClass M] [MulOneClass N] [FunLike F M N] [MonoidHomClass F M N] : CoeTC F (M →* N)

Any type satisfying MonoidHomClass can be cast into MonoidHom via MonoidHomClass.toMonoidHom.

Equations

• instCoeTCMonoidHomOfMonoidHomClass = { coe := MonoidHomClass.toMonoidHom }

instance instCoeTCAddMonoidHomOfAddMonoidHomClass {M : Type u_4} {N : Type u_5} {F : Type u_9} [AddZeroClass M] [AddZeroClass N] [FunLike F M N] [AddMonoidHomClass F M N] : CoeTC F (M →+ N)

Any type satisfying AddMonoidHomClass can be cast into AddMonoidHom via AddMonoidHomClass.toAddMonoidHom.

Equations

• instCoeTCAddMonoidHomOfAddMonoidHomClass = { coe := AddMonoidHomClass.toAddMonoidHom }

@[simp]

theorem MonoidHom.coe_coe {M : Type u_4} {N : Type u_5} {F : Type u_9} [MulOneClass M] [MulOneClass N] [FunLike F M N] [MonoidHomClass F M N] (f : F) : ⇑↑f = ⇑f

@[simp]

theorem AddMonoidHom.coe_coe {M : Type u_4} {N : Type u_5} {F : Type u_9} [AddZeroClass M] [AddZeroClass N] [FunLike F M N] [AddMonoidHomClass F M N] (f : F) : ⇑↑f = ⇑f

theorem map_mul_eq_one {M : Type u_4} {N : Type u_5} {F : Type u_9} [MulOneClass M] [MulOneClass N] [FunLike F M N] [MonoidHomClass F M N] (f : F) {a b : M} (h : a * b = 1) : f a * f b = 1

theorem map_add_eq_zero {M : Type u_4} {N : Type u_5} {F : Type u_9} [AddZeroClass M] [AddZeroClass N] [FunLike F M N] [AddMonoidHomClass F M N] (f : F) {a b : M} (h : a + b = 0) : f a + f b = 0

theorem map_div' {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [DivInvMonoid G] [DivInvMonoid H] [MonoidHomClass F G H] (f : F) (hf : ∀ (a : G), f a⁻¹ = (f a)⁻¹) (a b : G) : f (a / b) = f a / f b

theorem map_sub' {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [SubNegMonoid G] [SubNegMonoid H] [AddMonoidHomClass F G H] (f : F) (hf : ∀ (a : G), f (-a) = -f a) (a b : G) : f (a - b) = f a - f b

theorem map_comp_div' {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [DivInvMonoid G] [DivInvMonoid H] [MonoidHomClass F G H] (f : F) (hf : ∀ (a : G), f a⁻¹ = (f a)⁻¹) (g h : ι → G) : ⇑f ∘ (g / h) = ⇑f ∘ g / ⇑f ∘ h

theorem map_comp_sub' {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [SubNegMonoid G] [SubNegMonoid H] [AddMonoidHomClass F G H] (f : F) (hf : ∀ (a : G), f (-a) = -f a) (g h : ι → G) : ⇑f ∘ (g - h) = ⇑f ∘ g - ⇑f ∘ h

@[simp]

theorem map_inv {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (a : G) : f a⁻¹ = (f a)⁻¹

Group homomorphisms preserve inverse.

See note [low priority simp lemmas]

@[simp]

theorem map_neg {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [AddGroup G] [SubtractionMonoid H] [AddMonoidHomClass F G H] (f : F) (a : G) : f (-a) = -f a

Additive group homomorphisms preserve negation.

@[simp]

theorem map_comp_inv {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (g : ι → G) : ⇑f ∘ g⁻¹ = (⇑f ∘ g)⁻¹

@[simp]

theorem map_comp_neg {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [AddGroup G] [SubtractionMonoid H] [AddMonoidHomClass F G H] (f : F) (g : ι → G) : ⇑f ∘ (-g) = -⇑f ∘ g

theorem map_mul_inv {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (a b : G) : f (a * b⁻¹) = f a * (f b)⁻¹

Group homomorphisms preserve division.

theorem map_add_neg {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [AddGroup G] [SubtractionMonoid H] [AddMonoidHomClass F G H] (f : F) (a b : G) : f (a + -b) = f a + -f b

Additive group homomorphisms preserve subtraction.

theorem map_comp_mul_inv {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (g h : ι → G) : ⇑f ∘ (g * h⁻¹) = ⇑f ∘ g * (⇑f ∘ h)⁻¹

theorem map_comp_add_neg {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [AddGroup G] [SubtractionMonoid H] [AddMonoidHomClass F G H] (f : F) (g h : ι → G) : ⇑f ∘ (g + -h) = ⇑f ∘ g + -⇑f ∘ h

@[simp]

theorem map_div {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (a b : G) : f (a / b) = f a / f b

Group homomorphisms preserve division.

See note [low priority simp lemmas]

@[simp]

theorem map_sub {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [AddGroup G] [SubtractionMonoid H] [AddMonoidHomClass F G H] (f : F) (a b : G) : f (a - b) = f a - f b

Additive group homomorphisms preserve subtraction.

@[simp]

theorem map_comp_div {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (g h : ι → G) : ⇑f ∘ (g / h) = ⇑f ∘ g / ⇑f ∘ h

@[simp]

theorem map_comp_sub {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [AddGroup G] [SubtractionMonoid H] [AddMonoidHomClass F G H] (f : F) (g h : ι → G) : ⇑f ∘ (g - h) = ⇑f ∘ g - ⇑f ∘ h

@[simp]

theorem map_pow {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [Monoid G] [Monoid H] [MonoidHomClass F G H] (f : F) (a : G) (n : ℕ) : f (a ^ n) = f a ^ n

See note [low priority simp lemmas]

@[simp]

theorem map_nsmul {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [AddMonoid G] [AddMonoid H] [AddMonoidHomClass F G H] (f : F) (n : ℕ) (a : G) : f (n • a) = n • f a

@[simp]

theorem map_comp_pow {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [Monoid G] [Monoid H] [MonoidHomClass F G H] (f : F) (g : ι → G) (n : ℕ) : ⇑f ∘ (g ^ n) = ⇑f ∘ g ^ n

@[simp]

theorem map_comp_nsmul {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [AddMonoid G] [AddMonoid H] [AddMonoidHomClass F G H] (f : F) (g : ι → G) (n : ℕ) : ⇑f ∘ (n • g) = n • ⇑f ∘ g

theorem map_zpow' {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [DivInvMonoid G] [DivInvMonoid H] [MonoidHomClass F G H] (f : F) (hf : ∀ (x : G), f x⁻¹ = (f x)⁻¹) (a : G) (n : ℤ) : f (a ^ n) = f a ^ n

theorem map_zsmul' {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [SubNegMonoid G] [SubNegMonoid H] [AddMonoidHomClass F G H] (f : F) (hf : ∀ (x : G), f (-x) = -f x) (a : G) (n : ℤ) : f (n • a) = n • f a

@[simp]

theorem map_comp_zpow' {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [DivInvMonoid G] [DivInvMonoid H] [MonoidHomClass F G H] (f : F) (hf : ∀ (x : G), f x⁻¹ = (f x)⁻¹) (g : ι → G) (n : ℤ) : ⇑f ∘ (g ^ n) = ⇑f ∘ g ^ n

@[simp]

theorem map_comp_zsmul' {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [SubNegMonoid G] [SubNegMonoid H] [AddMonoidHomClass F G H] (f : F) (hf : ∀ (x : G), f (-x) = -f x) (g : ι → G) (n : ℤ) : ⇑f ∘ (n • g) = n • ⇑f ∘ g

@[simp]

theorem map_zpow {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (g : G) (n : ℤ) : f (g ^ n) = f g ^ n

Group homomorphisms preserve integer power.

See note [low priority simp lemmas]

@[simp]

theorem map_zsmul {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [AddGroup G] [SubtractionMonoid H] [AddMonoidHomClass F G H] (f : F) (n : ℤ) (g : G) : f (n • g) = n • f g

Additive group homomorphisms preserve integer scaling.

theorem map_comp_zpow {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (g : ι → G) (n : ℤ) : ⇑f ∘ (g ^ n) = ⇑f ∘ g ^ n

theorem map_comp_zsmul {ι : Type u_1} {G : Type u_7} {H : Type u_8} {F : Type u_9} [FunLike F G H] [AddGroup G] [SubtractionMonoid H] [AddMonoidHomClass F G H] (f : F) (g : ι → G) (n : ℤ) : ⇑f ∘ (n • g) = n • ⇑f ∘ g

Bundled morphisms can be down-cast to weaker bundlings

instance MonoidHom.coeToOneHom {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] : Coe (M →* N) (OneHom M N)

MonoidHom down-cast to a OneHom, forgetting the multiplicative property.

Equations

• MonoidHom.coeToOneHom = { coe := MonoidHom.toOneHom }

instance AddMonoidHom.coeToZeroHom {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] : Coe (M →+ N) (ZeroHom M N)

AddMonoidHom down-cast to a ZeroHom, forgetting the additive property

Equations

• AddMonoidHom.coeToZeroHom = { coe := AddMonoidHom.toZeroHom }

instance MonoidHom.coeToMulHom {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] : Coe (M →* N) (M →ₙ* N)

MonoidHom down-cast to a MulHom, forgetting the 1-preserving property.

Equations

• MonoidHom.coeToMulHom = { coe := MonoidHom.toMulHom }

instance AddMonoidHom.coeToAddHom {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] : Coe (M →+ N) (M →ₙ+ N)

AddMonoidHom down-cast to an AddHom, forgetting the 0-preserving property.

Equations

• AddMonoidHom.coeToAddHom = { coe := AddMonoidHom.toAddHom }

@[simp]

theorem OneHom.coe_mk {M : Type u_4} {N : Type u_5} [One M] [One N] (f : M → N) (h1 : f 1 = 1) : ⇑{ toFun := f, map_one' := h1 } = f

@[simp]

theorem ZeroHom.coe_mk {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] (f : M → N) (h1 : f 0 = 0) : ⇑{ toFun := f, map_zero' := h1 } = f

@[simp]

theorem OneHom.toFun_eq_coe {M : Type u_4} {N : Type u_5} [One M] [One N] (f : OneHom M N) : f.toFun = ⇑f

@[simp]

theorem ZeroHom.toFun_eq_coe {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] (f : ZeroHom M N) : f.toFun = ⇑f

@[simp]

theorem MulHom.coe_mk {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M → N) (hmul : ∀ (x y : M), f (x * y) = f x * f y) : ⇑{ toFun := f, map_mul' := hmul } = f

@[simp]

theorem AddHom.coe_mk {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M → N) (hmul : ∀ (x y : M), f (x + y) = f x + f y) : ⇑{ toFun := f, map_add' := hmul } = f

@[simp]

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

@[simp]

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

@[simp]

theorem MonoidHom.coe_mk {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : OneHom M N) (hmul : ∀ (x y : M), f.toFun (x * y) = f.toFun x * f.toFun y) : ⇑{ toOneHom := f, map_mul' := hmul } = ⇑f

@[simp]

theorem AddMonoidHom.coe_mk {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : ZeroHom M N) (hmul : ∀ (x y : M), f.toFun (x + y) = f.toFun x + f.toFun y) : ⇑{ toZeroHom := f, map_add' := hmul } = ⇑f

@[simp]

theorem MonoidHom.toOneHom_coe {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) : ⇑↑f = ⇑f

@[simp]

theorem AddMonoidHom.toZeroHom_coe {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) : ⇑↑f = ⇑f

@[simp]

theorem MonoidHom.toMulHom_coe {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) : (↑f).toFun = ⇑f

@[simp]

theorem AddMonoidHom.toAddHom_coe {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) : (↑f).toFun = ⇑f

theorem MonoidHom.toFun_eq_coe {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) : (↑f).toFun = ⇑f

theorem AddMonoidHom.toFun_eq_coe {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) : (↑f).toFun = ⇑f

theorem ZeroHom.ext {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] ⦃f g : ZeroHom M N⦄ (h : ∀ (x : M), f x = g x) : f = g

theorem OneHom.ext {M : Type u_4} {N : Type u_5} [One M] [One N] ⦃f g : OneHom M N⦄ (h : ∀ (x : M), f x = g x) : f = g

theorem AddHom.ext {M : Type u_4} {N : Type u_5} [Add M] [Add N] ⦃f g : M →ₙ+ N⦄ (h : ∀ (x : M), f x = g x) : f = g

theorem MulHom.ext {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] ⦃f g : M →ₙ* N⦄ (h : ∀ (x : M), f x = g x) : f = g

theorem AddMonoidHom.ext {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] ⦃f g : M →+ N⦄ (h : ∀ (x : M), f x = g x) : f = g

theorem MonoidHom.ext {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] ⦃f g : M →* N⦄ (h : ∀ (x : M), f x = g x) : f = g

def MonoidHom.mk' {M : Type u_4} {G : Type u_7} [Group G] [MulOneClass M] (f : M → G) (map_mul : ∀ (a b : M), f (a * b) = f a * f b) : M →* G

Makes a group homomorphism from a proof that the map preserves multiplication.

Equations

• MonoidHom.mk' f map_mul = { toFun := f, map_one' := ⋯, map_mul' := map_mul }

def AddMonoidHom.mk' {M : Type u_4} {G : Type u_7} [AddGroup G] [AddZeroClass M] (f : M → G) (map_mul : ∀ (a b : M), f (a + b) = f a + f b) : M →+ G

Makes an additive group homomorphism from a proof that the map preserves addition.

Equations

• AddMonoidHom.mk' f map_mul = { toFun := f, map_zero' := ⋯, map_add' := map_mul }

@[simp]

theorem AddMonoidHom.mk'_apply {M : Type u_4} {G : Type u_7} [AddGroup G] [AddZeroClass M] (f : M → G) (map_mul : ∀ (a b : M), f (a + b) = f a + f b) : ⇑(AddMonoidHom.mk' f map_mul) = f

@[simp]

theorem MonoidHom.mk'_apply {M : Type u_4} {G : Type u_7} [Group G] [MulOneClass M] (f : M → G) (map_mul : ∀ (a b : M), f (a * b) = f a * f b) : ⇑(MonoidHom.mk' f map_mul) = f

@[simp]

theorem OneHom.mk_coe {M : Type u_4} {N : Type u_5} [One M] [One N] (f : OneHom M N) (h1 : f 1 = 1) : { toFun := ⇑f, map_one' := h1 } = f

@[simp]

theorem ZeroHom.mk_coe {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] (f : ZeroHom M N) (h1 : f 0 = 0) : { toFun := ⇑f, map_zero' := h1 } = f

@[simp]

theorem MulHom.mk_coe {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M →ₙ* N) (hmul : ∀ (x y : M), f (x * y) = f x * f y) : { toFun := ⇑f, map_mul' := hmul } = f

@[simp]

theorem AddHom.mk_coe {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M →ₙ+ N) (hmul : ∀ (x y : M), f (x + y) = f x + f y) : { toFun := ⇑f, map_add' := hmul } = f

@[simp]

theorem MonoidHom.mk_coe {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) (hmul : ∀ (x y : M), (↑f).toFun (x * y) = (↑f).toFun x * (↑f).toFun y) : { toOneHom := ↑f, map_mul' := hmul } = f

@[simp]

theorem AddMonoidHom.mk_coe {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (hmul : ∀ (x y : M), (↑f).toFun (x + y) = (↑f).toFun x + (↑f).toFun y) : { toZeroHom := ↑f, map_add' := hmul } = f

def OneHom.copy {M : Type u_4} {N : Type u_5} [One M] [One N] (f : OneHom M N) (f' : M → N) (h : f' = ⇑f) : OneHom M N

Copy of a OneHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toFun := f', map_one' := ⋯ }

def ZeroHom.copy {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] (f : ZeroHom M N) (f' : M → N) (h : f' = ⇑f) : ZeroHom M N

Copy of a ZeroHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toFun := f', map_zero' := ⋯ }

@[simp]

theorem OneHom.coe_copy {M : Type u_4} {N : Type u_5} {x✝ : One M} {x✝¹ : One N} (f : OneHom M N) (f' : M → N) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

@[simp]

theorem ZeroHom.coe_copy {M : Type u_4} {N : Type u_5} {x✝ : Zero M} {x✝¹ : Zero N} (f : ZeroHom M N) (f' : M → N) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem OneHom.coe_copy_eq {M : Type u_4} {N : Type u_5} {x✝ : One M} {x✝¹ : One N} (f : OneHom M N) (f' : M → N) (h : f' = ⇑f) : f.copy f' h = f

theorem ZeroHom.coe_copy_eq {M : Type u_4} {N : Type u_5} {x✝ : Zero M} {x✝¹ : Zero N} (f : ZeroHom M N) (f' : M → N) (h : f' = ⇑f) : f.copy f' h = f

def MulHom.copy {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M →ₙ* N) (f' : M → N) (h : f' = ⇑f) : M →ₙ* N

Copy of a MulHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toFun := f', map_mul' := ⋯ }

def AddHom.copy {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M →ₙ+ N) (f' : M → N) (h : f' = ⇑f) : M →ₙ+ N

Copy of an AddHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toFun := f', map_add' := ⋯ }

@[simp]

theorem MulHom.coe_copy {M : Type u_4} {N : Type u_5} {x✝ : Mul M} {x✝¹ : Mul N} (f : M →ₙ* N) (f' : M → N) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

@[simp]

theorem AddHom.coe_copy {M : Type u_4} {N : Type u_5} {x✝ : Add M} {x✝¹ : Add N} (f : M →ₙ+ N) (f' : M → N) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem MulHom.coe_copy_eq {M : Type u_4} {N : Type u_5} {x✝ : Mul M} {x✝¹ : Mul N} (f : M →ₙ* N) (f' : M → N) (h : f' = ⇑f) : f.copy f' h = f

theorem AddHom.coe_copy_eq {M : Type u_4} {N : Type u_5} {x✝ : Add M} {x✝¹ : Add N} (f : M →ₙ+ N) (f' : M → N) (h : f' = ⇑f) : f.copy f' h = f

def MonoidHom.copy {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) (f' : M → N) (h : f' = ⇑f) : M →* N

Copy of a MonoidHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toOneHom := (↑f).copy f' h, map_mul' := ⋯ }

def AddMonoidHom.copy {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (f' : M → N) (h : f' = ⇑f) : M →+ N

Copy of an AddMonoidHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toZeroHom := (↑f).copy f' h, map_add' := ⋯ }

@[simp]

theorem MonoidHom.coe_copy {M : Type u_4} {N : Type u_5} {x✝ : MulOneClass M} {x✝¹ : MulOneClass N} (f : M →* N) (f' : M → N) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

@[simp]

theorem AddMonoidHom.coe_copy {M : Type u_4} {N : Type u_5} {x✝ : AddZeroClass M} {x✝¹ : AddZeroClass N} (f : M →+ N) (f' : M → N) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem MonoidHom.copy_eq {M : Type u_4} {N : Type u_5} {x✝ : MulOneClass M} {x✝¹ : MulOneClass N} (f : M →* N) (f' : M → N) (h : f' = ⇑f) : f.copy f' h = f

theorem AddMonoidHom.copy_eq {M : Type u_4} {N : Type u_5} {x✝ : AddZeroClass M} {x✝¹ : AddZeroClass N} (f : M →+ N) (f' : M → N) (h : f' = ⇑f) : f.copy f' h = f

theorem OneHom.map_one {M : Type u_4} {N : Type u_5} [One M] [One N] (f : OneHom M N) : f 1 = 1

theorem ZeroHom.map_zero {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] (f : ZeroHom M N) : f 0 = 0

theorem MonoidHom.map_one {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) : f 1 = 1

If f is a monoid homomorphism then f 1 = 1.

theorem AddMonoidHom.map_zero {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) : f 0 = 0

If f is an additive monoid homomorphism then f 0 = 0.

theorem MulHom.map_mul {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M →ₙ* N) (a b : M) : f (a * b) = f a * f b

theorem AddHom.map_add {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M →ₙ+ N) (a b : M) : f (a + b) = f a + f b

theorem MonoidHom.map_mul {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) (a b : M) : f (a * b) = f a * f b

If f is a monoid homomorphism then f (a * b) = f a * f b.

theorem AddMonoidHom.map_add {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (a b : M) : f (a + b) = f a + f b

If f is an additive monoid homomorphism then f (a + b) = f a + f b.

theorem MonoidHom.map_exists_right_inv {M : Type u_4} {N : Type u_5} {F : Type u_9} [MulOneClass M] [MulOneClass N] [FunLike F M N] [MonoidHomClass F M N] (f : F) {x : M} (hx : ∃ (y : M), x * y = 1) : ∃ (y : N), f x * y = 1

Given a monoid homomorphism f : M →* N and an element x : M, if x has a right inverse, then f x has a right inverse too. For elements invertible on both sides see IsUnit.map.

theorem AddMonoidHom.map_exists_right_neg {M : Type u_4} {N : Type u_5} {F : Type u_9} [AddZeroClass M] [AddZeroClass N] [FunLike F M N] [AddMonoidHomClass F M N] (f : F) {x : M} (hx : ∃ (y : M), x + y = 0) : ∃ (y : N), f x + y = 0

Given an AddMonoid homomorphism f : M →+ N and an element x : M, if x has a right inverse, then f x has a right inverse too.

theorem MonoidHom.map_exists_left_inv {M : Type u_4} {N : Type u_5} {F : Type u_9} [MulOneClass M] [MulOneClass N] [FunLike F M N] [MonoidHomClass F M N] (f : F) {x : M} (hx : ∃ (y : M), y * x = 1) : ∃ (y : N), y * f x = 1

Given a monoid homomorphism f : M →* N and an element x : M, if x has a left inverse, then f x has a left inverse too. For elements invertible on both sides see IsUnit.map.

theorem AddMonoidHom.map_exists_left_neg {M : Type u_4} {N : Type u_5} {F : Type u_9} [AddZeroClass M] [AddZeroClass N] [FunLike F M N] [AddMonoidHomClass F M N] (f : F) {x : M} (hx : ∃ (y : M), y + x = 0) : ∃ (y : N), y + f x = 0

Given an AddMonoid homomorphism f : M →+ N and an element x : M, if x has a left inverse, then f x has a left inverse too. For elements invertible on both sides see IsAddUnit.map.

def OneHom.id (M : Type u_10) [One M] : OneHom M M

The identity map from a type with 1 to itself.

Equations

• OneHom.id M = { toFun := fun (x : M) => x, map_one' := ⋯ }

def ZeroHom.id (M : Type u_10) [Zero M] : ZeroHom M M

The identity map from a type with zero to itself.

Equations

• ZeroHom.id M = { toFun := fun (x : M) => x, map_zero' := ⋯ }

@[simp]

theorem ZeroHom.id_apply (M : Type u_10) [Zero M] (x : M) : (ZeroHom.id M) x = x

@[simp]

theorem OneHom.id_apply (M : Type u_10) [One M] (x : M) : (OneHom.id M) x = x

def MulHom.id (M : Type u_10) [Mul M] : M →ₙ* M

The identity map from a type with multiplication to itself.

Equations

• MulHom.id M = { toFun := fun (x : M) => x, map_mul' := ⋯ }

def AddHom.id (M : Type u_10) [Add M] : M →ₙ+ M

The identity map from a type with addition to itself.

Equations

• AddHom.id M = { toFun := fun (x : M) => x, map_add' := ⋯ }

@[simp]

theorem AddHom.id_apply (M : Type u_10) [Add M] (x : M) : (AddHom.id M) x = x

@[simp]

theorem MulHom.id_apply (M : Type u_10) [Mul M] (x : M) : (MulHom.id M) x = x

def MonoidHom.id (M : Type u_10) [MulOneClass M] : M →* M

The identity map from a monoid to itself.

Equations

• MonoidHom.id M = { toFun := fun (x : M) => x, map_one' := ⋯, map_mul' := ⋯ }

def AddMonoidHom.id (M : Type u_10) [AddZeroClass M] : M →+ M

The identity map from an additive monoid to itself.

Equations

• AddMonoidHom.id M = { toFun := fun (x : M) => x, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem AddMonoidHom.id_apply (M : Type u_10) [AddZeroClass M] (x : M) : (AddMonoidHom.id M) x = x

@[simp]

theorem MonoidHom.id_apply (M : Type u_10) [MulOneClass M] (x : M) : (MonoidHom.id M) x = x

def OneHom.comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [One M] [One N] [One P] (hnp : OneHom N P) (hmn : OneHom M N) : OneHom M P

Composition of OneHoms as a OneHom.

Equations

• hnp.comp hmn = { toFun := ⇑hnp ∘ ⇑hmn, map_one' := ⋯ }

def ZeroHom.comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [Zero M] [Zero N] [Zero P] (hnp : ZeroHom N P) (hmn : ZeroHom M N) : ZeroHom M P

Composition of ZeroHoms as a ZeroHom.

Equations

• hnp.comp hmn = { toFun := ⇑hnp ∘ ⇑hmn, map_zero' := ⋯ }

def MulHom.comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [Mul M] [Mul N] [Mul P] (hnp : N →ₙ* P) (hmn : M →ₙ* N) : M →ₙ* P

Composition of MulHoms as a MulHom.

Equations

• hnp.comp hmn = { toFun := ⇑hnp ∘ ⇑hmn, map_mul' := ⋯ }

def AddHom.comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [Add M] [Add N] [Add P] (hnp : N →ₙ+ P) (hmn : M →ₙ+ N) : M →ₙ+ P

Composition of AddHoms as an AddHom.

Equations

• hnp.comp hmn = { toFun := ⇑hnp ∘ ⇑hmn, map_add' := ⋯ }

def MonoidHom.comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [MulOneClass M] [MulOneClass N] [MulOneClass P] (hnp : N →* P) (hmn : M →* N) : M →* P

Composition of monoid morphisms as a monoid morphism.

Equations

• hnp.comp hmn = { toFun := ⇑hnp ∘ ⇑hmn, map_one' := ⋯, map_mul' := ⋯ }

def AddMonoidHom.comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] (hnp : N →+ P) (hmn : M →+ N) : M →+ P

Composition of additive monoid morphisms as an additive monoid morphism.

Equations

• hnp.comp hmn = { toFun := ⇑hnp ∘ ⇑hmn, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem OneHom.coe_comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [One M] [One N] [One P] (g : OneHom N P) (f : OneHom M N) : ⇑(g.comp f) = ⇑g ∘ ⇑f

@[simp]

theorem ZeroHom.coe_comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [Zero M] [Zero N] [Zero P] (g : ZeroHom N P) (f : ZeroHom M N) : ⇑(g.comp f) = ⇑g ∘ ⇑f

@[simp]

theorem MulHom.coe_comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [Mul M] [Mul N] [Mul P] (g : N →ₙ* P) (f : M →ₙ* N) : ⇑(g.comp f) = ⇑g ∘ ⇑f

@[simp]

theorem AddHom.coe_comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [Add M] [Add N] [Add P] (g : N →ₙ+ P) (f : M →ₙ+ N) : ⇑(g.comp f) = ⇑g ∘ ⇑f

@[simp]

theorem MonoidHom.coe_comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [MulOneClass M] [MulOneClass N] [MulOneClass P] (g : N →* P) (f : M →* N) : ⇑(g.comp f) = ⇑g ∘ ⇑f

@[simp]

theorem AddMonoidHom.coe_comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] (g : N →+ P) (f : M →+ N) : ⇑(g.comp f) = ⇑g ∘ ⇑f

theorem OneHom.comp_apply {M : Type u_4} {N : Type u_5} {P : Type u_6} [One M] [One N] [One P] (g : OneHom N P) (f : OneHom M N) (x : M) : (g.comp f) x = g (f x)

theorem ZeroHom.comp_apply {M : Type u_4} {N : Type u_5} {P : Type u_6} [Zero M] [Zero N] [Zero P] (g : ZeroHom N P) (f : ZeroHom M N) (x : M) : (g.comp f) x = g (f x)

theorem MulHom.comp_apply {M : Type u_4} {N : Type u_5} {P : Type u_6} [Mul M] [Mul N] [Mul P] (g : N →ₙ* P) (f : M →ₙ* N) (x : M) : (g.comp f) x = g (f x)

theorem AddHom.comp_apply {M : Type u_4} {N : Type u_5} {P : Type u_6} [Add M] [Add N] [Add P] (g : N →ₙ+ P) (f : M →ₙ+ N) (x : M) : (g.comp f) x = g (f x)

theorem MonoidHom.comp_apply {M : Type u_4} {N : Type u_5} {P : Type u_6} [MulOneClass M] [MulOneClass N] [MulOneClass P] (g : N →* P) (f : M →* N) (x : M) : (g.comp f) x = g (f x)

theorem AddMonoidHom.comp_apply {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] (g : N →+ P) (f : M →+ N) (x : M) : (g.comp f) x = g (f x)

theorem OneHom.comp_assoc {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_10} [One M] [One N] [One P] [One Q] (f : OneHom M N) (g : OneHom N P) (h : OneHom P Q) : (h.comp g).comp f = h.comp (g.comp f)

Composition of monoid homomorphisms is associative.

theorem ZeroHom.comp_assoc {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_10} [Zero M] [Zero N] [Zero P] [Zero Q] (f : ZeroHom M N) (g : ZeroHom N P) (h : ZeroHom P Q) : (h.comp g).comp f = h.comp (g.comp f)

Composition of additive monoid homomorphisms is associative.

theorem MulHom.comp_assoc {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_10} [Mul M] [Mul N] [Mul P] [Mul Q] (f : M →ₙ* N) (g : N →ₙ* P) (h : P →ₙ* Q) : (h.comp g).comp f = h.comp (g.comp f)

theorem AddHom.comp_assoc {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_10} [Add M] [Add N] [Add P] [Add Q] (f : M →ₙ+ N) (g : N →ₙ+ P) (h : P →ₙ+ Q) : (h.comp g).comp f = h.comp (g.comp f)

theorem MonoidHom.comp_assoc {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_10} [MulOneClass M] [MulOneClass N] [MulOneClass P] [MulOneClass Q] (f : M →* N) (g : N →* P) (h : P →* Q) : (h.comp g).comp f = h.comp (g.comp f)

theorem AddMonoidHom.comp_assoc {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_10} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] [AddZeroClass Q] (f : M →+ N) (g : N →+ P) (h : P →+ Q) : (h.comp g).comp f = h.comp (g.comp f)

theorem OneHom.cancel_right {M : Type u_4} {N : Type u_5} {P : Type u_6} [One M] [One N] [One P] {g₁ g₂ : OneHom N P} {f : OneHom M N} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

theorem ZeroHom.cancel_right {M : Type u_4} {N : Type u_5} {P : Type u_6} [Zero M] [Zero N] [Zero P] {g₁ g₂ : ZeroHom N P} {f : ZeroHom M N} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

theorem MulHom.cancel_right {M : Type u_4} {N : Type u_5} {P : Type u_6} [Mul M] [Mul N] [Mul P] {g₁ g₂ : N →ₙ* P} {f : M →ₙ* N} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

theorem AddHom.cancel_right {M : Type u_4} {N : Type u_5} {P : Type u_6} [Add M] [Add N] [Add P] {g₁ g₂ : N →ₙ+ P} {f : M →ₙ+ N} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

theorem MonoidHom.cancel_right {M : Type u_4} {N : Type u_5} {P : Type u_6} [MulOneClass M] [MulOneClass N] [MulOneClass P] {g₁ g₂ : N →* P} {f : M →* N} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

theorem AddMonoidHom.cancel_right {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] {g₁ g₂ : N →+ P} {f : M →+ N} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

theorem OneHom.cancel_left {M : Type u_4} {N : Type u_5} {P : Type u_6} [One M] [One N] [One P] {g : OneHom N P} {f₁ f₂ : OneHom M N} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

theorem ZeroHom.cancel_left {M : Type u_4} {N : Type u_5} {P : Type u_6} [Zero M] [Zero N] [Zero P] {g : ZeroHom N P} {f₁ f₂ : ZeroHom M N} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

theorem MulHom.cancel_left {M : Type u_4} {N : Type u_5} {P : Type u_6} [Mul M] [Mul N] [Mul P] {g : N →ₙ* P} {f₁ f₂ : M →ₙ* N} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

theorem AddHom.cancel_left {M : Type u_4} {N : Type u_5} {P : Type u_6} [Add M] [Add N] [Add P] {g : N →ₙ+ P} {f₁ f₂ : M →ₙ+ N} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

theorem MonoidHom.cancel_left {M : Type u_4} {N : Type u_5} {P : Type u_6} [MulOneClass M] [MulOneClass N] [MulOneClass P] {g : N →* P} {f₁ f₂ : M →* N} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

theorem AddMonoidHom.cancel_left {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] {g : N →+ P} {f₁ f₂ : M →+ N} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

theorem MonoidHom.toOneHom_injective {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] : Function.Injective MonoidHom.toOneHom

theorem AddMonoidHom.toZeroHom_injective {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] : Function.Injective AddMonoidHom.toZeroHom

theorem MonoidHom.toMulHom_injective {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] : Function.Injective MonoidHom.toMulHom

theorem AddMonoidHom.toAddHom_injective {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] : Function.Injective AddMonoidHom.toAddHom

@[simp]

theorem OneHom.comp_id {M : Type u_4} {N : Type u_5} [One M] [One N] (f : OneHom M N) : f.comp (OneHom.id M) = f

@[simp]

theorem ZeroHom.comp_id {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] (f : ZeroHom M N) : f.comp (ZeroHom.id M) = f

@[simp]

theorem MulHom.comp_id {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M →ₙ* N) : f.comp (MulHom.id M) = f

@[simp]

theorem AddHom.comp_id {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M →ₙ+ N) : f.comp (AddHom.id M) = f

@[simp]

theorem MonoidHom.comp_id {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) : f.comp (MonoidHom.id M) = f

@[simp]

theorem AddMonoidHom.comp_id {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) : f.comp (AddMonoidHom.id M) = f

@[simp]

theorem OneHom.id_comp {M : Type u_4} {N : Type u_5} [One M] [One N] (f : OneHom M N) : (OneHom.id N).comp f = f

@[simp]

theorem ZeroHom.id_comp {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] (f : ZeroHom M N) : (ZeroHom.id N).comp f = f

@[simp]

theorem MulHom.id_comp {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M →ₙ* N) : (MulHom.id N).comp f = f

@[simp]

theorem AddHom.id_comp {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M →ₙ+ N) : (AddHom.id N).comp f = f

@[simp]

theorem MonoidHom.id_comp {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (f : M →* N) : (MonoidHom.id N).comp f = f

@[simp]

theorem AddMonoidHom.id_comp {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) : (AddMonoidHom.id N).comp f = f

theorem MonoidHom.map_pow {M : Type u_4} {N : Type u_5} [Monoid M] [Monoid N] (f : M →* N) (a : M) (n : ℕ) : f (a ^ n) = f a ^ n

theorem AddMonoidHom.map_nsmul {M : Type u_4} {N : Type u_5} [AddMonoid M] [AddMonoid N] (f : M →+ N) (a : M) (n : ℕ) : f (n • a) = n • f a

theorem MonoidHom.map_zpow' {M : Type u_4} {N : Type u_5} [DivInvMonoid M] [DivInvMonoid N] (f : M →* N) (hf : ∀ (x : M), f x⁻¹ = (f x)⁻¹) (a : M) (n : ℤ) : f (a ^ n) = f a ^ n

theorem AddMonoidHom.map_zsmul' {M : Type u_4} {N : Type u_5} [SubNegMonoid M] [SubNegMonoid N] (f : M →+ N) (hf : ∀ (x : M), f (-x) = -f x) (a : M) (n : ℤ) : f (n • a) = n • f a

def OneHom.inverse {M : Type u_4} {N : Type u_5} [One M] [One N] (f : OneHom M N) (g : N → M) (h₁ : Function.LeftInverse g ⇑f) : OneHom N M

Makes a OneHom inverse from the bijective inverse of a OneHom

Equations

• f.inverse g h₁ = { toFun := g, map_one' := ⋯ }

def ZeroHom.inverse {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] (f : ZeroHom M N) (g : N → M) (h₁ : Function.LeftInverse g ⇑f) : ZeroHom N M

Make a ZeroHom inverse from the bijective inverse of a ZeroHom

Equations

• f.inverse g h₁ = { toFun := g, map_zero' := ⋯ }

@[simp]

theorem OneHom.inverse_apply {M : Type u_4} {N : Type u_5} [One M] [One N] (f : OneHom M N) (g : N → M) (h₁ : Function.LeftInverse g ⇑f) (a✝ : N) : (f.inverse g h₁) a✝ = g a✝

@[simp]

theorem ZeroHom.inverse_apply {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] (f : ZeroHom M N) (g : N → M) (h₁ : Function.LeftInverse g ⇑f) (a✝ : N) : (f.inverse g h₁) a✝ = g a✝

def MulHom.inverse {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M →ₙ* N) (g : N → M) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) : N →ₙ* M

Makes a multiplicative inverse from a bijection which preserves multiplication.

Equations

• f.inverse g h₁ h₂ = { toFun := g, map_mul' := ⋯ }

def AddHom.inverse {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M →ₙ+ N) (g : N → M) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) : N →ₙ+ M

Makes an additive inverse from a bijection which preserves addition.

Equations

• f.inverse g h₁ h₂ = { toFun := g, map_add' := ⋯ }

@[simp]

theorem MulHom.inverse_apply {M : Type u_4} {N : Type u_5} [Mul M] [Mul N] (f : M →ₙ* N) (g : N → M) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) (a✝ : N) : (f.inverse g h₁ h₂) a✝ = g a✝

@[simp]

theorem AddHom.inverse_apply {M : Type u_4} {N : Type u_5} [Add M] [Add N] (f : M →ₙ+ N) (g : N → M) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) (a✝ : N) : (f.inverse g h₁ h₂) a✝ = g a✝

def MonoidHom.inverse {A : Type u_10} {B : Type u_11} [Monoid A] [Monoid B] (f : A →* B) (g : B → A) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) : B →* A

The inverse of a bijective MonoidHom is a MonoidHom.

Equations

• f.inverse g h₁ h₂ = { toFun := g, map_one' := ⋯, map_mul' := ⋯ }

def AddMonoidHom.inverse {A : Type u_10} {B : Type u_11} [AddMonoid A] [AddMonoid B] (f : A →+ B) (g : B → A) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) : B →+ A

The inverse of a bijective AddMonoidHom is an AddMonoidHom.

Equations

• f.inverse g h₁ h₂ = { toFun := g, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem MonoidHom.inverse_apply {A : Type u_10} {B : Type u_11} [Monoid A] [Monoid B] (f : A →* B) (g : B → A) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) (a✝ : B) : (f.inverse g h₁ h₂) a✝ = g a✝

@[simp]

theorem AddMonoidHom.inverse_apply {A : Type u_10} {B : Type u_11} [AddMonoid A] [AddMonoid B] (f : A →+ B) (g : B → A) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) (a✝ : B) : (f.inverse g h₁ h₂) a✝ = g a✝

def Monoid.End (M : Type u_4) [MulOneClass M] : Type u_4

The monoid of endomorphisms.

Equations

• Monoid.End M = (M →* M)

instance Monoid.End.instFunLike (M : Type u_4) [MulOneClass M] : FunLike (Monoid.End M) M M

Equations

• Monoid.End.instFunLike M = MonoidHom.instFunLike

theorem Monoid.End.instMonoidHomClass (M : Type u_4) [MulOneClass M] : MonoidHomClass (Monoid.End M) M M

instance Monoid.End.instOne (M : Type u_4) [MulOneClass M] : One (Monoid.End M)

Equations

• Monoid.End.instOne M = { one := MonoidHom.id M }

instance Monoid.End.instMul (M : Type u_4) [MulOneClass M] : Mul (Monoid.End M)

Equations

• Monoid.End.instMul M = { mul := MonoidHom.comp }

instance Monoid.End.inst (M : Type u_4) [MulOneClass M] : Monoid (Monoid.End M)

Equations

• Monoid.End.inst M = Monoid.mk ⋯ ⋯ (fun (n : ℕ) (f : Monoid.End M) => MonoidHom.copy (npowRec n f) (⇑f)^[n] ⋯) ⋯ ⋯

instance Monoid.End.instInhabited (M : Type u_4) [MulOneClass M] : Inhabited (Monoid.End M)

Equations

• Monoid.End.instInhabited M = { default := 1 }

@[simp]

theorem Monoid.End.coe_pow (M : Type u_4) [MulOneClass M] (f : Monoid.End M) (n : ℕ) : ⇑(f ^ n) = (⇑f)^[n]

@[simp]

theorem Monoid.coe_one (M : Type u_4) [MulOneClass M] : ⇑1 = id

@[simp]

theorem Monoid.coe_mul (M : Type u_4) [MulOneClass M] (f g : Monoid.End M) : ⇑(f * g) = ⇑f ∘ ⇑g

def AddMonoid.End (A : Type u_10) [AddZeroClass A] : Type u_10

The monoid of endomorphisms.

Equations

• AddMonoid.End A = (A →+ A)

instance AddMonoid.End.instFunLike (A : Type u_10) [AddZeroClass A] : FunLike (AddMonoid.End A) A A

Equations

• AddMonoid.End.instFunLike A = AddMonoidHom.instFunLike

theorem AddMonoid.End.instAddMonoidHomClass (A : Type u_10) [AddZeroClass A] : AddMonoidHomClass (AddMonoid.End A) A A

instance AddMonoid.End.instOne (A : Type u_10) [AddZeroClass A] : One (AddMonoid.End A)

Equations

• AddMonoid.End.instOne A = { one := AddMonoidHom.id A }

instance AddMonoid.End.instMul (A : Type u_10) [AddZeroClass A] : Mul (AddMonoid.End A)

Equations

• AddMonoid.End.instMul A = { mul := AddMonoidHom.comp }

@[simp]

theorem AddMonoid.End.coe_one (A : Type u_10) [AddZeroClass A] : ⇑1 = id

@[simp]

theorem AddMonoid.End.coe_mul (A : Type u_10) [AddZeroClass A] (f g : AddMonoid.End A) : ⇑(f * g) = ⇑f ∘ ⇑g

instance AddMonoid.End.monoid (A : Type u_10) [AddZeroClass A] : Monoid (AddMonoid.End A)

Equations

• AddMonoid.End.monoid A = Monoid.mk ⋯ ⋯ (fun (n : ℕ) (f : AddMonoid.End A) => AddMonoidHom.copy (npowRec n f) (⇑f)^[n] ⋯) ⋯ ⋯

@[simp]

theorem AddMonoid.End.coe_pow (A : Type u_10) [AddZeroClass A] (f : AddMonoid.End A) (n : ℕ) : ⇑(f ^ n) = (⇑f)^[n]

instance AddMonoid.End.instInhabited (A : Type u_10) [AddZeroClass A] : Inhabited (AddMonoid.End A)

Equations

• AddMonoid.End.instInhabited A = { default := 1 }

instance instOneOneHom {M : Type u_4} {N : Type u_5} [One M] [One N] : One (OneHom M N)

1 is the homomorphism sending all elements to 1.

Equations

• instOneOneHom = { one := { toFun := fun (x : M) => 1, map_one' := ⋯ } }

instance instZeroZeroHom {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] : Zero (ZeroHom M N)

0 is the homomorphism sending all elements to 0.

Equations

• instZeroZeroHom = { zero := { toFun := fun (x : M) => 0, map_zero' := ⋯ } }

instance instOneMulHom {M : Type u_4} {N : Type u_5} [Mul M] [MulOneClass N] : One (M →ₙ* N)

1 is the multiplicative homomorphism sending all elements to 1.

Equations

• instOneMulHom = { one := { toFun := fun (x : M) => 1, map_mul' := ⋯ } }

instance instZeroAddHom {M : Type u_4} {N : Type u_5} [Add M] [AddZeroClass N] : Zero (M →ₙ+ N)

0 is the additive homomorphism sending all elements to 0

Equations

• instZeroAddHom = { zero := { toFun := fun (x : M) => 0, map_add' := ⋯ } }

instance instOneMonoidHom {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] : One (M →* N)

1 is the monoid homomorphism sending all elements to 1.

Equations

• instOneMonoidHom = { one := { toFun := fun (x : M) => 1, map_one' := ⋯, map_mul' := ⋯ } }

instance instZeroAddMonoidHom {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] : Zero (M →+ N)

0 is the additive monoid homomorphism sending all elements to 0.

Equations

• instZeroAddMonoidHom = { zero := { toFun := fun (x : M) => 0, map_zero' := ⋯, map_add' := ⋯ } }

@[simp]

theorem OneHom.one_apply {M : Type u_4} {N : Type u_5} [One M] [One N] (x : M) : 1 x = 1

@[simp]

theorem ZeroHom.zero_apply {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] (x : M) : 0 x = 0

@[simp]

theorem MonoidHom.one_apply {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] (x : M) : 1 x = 1

@[simp]

theorem AddMonoidHom.zero_apply {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] (x : M) : 0 x = 0

@[simp]

theorem OneHom.one_comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [One M] [One N] [One P] (f : OneHom M N) : OneHom.comp 1 f = 1

@[simp]

theorem ZeroHom.zero_comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [Zero M] [Zero N] [Zero P] (f : ZeroHom M N) : ZeroHom.comp 0 f = 0

@[simp]

theorem OneHom.comp_one {M : Type u_4} {N : Type u_5} {P : Type u_6} [One M] [One N] [One P] (f : OneHom N P) : f.comp 1 = 1

@[simp]

theorem ZeroHom.comp_zero {M : Type u_4} {N : Type u_5} {P : Type u_6} [Zero M] [Zero N] [Zero P] (f : ZeroHom N P) : f.comp 0 = 0

instance instInhabitedOneHom {M : Type u_4} {N : Type u_5} [One M] [One N] : Inhabited (OneHom M N)

Equations

• instInhabitedOneHom = { default := 1 }

instance instInhabitedZeroHom {M : Type u_4} {N : Type u_5} [Zero M] [Zero N] : Inhabited (ZeroHom M N)

Equations

• instInhabitedZeroHom = { default := 0 }

instance instInhabitedMulHom {M : Type u_4} {N : Type u_5} [Mul M] [MulOneClass N] : Inhabited (M →ₙ* N)

Equations

• instInhabitedMulHom = { default := 1 }

instance instInhabitedAddHom {M : Type u_4} {N : Type u_5} [Add M] [AddZeroClass N] : Inhabited (M →ₙ+ N)

Equations

• instInhabitedAddHom = { default := 0 }

instance instInhabitedMonoidHom {M : Type u_4} {N : Type u_5} [MulOneClass M] [MulOneClass N] : Inhabited (M →* N)

Equations

• instInhabitedMonoidHom = { default := 1 }

instance instInhabitedAddMonoidHom {M : Type u_4} {N : Type u_5} [AddZeroClass M] [AddZeroClass N] : Inhabited (M →+ N)

Equations

• instInhabitedAddMonoidHom = { default := 0 }

@[simp]

theorem MonoidHom.one_comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [MulOneClass M] [MulOneClass N] [MulOneClass P] (f : M →* N) : MonoidHom.comp 1 f = 1

@[simp]

theorem AddMonoidHom.zero_comp {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] (f : M →+ N) : AddMonoidHom.comp 0 f = 0

@[simp]

theorem MonoidHom.comp_one {M : Type u_4} {N : Type u_5} {P : Type u_6} [MulOneClass M] [MulOneClass N] [MulOneClass P] (f : N →* P) : f.comp 1 = 1

@[simp]

theorem AddMonoidHom.comp_zero {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddZeroClass M] [AddZeroClass N] [AddZeroClass P] (f : N →+ P) : f.comp 0 = 0

theorem MonoidHom.map_inv {α : Type u_2} {β : Type u_3} [Group α] [DivisionMonoid β] (f : α →* β) (a : α) : f a⁻¹ = (f a)⁻¹

Group homomorphisms preserve inverse.

theorem AddMonoidHom.map_neg {α : Type u_2} {β : Type u_3} [AddGroup α] [SubtractionMonoid β] (f : α →+ β) (a : α) : f (-a) = -f a

Additive group homomorphisms preserve negation.

theorem MonoidHom.map_zpow {α : Type u_2} {β : Type u_3} [Group α] [DivisionMonoid β] (f : α →* β) (g : α) (n : ℤ) : f (g ^ n) = f g ^ n

Group homomorphisms preserve integer power.

theorem AddMonoidHom.map_zsmul {α : Type u_2} {β : Type u_3} [AddGroup α] [SubtractionMonoid β] (f : α →+ β) (g : α) (n : ℤ) : f (n • g) = n • f g

Additive group homomorphisms preserve integer scaling.

theorem MonoidHom.map_div {α : Type u_2} {β : Type u_3} [Group α] [DivisionMonoid β] (f : α →* β) (g h : α) : f (g / h) = f g / f h

Group homomorphisms preserve division.

theorem AddMonoidHom.map_sub {α : Type u_2} {β : Type u_3} [AddGroup α] [SubtractionMonoid β] (f : α →+ β) (g h : α) : f (g - h) = f g - f h

Additive group homomorphisms preserve subtraction.

theorem MonoidHom.map_mul_inv {α : Type u_2} {β : Type u_3} [Group α] [DivisionMonoid β] (f : α →* β) (g h : α) : f (g * h⁻¹) = f g * (f h)⁻¹

Group homomorphisms preserve division.

theorem AddMonoidHom.map_add_neg {α : Type u_2} {β : Type u_3} [AddGroup α] [SubtractionMonoid β] (f : α →+ β) (g h : α) : f (g + -h) = f g + -f h

Additive group homomorphisms preserve subtraction.