Definitions of group actions

This file defines a hierarchy of group action type-classes on top of the previously defined notation classes SMul and its additive version VAdd:

• MulAction M α and its additive version AddAction G P are typeclasses used for actions of multiplicative and additive monoids and groups; they extend notation classes SMul and VAdd that are defined in Algebra.Group.Defs;
• DistribMulAction M A is a typeclass for an action of a multiplicative monoid on an additive monoid such that a • (b + c) = a • b + a • c and a • 0 = 0.

The hierarchy is extended further by Module, defined elsewhere.

Also provided are typeclasses for faithful and transitive actions, and typeclasses regarding the interaction of different group actions,

• SMulCommClass M N α and its additive version VAddCommClass M N α;
• IsScalarTower M N α and its additive version VAddAssocClass M N α;
• IsCentralScalar M α and its additive version IsCentralVAdd M N α.

Notation

• a • b is used as notation for SMul.smul a b.
• a +ᵥ b is used as notation for VAdd.vadd a b.

Implementation details

This file should avoid depending on other parts of GroupTheory, to avoid import cycles. More sophisticated lemmas belong in GroupTheory.GroupAction.

Tags

group action

Faithful actions

class FaithfulVAdd (G : Type u_11) (P : Type u_12) [VAdd G P] : Prop

Typeclass for faithful actions.

• eq_of_vadd_eq_vadd : ∀ {g₁ g₂ : G}, (∀ (p : P), g₁ +ᵥ p = g₂ +ᵥ p) → g₁ = g₂

  Two elements g₁ and g₂ are equal whenever they act in the same way on all points.

theorem FaithfulVAdd.eq_of_vadd_eq_vadd {G : Type u_11} {P : Type u_12} : ∀ {inst : VAdd G P} [self : FaithfulVAdd G P] {g₁ g₂ : G}, (∀ (p : P), g₁ +ᵥ p = g₂ +ᵥ p) → g₁ = g₂

Two elements g₁ and g₂ are equal whenever they act in the same way on all points.

class FaithfulSMul (M : Type u_11) (α : Type u_12) [SMul M α] : Prop

Typeclass for faithful actions.

• eq_of_smul_eq_smul : ∀ {m₁ m₂ : M}, (∀ (a : α), m₁ • a = m₂ • a) → m₁ = m₂

  Two elements m₁ and m₂ are equal whenever they act in the same way on all points.

theorem FaithfulSMul.eq_of_smul_eq_smul {M : Type u_11} {α : Type u_12} : ∀ {inst : SMul M α} [self : FaithfulSMul M α] {m₁ m₂ : M}, (∀ (a : α), m₁ • a = m₂ • a) → m₁ = m₂

Two elements m₁ and m₂ are equal whenever they act in the same way on all points.

theorem vadd_left_injective' {M : Type u_1} {α : Type u_7} [VAdd M α] [FaithfulVAdd M α] : Function.Injective fun (x1 : M) (x2 : α) => x1 +ᵥ x2

theorem smul_left_injective' {M : Type u_1} {α : Type u_7} [SMul M α] [FaithfulSMul M α] : Function.Injective fun (x1 : M) (x2 : α) => x1 • x2

@[instance 910]

instance Add.toVAdd (α : Type u_11) [Add α] : VAdd α α

See also AddMonoid.toAddAction

Equations

• Add.toVAdd α = { vadd := fun (x1 x2 : α) => x1 + x2 }

@[instance 910]

instance Mul.toSMul (α : Type u_11) [Mul α] : SMul α α

See also Monoid.toMulAction and MulZeroClass.toSMulWithZero.

Equations

• Mul.toSMul α = { smul := fun (x1 x2 : α) => x1 * x2 }

@[simp]

theorem vadd_eq_add (α : Type u_11) [Add α] {a : α} {a' : α} : a +ᵥ a' = a + a'

@[simp]

theorem smul_eq_mul (α : Type u_11) [Mul α] {a : α} {a' : α} : a • a' = a * a'

instance AddRightCancelMonoid.faithfulVAdd {α : Type u_7} [AddRightCancelMonoid α] : FaithfulVAdd α α

AddMonoid.toAddAction is faithful on additive cancellative monoids.

Equations

• ⋯ = ⋯

instance RightCancelMonoid.faithfulSMul {α : Type u_7} [RightCancelMonoid α] : FaithfulSMul α α

Monoid.toMulAction is faithful on cancellative monoids.

Equations

• ⋯ = ⋯

class AddAction (G : Type u_11) (P : Type u_12) [AddMonoid G] extends VAdd : Type (max u_11 u_12)

Type class for additive monoid actions.

• vadd : G → P → P
• zero_vadd : ∀ (p : P), 0 +ᵥ p = p

  Zero is a neutral element for +ᵥ

• add_vadd : ∀ (g₁ g₂ : G) (p : P), g₁ + g₂ +ᵥ p = g₁ +ᵥ (g₂ +ᵥ p)

  Associativity of + and +ᵥ

theorem AddAction.zero_vadd {G : Type u_11} {P : Type u_12} : ∀ {inst : AddMonoid G} [self : AddAction G P] (p : P), 0 +ᵥ p = p

Zero is a neutral element for +ᵥ

theorem AddAction.add_vadd {G : Type u_11} {P : Type u_12} : ∀ {inst : AddMonoid G} [self : AddAction G P] (g₁ g₂ : G) (p : P), g₁ + g₂ +ᵥ p = g₁ +ᵥ (g₂ +ᵥ p)

Associativity of + and +ᵥ

theorem AddAction.ext_iff {G : Type u_11} {P : Type u_12} : ∀ {inst : AddMonoid G} {x y : AddAction G P}, x = y ↔ VAdd.vadd = VAdd.vadd

theorem MulAction.ext_iff {α : Type u_11} {β : Type u_12} : ∀ {inst : Monoid α} {x y : MulAction α β}, x = y ↔ SMul.smul = SMul.smul

theorem AddAction.ext {G : Type u_11} {P : Type u_12} : ∀ {inst : AddMonoid G} {x y : AddAction G P}, VAdd.vadd = VAdd.vadd → x = y

theorem MulAction.ext {α : Type u_11} {β : Type u_12} : ∀ {inst : Monoid α} {x y : MulAction α β}, SMul.smul = SMul.smul → x = y

class MulAction (α : Type u_11) (β : Type u_12) [Monoid α] extends SMul : Type (max u_11 u_12)

Typeclass for multiplicative actions by monoids. This generalizes group actions.

• smul : α → β → β
• one_smul : ∀ (b : β), 1 • b = b

  One is the neutral element for •

• mul_smul : ∀ (x y : α) (b : β), (x * y) • b = x • y • b

  Associativity of • and *

theorem MulAction.one_smul {α : Type u_11} {β : Type u_12} : ∀ {inst : Monoid α} [self : MulAction α β] (b : β), 1 • b = b

One is the neutral element for •

theorem MulAction.mul_smul {α : Type u_11} {β : Type u_12} : ∀ {inst : Monoid α} [self : MulAction α β] (x y : α) (b : β), (x * y) • b = x • y • b

Associativity of • and *

(Pre)transitive action

M acts pretransitively on α if for any x y there is g such that g • x = y (or g +ᵥ x = y for an additive action). A transitive action should furthermore have α nonempty.

In this section we define typeclasses MulAction.IsPretransitive and AddAction.IsPretransitive and provide MulAction.exists_smul_eq/AddAction.exists_vadd_eq, MulAction.surjective_smul/AddAction.surjective_vadd as public interface to access this property. We do not provide typeclasses *Action.IsTransitive; users should assume [MulAction.IsPretransitive M α] [Nonempty α] instead.

class AddAction.IsPretransitive (M : Type u_11) (α : Type u_12) [VAdd M α] : Prop

M acts pretransitively on α if for any x y there is g such that g +ᵥ x = y. A transitive action should furthermore have α nonempty.

• exists_vadd_eq : ∀ (x y : α), ∃ (g : M), g +ᵥ x = y

  There is g such that g +ᵥ x = y.

theorem AddAction.IsPretransitive.exists_vadd_eq {M : Type u_11} {α : Type u_12} : ∀ {inst : VAdd M α} [self : AddAction.IsPretransitive M α] (x y : α), ∃ (g : M), g +ᵥ x = y

There is g such that g +ᵥ x = y.

class MulAction.IsPretransitive (M : Type u_11) (α : Type u_12) [SMul M α] : Prop

M acts pretransitively on α if for any x y there is g such that g • x = y. A transitive action should furthermore have α nonempty.

• exists_smul_eq : ∀ (x y : α), ∃ (g : M), g • x = y

  There is g such that g • x = y.

theorem MulAction.IsPretransitive.exists_smul_eq {M : Type u_11} {α : Type u_12} : ∀ {inst : SMul M α} [self : MulAction.IsPretransitive M α] (x y : α), ∃ (g : M), g • x = y

There is g such that g • x = y.

theorem AddAction.exists_vadd_eq (M : Type u_1) {α : Type u_7} [VAdd M α] [AddAction.IsPretransitive M α] (x : α) (y : α) : ∃ (m : M), m +ᵥ x = y

theorem MulAction.exists_smul_eq (M : Type u_1) {α : Type u_7} [SMul M α] [MulAction.IsPretransitive M α] (x : α) (y : α) : ∃ (m : M), m • x = y

theorem AddAction.surjective_vadd (M : Type u_1) {α : Type u_7} [VAdd M α] [AddAction.IsPretransitive M α] (x : α) : Function.Surjective fun (c : M) => c +ᵥ x

theorem MulAction.surjective_smul (M : Type u_1) {α : Type u_7} [SMul M α] [MulAction.IsPretransitive M α] (x : α) : Function.Surjective fun (c : M) => c • x

instance AddAction.Regular.isPretransitive {G : Type u_3} [AddGroup G] : AddAction.IsPretransitive G G

The regular action of a group on itself is transitive.

Equations

• ⋯ = ⋯

instance MulAction.Regular.isPretransitive {G : Type u_3} [Group G] : MulAction.IsPretransitive G G

The regular action of a group on itself is transitive.

Equations

• ⋯ = ⋯

Scalar tower and commuting actions

class VAddCommClass (M : Type u_11) (N : Type u_12) (α : Type u_13) [VAdd M α] [VAdd N α] : Prop

A typeclass mixin saying that two additive actions on the same space commute.

• vadd_comm : ∀ (m : M) (n : N) (a : α), m +ᵥ (n +ᵥ a) = n +ᵥ (m +ᵥ a)

  +ᵥ is left commutative

theorem VAddCommClass.vadd_comm {M : Type u_11} {N : Type u_12} {α : Type u_13} : ∀ {inst : VAdd M α} {inst_1 : VAdd N α} [self : VAddCommClass M N α] (m : M) (n : N) (a : α), m +ᵥ (n +ᵥ a) = n +ᵥ (m +ᵥ a)

+ᵥ is left commutative

class SMulCommClass (M : Type u_11) (N : Type u_12) (α : Type u_13) [SMul M α] [SMul N α] : Prop

A typeclass mixin saying that two multiplicative actions on the same space commute.

• smul_comm : ∀ (m : M) (n : N) (a : α), m • n • a = n • m • a

  • is left commutative

theorem SMulCommClass.smul_comm {M : Type u_11} {N : Type u_12} {α : Type u_13} : ∀ {inst : SMul M α} {inst_1 : SMul N α} [self : SMulCommClass M N α] (m : M) (n : N) (a : α), m • n • a = n • m • a

• is left commutative

theorem VAddCommClass.symm (M : Type u_11) (N : Type u_12) (α : Type u_13) [VAdd M α] [VAdd N α] [VAddCommClass M N α] : VAddCommClass N M α

Commutativity of additive actions is a symmetric relation. This lemma can't be an instance because this would cause a loop in the instance search graph.

theorem SMulCommClass.symm (M : Type u_11) (N : Type u_12) (α : Type u_13) [SMul M α] [SMul N α] [SMulCommClass M N α] : SMulCommClass N M α

Commutativity of actions is a symmetric relation. This lemma can't be an instance because this would cause a loop in the instance search graph.

theorem Function.Injective.vaddCommClass {M : Type u_1} {N : Type u_2} {α : Type u_7} {β : Type u_8} [VAdd M α] [VAdd N α] [VAdd M β] [VAdd N β] [VAddCommClass M N β] {f : α → β} (hf : Function.Injective f) (h₁ : ∀ (c : M) (x : α), f (c +ᵥ x) = c +ᵥ f x) (h₂ : ∀ (c : N) (x : α), f (c +ᵥ x) = c +ᵥ f x) : VAddCommClass M N α

theorem Function.Injective.smulCommClass {M : Type u_1} {N : Type u_2} {α : Type u_7} {β : Type u_8} [SMul M α] [SMul N α] [SMul M β] [SMul N β] [SMulCommClass M N β] {f : α → β} (hf : Function.Injective f) (h₁ : ∀ (c : M) (x : α), f (c • x) = c • f x) (h₂ : ∀ (c : N) (x : α), f (c • x) = c • f x) : SMulCommClass M N α

theorem Function.Surjective.vaddCommClass {M : Type u_1} {N : Type u_2} {α : Type u_7} {β : Type u_8} [VAdd M α] [VAdd N α] [VAdd M β] [VAdd N β] [VAddCommClass M N α] {f : α → β} (hf : Function.Surjective f) (h₁ : ∀ (c : M) (x : α), f (c +ᵥ x) = c +ᵥ f x) (h₂ : ∀ (c : N) (x : α), f (c +ᵥ x) = c +ᵥ f x) : VAddCommClass M N β

theorem Function.Surjective.smulCommClass {M : Type u_1} {N : Type u_2} {α : Type u_7} {β : Type u_8} [SMul M α] [SMul N α] [SMul M β] [SMul N β] [SMulCommClass M N α] {f : α → β} (hf : Function.Surjective f) (h₁ : ∀ (c : M) (x : α), f (c • x) = c • f x) (h₂ : ∀ (c : N) (x : α), f (c • x) = c • f x) : SMulCommClass M N β

instance vaddCommClass_self (M : Type u_11) (α : Type u_12) [AddCommMonoid M] [AddAction M α] : VAddCommClass M M α

Equations

• ⋯ = ⋯

instance smulCommClass_self (M : Type u_11) (α : Type u_12) [CommMonoid M] [MulAction M α] : SMulCommClass M M α

Equations

• ⋯ = ⋯

class VAddAssocClass (M : Type u_11) (N : Type u_12) (α : Type u_13) [VAdd M N] [VAdd N α] [VAdd M α] : Prop

An instance of VAddAssocClass M N α states that the additive action of M on α is determined by the additive actions of M on N and N on α.

• vadd_assoc : ∀ (x : M) (y : N) (z : α), x +ᵥ y +ᵥ z = x +ᵥ (y +ᵥ z)

  Associativity of +ᵥ

theorem VAddAssocClass.vadd_assoc {M : Type u_11} {N : Type u_12} {α : Type u_13} : ∀ {inst : VAdd M N} {inst_1 : VAdd N α} {inst_2 : VAdd M α} [self : VAddAssocClass M N α] (x : M) (y : N) (z : α), x +ᵥ y +ᵥ z = x +ᵥ (y +ᵥ z)

Associativity of +ᵥ

class IsScalarTower (M : Type u_11) (N : Type u_12) (α : Type u_13) [SMul M N] [SMul N α] [SMul M α] : Prop

An instance of IsScalarTower M N α states that the multiplicative action of M on α is determined by the multiplicative actions of M on N and N on α.

• smul_assoc : ∀ (x : M) (y : N) (z : α), (x • y) • z = x • y • z

  Associativity of •

theorem IsScalarTower.smul_assoc {M : Type u_11} {N : Type u_12} {α : Type u_13} : ∀ {inst : SMul M N} {inst_1 : SMul N α} {inst_2 : SMul M α} [self : IsScalarTower M N α] (x : M) (y : N) (z : α), (x • y) • z = x • y • z

Associativity of •

@[simp]

theorem vadd_assoc {α : Type u_7} {M : Type u_11} {N : Type u_12} [VAdd M N] [VAdd N α] [VAdd M α] [VAddAssocClass M N α] (x : M) (y : N) (z : α) : x +ᵥ y +ᵥ z = x +ᵥ (y +ᵥ z)

@[simp]

theorem smul_assoc {α : Type u_7} {M : Type u_11} {N : Type u_12} [SMul M N] [SMul N α] [SMul M α] [IsScalarTower M N α] (x : M) (y : N) (z : α) : (x • y) • z = x • y • z

instance AddSemigroup.isScalarTower {α : Type u_7} [AddSemigroup α] : VAddAssocClass α α α

Equations

• ⋯ = ⋯

instance Semigroup.isScalarTower {α : Type u_7} [Semigroup α] : IsScalarTower α α α

Equations

• ⋯ = ⋯

class IsCentralVAdd (M : Type u_11) (α : Type u_12) [VAdd M α] [VAdd Mᵃᵒᵖ α] : Prop

A typeclass indicating that the right (aka AddOpposite) and left actions by M on α are equal, that is that M acts centrally on α. This can be thought of as a version of commutativity for +ᵥ.

• op_vadd_eq_vadd : ∀ (m : M) (a : α), AddOpposite.op m +ᵥ a = m +ᵥ a

  The right and left actions of M on α are equal.

theorem IsCentralVAdd.op_vadd_eq_vadd {M : Type u_11} {α : Type u_12} : ∀ {inst : VAdd M α} {inst_1 : VAdd Mᵃᵒᵖ α} [self : IsCentralVAdd M α] (m : M) (a : α), AddOpposite.op m +ᵥ a = m +ᵥ a

The right and left actions of M on α are equal.

class IsCentralScalar (M : Type u_11) (α : Type u_12) [SMul M α] [SMul Mᵐᵒᵖ α] : Prop

A typeclass indicating that the right (aka MulOpposite) and left actions by M on α are equal, that is that M acts centrally on α. This can be thought of as a version of commutativity for •.

• op_smul_eq_smul : ∀ (m : M) (a : α), MulOpposite.op m • a = m • a

  The right and left actions of M on α are equal.

@[simp]

theorem IsCentralScalar.op_smul_eq_smul {M : Type u_11} {α : Type u_12} : ∀ {inst : SMul M α} {inst_1 : SMul Mᵐᵒᵖ α} [self : IsCentralScalar M α] (m : M) (a : α), MulOpposite.op m • a = m • a

The right and left actions of M on α are equal.

theorem IsCentralVAdd.unop_vadd_eq_vadd {M : Type u_11} {α : Type u_12} [VAdd M α] [VAdd Mᵃᵒᵖ α] [IsCentralVAdd M α] (m : Mᵃᵒᵖ) (a : α) : AddOpposite.unop m +ᵥ a = m +ᵥ a

theorem IsCentralScalar.unop_smul_eq_smul {M : Type u_11} {α : Type u_12} [SMul M α] [SMul Mᵐᵒᵖ α] [IsCentralScalar M α] (m : Mᵐᵒᵖ) (a : α) : MulOpposite.unop m • a = m • a

@[instance 50]

instance VAddCommClass.op_left {M : Type u_1} {N : Type u_2} {α : Type u_7} [VAdd M α] [VAdd Mᵃᵒᵖ α] [IsCentralVAdd M α] [VAdd N α] [VAddCommClass M N α] : VAddCommClass Mᵃᵒᵖ N α

Equations

• ⋯ = ⋯

@[instance 50]

instance SMulCommClass.op_left {M : Type u_1} {N : Type u_2} {α : Type u_7} [SMul M α] [SMul Mᵐᵒᵖ α] [IsCentralScalar M α] [SMul N α] [SMulCommClass M N α] : SMulCommClass Mᵐᵒᵖ N α

Equations

• ⋯ = ⋯

@[instance 50]

instance VAddCommClass.op_right {M : Type u_1} {N : Type u_2} {α : Type u_7} [VAdd M α] [VAdd N α] [VAdd Nᵃᵒᵖ α] [IsCentralVAdd N α] [VAddCommClass M N α] : VAddCommClass M Nᵃᵒᵖ α

Equations

• ⋯ = ⋯

@[instance 50]

instance SMulCommClass.op_right {M : Type u_1} {N : Type u_2} {α : Type u_7} [SMul M α] [SMul N α] [SMul Nᵐᵒᵖ α] [IsCentralScalar N α] [SMulCommClass M N α] : SMulCommClass M Nᵐᵒᵖ α

Equations

• ⋯ = ⋯

@[instance 50]

instance VAddAssocClass.op_left {M : Type u_1} {N : Type u_2} {α : Type u_7} [VAdd M α] [VAdd Mᵃᵒᵖ α] [IsCentralVAdd M α] [VAdd M N] [VAdd Mᵃᵒᵖ N] [IsCentralVAdd M N] [VAdd N α] [VAddAssocClass M N α] : VAddAssocClass Mᵃᵒᵖ N α

Equations

• ⋯ = ⋯

@[instance 50]

instance IsScalarTower.op_left {M : Type u_1} {N : Type u_2} {α : Type u_7} [SMul M α] [SMul Mᵐᵒᵖ α] [IsCentralScalar M α] [SMul M N] [SMul Mᵐᵒᵖ N] [IsCentralScalar M N] [SMul N α] [IsScalarTower M N α] : IsScalarTower Mᵐᵒᵖ N α

Equations

• ⋯ = ⋯

@[instance 50]

instance VAddAssocClass.op_right {M : Type u_1} {N : Type u_2} {α : Type u_7} [VAdd M α] [VAdd M N] [VAdd N α] [VAdd Nᵃᵒᵖ α] [IsCentralVAdd N α] [VAddAssocClass M N α] : VAddAssocClass M Nᵃᵒᵖ α

Equations

• ⋯ = ⋯

@[instance 50]

instance IsScalarTower.op_right {M : Type u_1} {N : Type u_2} {α : Type u_7} [SMul M α] [SMul M N] [SMul N α] [SMul Nᵐᵒᵖ α] [IsCentralScalar N α] [IsScalarTower M N α] : IsScalarTower M Nᵐᵒᵖ α

Equations

• ⋯ = ⋯

def VAdd.comp.vadd {M : Type u_1} {N : Type u_2} {α : Type u_7} [VAdd M α] (g : N → M) (n : N) (a : α) : α

Auxiliary definition for VAdd.comp, AddAction.compHom, etc.

Equations

• VAdd.comp.vadd g n a = g n +ᵥ a

def SMul.comp.smul {M : Type u_1} {N : Type u_2} {α : Type u_7} [SMul M α] (g : N → M) (n : N) (a : α) : α

Auxiliary definition for SMul.comp, MulAction.compHom, DistribMulAction.compHom, Module.compHom, etc.

Equations

• SMul.comp.smul g n a = g n • a

@[reducible, inline]

abbrev VAdd.comp {M : Type u_1} {N : Type u_2} (α : Type u_7) [VAdd M α] (g : N → M) : VAdd N α

An additive action of M on α and a function N → M induces an additive action of N on α.

Equations

• VAdd.comp α g = { vadd := VAdd.comp.vadd g }

@[reducible, inline]

abbrev SMul.comp {M : Type u_1} {N : Type u_2} (α : Type u_7) [SMul M α] (g : N → M) : SMul N α

An action of M on α and a function N → M induces an action of N on α.

Equations

• SMul.comp α g = { smul := SMul.comp.smul g }

theorem VAdd.comp.isScalarTower {M : Type u_1} {N : Type u_2} {α : Type u_7} {β : Type u_8} [VAdd M α] [VAdd M β] [VAdd α β] [VAddAssocClass M α β] (g : N → M) : VAddAssocClass N α β

Given a tower of additive actions M → α → β, if we use SMul.comp to pull back both of M's actions by a map g : N → M, then we obtain a new tower of scalar actions N → α → β.

This cannot be an instance because it can cause infinite loops whenever the SMul arguments are still metavariables.

theorem SMul.comp.isScalarTower {M : Type u_1} {N : Type u_2} {α : Type u_7} {β : Type u_8} [SMul M α] [SMul M β] [SMul α β] [IsScalarTower M α β] (g : N → M) : IsScalarTower N α β

Given a tower of scalar actions M → α → β, if we use SMul.comp to pull back both of M's actions by a map g : N → M, then we obtain a new tower of scalar actions N → α → β.

This cannot be an instance because it can cause infinite loops whenever the SMul arguments are still metavariables.

theorem VAdd.comp.vaddCommClass {M : Type u_1} {N : Type u_2} {α : Type u_7} {β : Type u_8} [VAdd M α] [VAdd β α] [VAddCommClass M β α] (g : N → M) : VAddCommClass N β α

This cannot be an instance because it can cause infinite loops whenever the VAdd arguments are still metavariables.

theorem SMul.comp.smulCommClass {M : Type u_1} {N : Type u_2} {α : Type u_7} {β : Type u_8} [SMul M α] [SMul β α] [SMulCommClass M β α] (g : N → M) : SMulCommClass N β α

This cannot be an instance because it can cause infinite loops whenever the SMul arguments are still metavariables.

theorem VAdd.comp.vaddCommClass' {M : Type u_1} {N : Type u_2} {α : Type u_7} {β : Type u_8} [VAdd M α] [VAdd β α] [VAddCommClass β M α] (g : N → M) : VAddCommClass β N α

This cannot be an instance because it can cause infinite loops whenever the VAdd arguments are still metavariables.

theorem SMul.comp.smulCommClass' {M : Type u_1} {N : Type u_2} {α : Type u_7} {β : Type u_8} [SMul M α] [SMul β α] [SMulCommClass β M α] (g : N → M) : SMulCommClass β N α

This cannot be an instance because it can cause infinite loops whenever the SMul arguments are still metavariables.

theorem add_vadd_comm {α : Type u_7} {β : Type u_8} [Add β] [VAdd α β] [VAddCommClass α β β] (s : α) (x : β) (y : β) : x + (s +ᵥ y) = s +ᵥ (x + y)

theorem mul_smul_comm {α : Type u_7} {β : Type u_8} [Mul β] [SMul α β] [SMulCommClass α β β] (s : α) (x : β) (y : β) : x * s • y = s • (x * y)

Note that the SMulCommClass α β β typeclass argument is usually satisfied by Algebra α β.

theorem vadd_add_assoc {α : Type u_7} {β : Type u_8} [Add β] [VAdd α β] [VAddAssocClass α β β] (r : α) (x : β) (y : β) : r +ᵥ x + y = r +ᵥ (x + y)

theorem smul_mul_assoc {α : Type u_7} {β : Type u_8} [Mul β] [SMul α β] [IsScalarTower α β β] (r : α) (x : β) (y : β) : r • x * y = r • (x * y)

Note that the IsScalarTower α β β typeclass argument is usually satisfied by Algebra α β.

theorem vadd_sub_assoc {α : Type u_7} {β : Type u_8} [SubNegMonoid β] [VAdd α β] [VAddAssocClass α β β] (r : α) (x : β) (y : β) : r +ᵥ x - y = r +ᵥ (x - y)

theorem smul_div_assoc {α : Type u_7} {β : Type u_8} [DivInvMonoid β] [SMul α β] [IsScalarTower α β β] (r : α) (x : β) (y : β) : r • x / y = r • (x / y)

Note that the IsScalarTower α β β typeclass argument is usually satisfied by Algebra α β.

theorem vadd_vadd_vadd_comm {α : Type u_7} {β : Type u_8} {γ : Type u_9} {δ : Type u_10} [VAdd α β] [VAdd α γ] [VAdd β δ] [VAdd α δ] [VAdd γ δ] [VAddAssocClass α β δ] [VAddAssocClass α γ δ] [VAddCommClass β γ δ] (a : α) (b : β) (c : γ) (d : δ) : a +ᵥ b +ᵥ (c +ᵥ d) = a +ᵥ c +ᵥ (b +ᵥ d)

theorem smul_smul_smul_comm {α : Type u_7} {β : Type u_8} {γ : Type u_9} {δ : Type u_10} [SMul α β] [SMul α γ] [SMul β δ] [SMul α δ] [SMul γ δ] [IsScalarTower α β δ] [IsScalarTower α γ δ] [SMulCommClass β γ δ] (a : α) (b : β) (c : γ) (d : δ) : (a • b) • c • d = (a • c) • b • d

theorem vadd_add_vadd_comm {α : Type u_7} {β : Type u_8} [Add α] [Add β] [VAdd α β] [VAddAssocClass α β β] [VAddAssocClass α α β] [VAddCommClass α β β] (a : α) (b : β) (c : α) (d : β) : a +ᵥ b + (c +ᵥ d) = a + c +ᵥ (b + d)

theorem smul_mul_smul_comm {α : Type u_7} {β : Type u_8} [Mul α] [Mul β] [SMul α β] [IsScalarTower α β β] [IsScalarTower α α β] [SMulCommClass α β β] (a : α) (b : β) (c : α) (d : β) : a • b * c • d = (a * c) • (b * d)

Note that the IsScalarTower α β β and SMulCommClass α β β typeclass arguments are usually satisfied by Algebra α β.

@[deprecated]

theorem smul_mul_smul {α : Type u_7} {β : Type u_8} [Mul α] [Mul β] [SMul α β] [IsScalarTower α β β] [IsScalarTower α α β] [SMulCommClass α β β] (a : α) (b : β) (c : α) (d : β) : a • b * c • d = (a * c) • (b * d)

Alias of smul_mul_smul_comm.

---

Note that the IsScalarTower α β β and SMulCommClass α β β typeclass arguments are usually satisfied by Algebra α β.

@[deprecated]

theorem vadd_add_vadd {α : Type u_7} {β : Type u_8} [Add α] [Add β] [VAdd α β] [VAddAssocClass α β β] [VAddAssocClass α α β] [VAddCommClass α β β] (a : α) (b : β) (c : α) (d : β) : a +ᵥ b + (c +ᵥ d) = a + c +ᵥ (b + d)

theorem add_vadd_add_comm {α : Type u_7} {β : Type u_8} [Add α] [Add β] [VAdd α β] [VAddAssocClass α β β] [VAddAssocClass α α β] [VAddCommClass α β β] (a : α) (b : α) (c : β) (d : β) : a + b +ᵥ (c + d) = a +ᵥ c + (b +ᵥ d)

theorem mul_smul_mul_comm {α : Type u_7} {β : Type u_8} [Mul α] [Mul β] [SMul α β] [IsScalarTower α β β] [IsScalarTower α α β] [SMulCommClass α β β] (a : α) (b : α) (c : β) (d : β) : (a * b) • (c * d) = a • c * b • d

Note that the IsScalarTower α β β and SMulCommClass α β β typeclass arguments are usually satisfied by Algebra α β.

theorem AddCommute.vadd_right {M : Type u_1} {α : Type u_7} [VAdd M α] [Add α] [VAddCommClass M α α] [VAddAssocClass M α α] {a : α} {b : α} (h : AddCommute a b) (r : M) : AddCommute a (r +ᵥ b)

theorem Commute.smul_right {M : Type u_1} {α : Type u_7} [SMul M α] [Mul α] [SMulCommClass M α α] [IsScalarTower M α α] {a : α} {b : α} (h : Commute a b) (r : M) : Commute a (r • b)

theorem AddCommute.vadd_left {M : Type u_1} {α : Type u_7} [VAdd M α] [Add α] [VAddCommClass M α α] [VAddAssocClass M α α] {a : α} {b : α} (h : AddCommute a b) (r : M) : AddCommute (r +ᵥ a) b

theorem Commute.smul_left {M : Type u_1} {α : Type u_7} [SMul M α] [Mul α] [SMulCommClass M α α] [IsScalarTower M α α] {a : α} {b : α} (h : Commute a b) (r : M) : Commute (r • a) b

theorem vadd_vadd {M : Type u_1} {α : Type u_7} [AddMonoid M] [AddAction M α] (a₁ : M) (a₂ : M) (b : α) : a₁ +ᵥ (a₂ +ᵥ b) = a₁ + a₂ +ᵥ b

theorem smul_smul {M : Type u_1} {α : Type u_7} [Monoid M] [MulAction M α] (a₁ : M) (a₂ : M) (b : α) : a₁ • a₂ • b = (a₁ * a₂) • b

@[simp]

theorem zero_vadd (M : Type u_1) {α : Type u_7} [AddMonoid M] [AddAction M α] (b : α) : 0 +ᵥ b = b

@[simp]

theorem one_smul (M : Type u_1) {α : Type u_7} [Monoid M] [MulAction M α] (b : α) : 1 • b = b

theorem zero_vadd_eq_id (M : Type u_1) {α : Type u_7} [AddMonoid M] [AddAction M α] : (fun (x : α) => 0 +ᵥ x) = id

VAdd version of zero_add_eq_id

theorem one_smul_eq_id (M : Type u_1) {α : Type u_7} [Monoid M] [MulAction M α] : (fun (x : α) => 1 • x) = id

SMul version of one_mul_eq_id

theorem comp_vadd_left (M : Type u_1) {α : Type u_7} [AddMonoid M] [AddAction M α] (a₁ : M) (a₂ : M) : ((fun (x : α) => a₁ +ᵥ x) ∘ fun (x : α) => a₂ +ᵥ x) = fun (x : α) => a₁ + a₂ +ᵥ x

VAdd version of comp_add_left

theorem comp_smul_left (M : Type u_1) {α : Type u_7} [Monoid M] [MulAction M α] (a₁ : M) (a₂ : M) : ((fun (x : α) => a₁ • x) ∘ fun (x : α) => a₂ • x) = fun (x : α) => (a₁ * a₂) • x

SMul version of comp_mul_left

theorem Function.Injective.addAction.proof_2 {M : Type u_2} {α : Type u_3} {β : Type u_1} [AddMonoid M] [AddAction M α] [VAdd M β] (f : β → α) (hf : Function.Injective f) (smul : ∀ (c : M) (x : β), f (c +ᵥ x) = c +ᵥ f x) (c₁ : M) (c₂ : M) (x : β) : c₁ + c₂ +ᵥ x = c₁ +ᵥ (c₂ +ᵥ x)

theorem Function.Injective.addAction.proof_1 {M : Type u_2} {α : Type u_3} {β : Type u_1} [AddMonoid M] [AddAction M α] [VAdd M β] (f : β → α) (hf : Function.Injective f) (smul : ∀ (c : M) (x : β), f (c +ᵥ x) = c +ᵥ f x) (x : β) : 0 +ᵥ x = x

@[reducible, inline]

abbrev Function.Injective.addAction {M : Type u_1} {α : Type u_7} {β : Type u_8} [AddMonoid M] [AddAction M α] [VAdd M β] (f : β → α) (hf : Function.Injective f) (smul : ∀ (c : M) (x : β), f (c +ᵥ x) = c +ᵥ f x) : AddAction M β

Pullback an additive action along an injective map respecting +ᵥ.

Equations

• Function.Injective.addAction f hf smul = AddAction.mk ⋯ ⋯

@[reducible, inline]

abbrev Function.Injective.mulAction {M : Type u_1} {α : Type u_7} {β : Type u_8} [Monoid M] [MulAction M α] [SMul M β] (f : β → α) (hf : Function.Injective f) (smul : ∀ (c : M) (x : β), f (c • x) = c • f x) : MulAction M β

Pullback a multiplicative action along an injective map respecting •. See note [reducible non-instances].

Equations

• Function.Injective.mulAction f hf smul = MulAction.mk ⋯ ⋯

theorem Function.Surjective.addAction.proof_2 {M : Type u_2} {α : Type u_3} {β : Type u_1} [AddMonoid M] [AddAction M α] [VAdd M β] (f : α → β) (hf : Function.Surjective f) (smul : ∀ (c : M) (x : α), f (c +ᵥ x) = c +ᵥ f x) (a : M) (a : M) (y : β) : a✝ + a +ᵥ y = a✝ +ᵥ (a +ᵥ y)

@[reducible, inline]

abbrev Function.Surjective.addAction {M : Type u_1} {α : Type u_7} {β : Type u_8} [AddMonoid M] [AddAction M α] [VAdd M β] (f : α → β) (hf : Function.Surjective f) (smul : ∀ (c : M) (x : α), f (c +ᵥ x) = c +ᵥ f x) : AddAction M β

Pushforward an additive action along a surjective map respecting +ᵥ.

Equations

• Function.Surjective.addAction f hf smul = AddAction.mk ⋯ ⋯

theorem Function.Surjective.addAction.proof_1 {M : Type u_2} {α : Type u_3} {β : Type u_1} [AddMonoid M] [AddAction M α] [VAdd M β] (f : α → β) (hf : Function.Surjective f) (smul : ∀ (c : M) (x : α), f (c +ᵥ x) = c +ᵥ f x) (y : β) : 0 +ᵥ y = y

@[reducible, inline]

abbrev Function.Surjective.mulAction {M : Type u_1} {α : Type u_7} {β : Type u_8} [Monoid M] [MulAction M α] [SMul M β] (f : α → β) (hf : Function.Surjective f) (smul : ∀ (c : M) (x : α), f (c • x) = c • f x) : MulAction M β

Pushforward a multiplicative action along a surjective map respecting •. See note [reducible non-instances].

Equations

• Function.Surjective.mulAction f hf smul = MulAction.mk ⋯ ⋯

@[reducible, inline]

abbrev Function.Surjective.addActionLeft {R : Type u_11} {S : Type u_12} {M : Type u_13} [AddMonoid R] [AddAction R M] [AddMonoid S] [VAdd S M] (f : R →+ S) (hf : Function.Surjective ⇑f) (hsmul : ∀ (c : R) (x : M), f c +ᵥ x = c +ᵥ x) : AddAction S M

Push forward the action of R on M along a compatible surjective map f : R →+ S.

Equations

• Function.Surjective.addActionLeft f hf hsmul = AddAction.mk ⋯ ⋯

theorem Function.Surjective.addActionLeft.proof_2 {R : Type u_3} {S : Type u_2} {M : Type u_1} [AddMonoid R] [AddAction R M] [AddMonoid S] [VAdd S M] (f : R →+ S) (hf : Function.Surjective ⇑f) (hsmul : ∀ (c : R) (x : M), f c +ᵥ x = c +ᵥ x) (y₁ : S) (y₂ : S) (b : M) : y₁ + y₂ +ᵥ b = y₁ +ᵥ (y₂ +ᵥ b)

theorem Function.Surjective.addActionLeft.proof_1 {R : Type u_3} {S : Type u_2} {M : Type u_1} [AddMonoid R] [AddAction R M] [AddMonoid S] [VAdd S M] (f : R →+ S) (hsmul : ∀ (c : R) (x : M), f c +ᵥ x = c +ᵥ x) (b : M) : 0 +ᵥ b = b

@[reducible, inline]

abbrev Function.Surjective.mulActionLeft {R : Type u_11} {S : Type u_12} {M : Type u_13} [Monoid R] [MulAction R M] [Monoid S] [SMul S M] (f : R →* S) (hf : Function.Surjective ⇑f) (hsmul : ∀ (c : R) (x : M), f c • x = c • x) : MulAction S M

Push forward the action of R on M along a compatible surjective map f : R →* S.

See also Function.Surjective.distribMulActionLeft and Function.Surjective.moduleLeft.

Equations

• Function.Surjective.mulActionLeft f hf hsmul = MulAction.mk ⋯ ⋯

theorem AddMonoid.toAddAction.proof_2 (M : Type u_1) [AddMonoid M] (a : M) (b : M) (c : M) : a + b + c = a + (b + c)

@[instance 910]

instance AddMonoid.toAddAction (M : Type u_1) [AddMonoid M] : AddAction M M

The regular action of a monoid on itself by left addition.

This is promoted to an AddTorsor by addGroup_is_addTorsor.

Equations

• AddMonoid.toAddAction M = AddAction.mk ⋯ ⋯

theorem AddMonoid.toAddAction.proof_1 (M : Type u_1) [AddMonoid M] (a : M) : 0 + a = a

@[instance 910]

instance Monoid.toMulAction (M : Type u_1) [Monoid M] : MulAction M M

The regular action of a monoid on itself by left multiplication.

This is promoted to a module by Semiring.toModule.

Equations

• Monoid.toMulAction M = MulAction.mk ⋯ ⋯

instance VAddAssocClass.left (M : Type u_1) {α : Type u_7} [AddMonoid M] [AddAction M α] : VAddAssocClass M M α

Equations

• ⋯ = ⋯

instance IsScalarTower.left (M : Type u_1) {α : Type u_7} [Monoid M] [MulAction M α] : IsScalarTower M M α

Equations

• ⋯ = ⋯

theorem smul_pow {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] [MulAction M N] [IsScalarTower M N N] [SMulCommClass M N N] (r : M) (x : N) (n : ℕ) : (r • x) ^ n = r ^ n • x ^ n

@[simp]

theorem neg_vadd_vadd {G : Type u_3} {α : Type u_7} [AddGroup G] [AddAction G α] (g : G) (a : α) : -g +ᵥ (g +ᵥ a) = a

@[simp]

theorem inv_smul_smul {G : Type u_3} {α : Type u_7} [Group G] [MulAction G α] (g : G) (a : α) : g⁻¹ • g • a = a

@[simp]

theorem vadd_neg_vadd {G : Type u_3} {α : Type u_7} [AddGroup G] [AddAction G α] (g : G) (a : α) : g +ᵥ (-g +ᵥ a) = a

@[simp]

theorem smul_inv_smul {G : Type u_3} {α : Type u_7} [Group G] [MulAction G α] (g : G) (a : α) : g • g⁻¹ • a = a

theorem neg_vadd_eq_iff {G : Type u_3} {α : Type u_7} [AddGroup G] [AddAction G α] {g : G} {a : α} {b : α} : -g +ᵥ a = b ↔ a = g +ᵥ b

theorem inv_smul_eq_iff {G : Type u_3} {α : Type u_7} [Group G] [MulAction G α] {g : G} {a : α} {b : α} : g⁻¹ • a = b ↔ a = g • b

theorem eq_neg_vadd_iff {G : Type u_3} {α : Type u_7} [AddGroup G] [AddAction G α] {g : G} {a : α} {b : α} : a = -g +ᵥ b ↔ g +ᵥ a = b

theorem eq_inv_smul_iff {G : Type u_3} {α : Type u_7} [Group G] [MulAction G α] {g : G} {a : α} {b : α} : a = g⁻¹ • b ↔ g • a = b

@[simp]

theorem Commute.smul_right_iff {G : Type u_3} {H : Type u_4} [Group G] [Mul H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] {g : G} {a : H} {b : H} : Commute a (g • b) ↔ Commute a b

@[simp]

theorem Commute.smul_left_iff {G : Type u_3} {H : Type u_4} [Group G] [Mul H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] {g : G} {a : H} {b : H} : Commute (g • a) b ↔ Commute a b

theorem smul_inv {G : Type u_3} {H : Type u_4} [Group G] [Group H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] (g : G) (a : H) : (g • a)⁻¹ = g⁻¹ • a⁻¹

theorem smul_zpow {G : Type u_3} {H : Type u_4} [Group G] [Group H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] (g : G) (a : H) (n : ℤ) : (g • a) ^ n = g ^ n • a ^ n

theorem SMulCommClass.of_commMonoid (A : Type u_11) (B : Type u_12) (G : Type u_13) [CommMonoid G] [SMul A G] [SMul B G] [IsScalarTower A G G] [IsScalarTower B G G] : SMulCommClass A B G

theorem AddAction.toFun.proof_1 (M : Type u_1) (α : Type u_2) [AddMonoid M] [AddAction M α] (y₁ : α) (y₂ : α) (H : (fun (y : α) (x : M) => x +ᵥ y) y₁ = (fun (y : α) (x : M) => x +ᵥ y) y₂) : y₁ = y₂

def AddAction.toFun (M : Type u_1) (α : Type u_7) [AddMonoid M] [AddAction M α] : α ↪ M → α

Embedding of α into functions M → α induced by an additive action of M on α.

Equations

• AddAction.toFun M α = { toFun := fun (y : α) (x : M) => x +ᵥ y, inj' := ⋯ }

def MulAction.toFun (M : Type u_1) (α : Type u_7) [Monoid M] [MulAction M α] : α ↪ M → α

Embedding of α into functions M → α induced by a multiplicative action of M on α.

Equations

• MulAction.toFun M α = { toFun := fun (y : α) (x : M) => x • y, inj' := ⋯ }

@[simp]

theorem AddAction.toFun_apply {M : Type u_1} {α : Type u_7} [AddMonoid M] [AddAction M α] (x : M) (y : α) : (AddAction.toFun M α) y x = x +ᵥ y

@[simp]

theorem MulAction.toFun_apply {M : Type u_1} {α : Type u_7} [Monoid M] [MulAction M α] (x : M) (y : α) : (MulAction.toFun M α) y x = x • y

@[reducible, inline]

abbrev AddAction.compHom {M : Type u_1} {N : Type u_2} (α : Type u_7) [AddMonoid M] [AddAction M α] [AddMonoid N] (g : N →+ M) : AddAction N α

An additive action of M on α and an additive monoid homomorphism N → M induce an additive action of N on α.

See note [reducible non-instances].

Equations

• AddAction.compHom α g = AddAction.mk ⋯ ⋯

theorem AddAction.compHom.proof_2 {M : Type u_3} {N : Type u_2} (α : Type u_1) [AddMonoid M] [AddAction M α] [AddMonoid N] (g : N →+ M) : ∀ (x x_1 : N) (x_2 : α), x + x_1 +ᵥ x_2 = x +ᵥ (x_1 +ᵥ x_2)

theorem AddAction.compHom.proof_1 {M : Type u_3} {N : Type u_2} (α : Type u_1) [AddMonoid M] [AddAction M α] [AddMonoid N] (g : N →+ M) : ∀ (x : α), 0 +ᵥ x = x

@[reducible, inline]

abbrev MulAction.compHom {M : Type u_1} {N : Type u_2} (α : Type u_7) [Monoid M] [MulAction M α] [Monoid N] (g : N →* M) : MulAction N α

A multiplicative action of M on α and a monoid homomorphism N → M induce a multiplicative action of N on α.

See note [reducible non-instances].

Equations

• MulAction.compHom α g = MulAction.mk ⋯ ⋯

theorem AddAction.compHom_vadd_def {E : Type u_11} {F : Type u_12} {G : Type u_13} [AddMonoid E] [AddMonoid F] [AddAction F G] (f : E →+ F) (a : E) (x : G) : a +ᵥ x = f a +ᵥ x

theorem MulAction.compHom_smul_def {E : Type u_11} {F : Type u_12} {G : Type u_13} [Monoid E] [Monoid F] [MulAction F G] (f : E →* F) (a : E) (x : G) : a • x = f a • x

theorem AddAction.isPretransitive_compHom {E : Type u_11} {F : Type u_12} {G : Type u_13} [AddMonoid E] [AddMonoid F] [AddAction F G] [AddAction.IsPretransitive F G] {f : E →+ F} (hf : Function.Surjective ⇑f) : AddAction.IsPretransitive E G

theorem MulAction.isPretransitive_compHom {E : Type u_11} {F : Type u_12} {G : Type u_13} [Monoid E] [Monoid F] [MulAction F G] [MulAction.IsPretransitive F G] {f : E →* F} (hf : Function.Surjective ⇑f) : MulAction.IsPretransitive E G

If an action is transitive, then composing this action with a surjective homomorphism gives again a transitive action.

theorem AddAction.IsPretransitive.of_vadd_eq {M : Type u_11} {N : Type u_12} {α : Type u_13} [VAdd M α] [VAdd N α] [AddAction.IsPretransitive M α] (f : M → N) (hf : ∀ {c : M} {x : α}, f c +ᵥ x = c +ᵥ x) : AddAction.IsPretransitive N α

theorem MulAction.IsPretransitive.of_smul_eq {M : Type u_11} {N : Type u_12} {α : Type u_13} [SMul M α] [SMul N α] [MulAction.IsPretransitive M α] (f : M → N) (hf : ∀ {c : M} {x : α}, f c • x = c • x) : MulAction.IsPretransitive N α

theorem AddAction.IsPretransitive.of_compHom {M : Type u_11} {N : Type u_12} {α : Type u_13} [AddMonoid M] [AddMonoid N] [AddAction N α] (f : M →+ N) [h : AddAction.IsPretransitive M α] : AddAction.IsPretransitive N α

theorem MulAction.IsPretransitive.of_compHom {M : Type u_11} {N : Type u_12} {α : Type u_13} [Monoid M] [Monoid N] [MulAction N α] (f : M →* N) [h : MulAction.IsPretransitive M α] : MulAction.IsPretransitive N α

theorem vadd_zero_vadd {α : Type u_7} {M : Type u_11} (N : Type u_12) [AddMonoid N] [VAdd M N] [AddAction N α] [VAdd M α] [VAddAssocClass M N α] (x : M) (y : α) : x +ᵥ 0 +ᵥ y = x +ᵥ y

theorem smul_one_smul {α : Type u_7} {M : Type u_11} (N : Type u_12) [Monoid N] [SMul M N] [MulAction N α] [SMul M α] [IsScalarTower M N α] (x : M) (y : α) : (x • 1) • y = x • y

theorem AddAction.IsPretransitive.of_isScalarTower (M : Type u_11) {N : Type u_12} {α : Type u_13} [AddMonoid N] [VAdd M N] [AddAction N α] [VAdd M α] [VAddAssocClass M N α] [AddAction.IsPretransitive M α] : AddAction.IsPretransitive N α

theorem MulAction.IsPretransitive.of_isScalarTower (M : Type u_11) {N : Type u_12} {α : Type u_13} [Monoid N] [SMul M N] [MulAction N α] [SMul M α] [IsScalarTower M N α] [MulAction.IsPretransitive M α] : MulAction.IsPretransitive N α

@[simp]

theorem vadd_zero_add {M : Type u_11} {N : Type u_12} [AddZeroClass N] [VAdd M N] [VAddAssocClass M N N] (x : M) (y : N) : x +ᵥ 0 + y = x +ᵥ y

@[simp]

theorem smul_one_mul {M : Type u_11} {N : Type u_12} [MulOneClass N] [SMul M N] [IsScalarTower M N N] (x : M) (y : N) : x • 1 * y = x • y

@[simp]

theorem add_vadd_zero {M : Type u_11} {N : Type u_12} [AddZeroClass N] [VAdd M N] [VAddCommClass M N N] (x : M) (y : N) : y + (x +ᵥ 0) = x +ᵥ y

@[simp]

theorem mul_smul_one {M : Type u_11} {N : Type u_12} [MulOneClass N] [SMul M N] [SMulCommClass M N N] (x : M) (y : N) : y * x • 1 = x • y

theorem VAddAssocClass.of_vadd_zero_add {M : Type u_11} {N : Type u_12} [AddMonoid N] [VAdd M N] (h : ∀ (x : M) (y : N), x +ᵥ 0 + y = x +ᵥ y) : VAddAssocClass M N N

theorem IsScalarTower.of_smul_one_mul {M : Type u_11} {N : Type u_12} [Monoid N] [SMul M N] (h : ∀ (x : M) (y : N), x • 1 * y = x • y) : IsScalarTower M N N

theorem VAddCommClass.of_add_vadd_zero {M : Type u_11} {N : Type u_12} [AddMonoid N] [VAdd M N] (H : ∀ (x : M) (y : N), y + (x +ᵥ 0) = x +ᵥ y) : VAddCommClass M N N

theorem SMulCommClass.of_mul_smul_one {M : Type u_11} {N : Type u_12} [Monoid N] [SMul M N] (H : ∀ (x : M) (y : N), y * x • 1 = x • y) : SMulCommClass M N N

theorem AddMonoidHom.vaddZeroHom.proof_1 {M : Type u_2} {N : Type u_1} [AddMonoid M] [AddZeroClass N] [AddAction M N] : 0 +ᵥ 0 = 0

theorem AddMonoidHom.vaddZeroHom.proof_2 {M : Type u_2} {N : Type u_1} [AddMonoid M] [AddZeroClass N] [AddAction M N] [VAddAssocClass M N N] (x : M) (y : M) : { toFun := fun (x : M) => x +ᵥ 0, map_zero' := ⋯ }.toFun (x + y) = { toFun := fun (x : M) => x +ᵥ 0, map_zero' := ⋯ }.toFun x + { toFun := fun (x : M) => x +ᵥ 0, map_zero' := ⋯ }.toFun y

def AddMonoidHom.vaddZeroHom {M : Type u_11} {N : Type u_12} [AddMonoid M] [AddZeroClass N] [AddAction M N] [VAddAssocClass M N N] : M →+ N

If the additive action of M on N is compatible with addition on N, then fun x ↦ x +ᵥ 0 is an additive monoid homomorphism from M to N.

Equations

• AddMonoidHom.vaddZeroHom = { toFun := fun (x : M) => x +ᵥ 0, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem MonoidHom.smulOneHom_apply {M : Type u_11} {N : Type u_12} [Monoid M] [MulOneClass N] [MulAction M N] [IsScalarTower M N N] (x : M) : MonoidHom.smulOneHom x = x • 1

@[simp]

theorem AddMonoidHom.vaddZeroHom_apply {M : Type u_11} {N : Type u_12} [AddMonoid M] [AddZeroClass N] [AddAction M N] [VAddAssocClass M N N] (x : M) : AddMonoidHom.vaddZeroHom x = x +ᵥ 0

def MonoidHom.smulOneHom {M : Type u_11} {N : Type u_12} [Monoid M] [MulOneClass N] [MulAction M N] [IsScalarTower M N N] : M →* N

If the multiplicative action of M on N is compatible with multiplication on N, then fun x ↦ x • 1 is a monoid homomorphism from M to N.

Equations

• MonoidHom.smulOneHom = { toFun := fun (x : M) => x • 1, map_one' := ⋯, map_mul' := ⋯ }

def addMonoidHomEquivAddActionIsScalarTower (M : Type u_11) (N : Type u_12) [AddMonoid M] [AddMonoid N] : (M →+ N) ≃ { _inst : AddAction M N // VAddAssocClass M N N }

A monoid homomorphism between two additive monoids M and N can be equivalently specified by an additive action of M on N that is compatible with the addition on N.

Equations

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

theorem addMonoidHomEquivAddActionIsScalarTower.proof_1 (M : Type u_1) (N : Type u_2) [AddMonoid M] [AddMonoid N] (f : M →+ N) : VAddAssocClass M N N

theorem addMonoidHomEquivAddActionIsScalarTower.proof_2 (M : Type u_1) (N : Type u_2) [AddMonoid M] [AddMonoid N] (f : M →+ N) : (fun (x : { _inst : AddAction M N // VAddAssocClass M N N }) => match x with | ⟨val, property⟩ => AddMonoidHom.vaddZeroHom) ((fun (f : M →+ N) => ⟨AddAction.compHom N f, ⋯⟩) f) = f

theorem addMonoidHomEquivAddActionIsScalarTower.proof_3 (M : Type u_1) (N : Type u_2) [AddMonoid M] [AddMonoid N] : ∀ (x : { _inst : AddAction M N // VAddAssocClass M N N }), (fun (f : M →+ N) => ⟨AddAction.compHom N f, ⋯⟩) ((fun (x : { _inst : AddAction M N // VAddAssocClass M N N }) => match x with | ⟨val, property⟩ => AddMonoidHom.vaddZeroHom) x) = x

def monoidHomEquivMulActionIsScalarTower (M : Type u_11) (N : Type u_12) [Monoid M] [Monoid N] : (M →* N) ≃ { _inst : MulAction M N // IsScalarTower M N N }

A monoid homomorphism between two monoids M and N can be equivalently specified by a multiplicative action of M on N that is compatible with the multiplication on N.

Equations

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

def Function.End (α : Type u_7) : Type u_7

The monoid of endomorphisms.

Note that this is generalized by CategoryTheory.End to categories other than Type u.

Equations

• Function.End α = (α → α)

instance instMonoidEnd (α : Type u_7) : Monoid (Function.End α)

Equations

• instMonoidEnd α = Monoid.mk ⋯ ⋯ (fun (n : ℕ) (f : Function.End α) => f^[n]) ⋯ ⋯

instance instInhabitedEnd (α : Type u_7) : Inhabited (Function.End α)

Equations

• instInhabitedEnd α = { default := 1 }

instance Function.End.applyMulAction {α : Type u_7} : MulAction (Function.End α) α

The tautological action by Function.End α on α.

This is generalized to bundled endomorphisms by:

• Equiv.Perm.applyMulAction
• AddMonoid.End.applyDistribMulAction
• AddMonoid.End.applyModule
• AddAut.applyDistribMulAction
• MulAut.applyMulDistribMulAction
• LinearEquiv.applyDistribMulAction
• LinearMap.applyModule
• RingHom.applyMulSemiringAction
• RingAut.applyMulSemiringAction
• AlgEquiv.applyMulSemiringAction

Equations

• Function.End.applyMulAction = MulAction.mk ⋯ ⋯

@[simp]

theorem Function.End.smul_def {α : Type u_7} (f : Function.End α) (a : α) : f • a = f a

theorem Function.End.mul_def {α : Type u_7} (f : Function.End α) (g : Function.End α) : f * g = f ∘ g

theorem Function.End.one_def {α : Type u_7} : 1 = id

instance Function.End.apply_FaithfulSMul {α : Type u_7} : FaithfulSMul (Function.End α) α

Function.End.applyMulAction is faithful.

Equations

• ⋯ = ⋯

def MulAction.toEndHom {M : Type u_1} {α : Type u_7} [Monoid M] [MulAction M α] : M →* Function.End α

The monoid hom representing a monoid action.

When M is a group, see MulAction.toPermHom.

Equations

• MulAction.toEndHom = { toFun := fun (x1 : M) (x2 : α) => x1 • x2, map_one' := ⋯, map_mul' := ⋯ }

@[reducible, inline]

abbrev MulAction.ofEndHom {M : Type u_1} {α : Type u_7} [Monoid M] (f : M →* Function.End α) : MulAction M α

The monoid action induced by a monoid hom to Function.End α

See note [reducible non-instances].

Equations

• MulAction.ofEndHom f = MulAction.compHom α f

Additive, Multiplicative

instance Additive.vadd {α : Type u_7} {β : Type u_8} [SMul α β] : VAdd (Additive α) β

Equations

• Additive.vadd = { vadd := fun (a : Additive α) (x : β) => Additive.toMul a • x }

instance Multiplicative.smul {α : Type u_7} {β : Type u_8} [VAdd α β] : SMul (Multiplicative α) β

Equations

• Multiplicative.smul = { smul := fun (a : Multiplicative α) (x : β) => Multiplicative.toAdd a +ᵥ x }

@[simp]

theorem toMul_smul {α : Type u_7} {β : Type u_8} [SMul α β] (a : Additive α) (b : β) : Additive.toMul a • b = a +ᵥ b

@[simp]

theorem ofMul_vadd {α : Type u_7} {β : Type u_8} [SMul α β] (a : α) (b : β) : Additive.ofMul a +ᵥ b = a • b

@[simp]

theorem toAdd_vadd {α : Type u_7} {β : Type u_8} [VAdd α β] (a : Multiplicative α) (b : β) : Multiplicative.toAdd a +ᵥ b = a • b

@[simp]

theorem ofAdd_smul {α : Type u_7} {β : Type u_8} [VAdd α β] (a : α) (b : β) : Multiplicative.ofAdd a • b = a +ᵥ b

instance Additive.addAction {α : Type u_7} {β : Type u_8} [Monoid α] [MulAction α β] : AddAction (Additive α) β

Equations

• Additive.addAction = AddAction.mk ⋯ ⋯

instance Multiplicative.mulAction {α : Type u_7} {β : Type u_8} [AddMonoid α] [AddAction α β] : MulAction (Multiplicative α) β

Equations

• Multiplicative.mulAction = MulAction.mk ⋯ ⋯

instance Additive.addAction_isPretransitive {α : Type u_7} {β : Type u_8} [Monoid α] [MulAction α β] [MulAction.IsPretransitive α β] : AddAction.IsPretransitive (Additive α) β

Equations

• ⋯ = ⋯

instance Multiplicative.mulAction_isPretransitive {α : Type u_7} {β : Type u_8} [AddMonoid α] [AddAction α β] [AddAction.IsPretransitive α β] : MulAction.IsPretransitive (Multiplicative α) β

Equations

• ⋯ = ⋯

instance Additive.vaddCommClass {α : Type u_7} {β : Type u_8} {γ : Type u_9} [SMul α γ] [SMul β γ] [SMulCommClass α β γ] : VAddCommClass (Additive α) (Additive β) γ

Equations

• ⋯ = ⋯

instance Multiplicative.smulCommClass {α : Type u_7} {β : Type u_8} {γ : Type u_9} [VAdd α γ] [VAdd β γ] [VAddCommClass α β γ] : SMulCommClass (Multiplicative α) (Multiplicative β) γ

Equations

• ⋯ = ⋯

instance AddAction.functionEnd {α : Type u_7} : AddAction (Additive (Function.End α)) α

The tautological additive action by Additive (Function.End α) on α.

Equations

• AddAction.functionEnd = inferInstance

def AddAction.toEndHom {M : Type u_1} {α : Type u_7} [AddMonoid M] [AddAction M α] : M →+ Additive (Function.End α)

The additive monoid hom representing an additive monoid action.

When M is a group, see AddAction.toPermHom.

Equations

• AddAction.toEndHom = MonoidHom.toAdditive'' MulAction.toEndHom

@[reducible, inline]

abbrev AddAction.ofEndHom {M : Type u_1} {α : Type u_7} [AddMonoid M] (f : M →+ Additive (Function.End α)) : AddAction M α

The additive action induced by a hom to Additive (Function.End α)

See note [reducible non-instances].

Equations

• AddAction.ofEndHom f = AddAction.compHom α f