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

class SMulZeroClass (M : Type u_10) (A : Type u_11) [Zero A] extends SMul : Type (max u_10 u_11)

Typeclass for scalar multiplication that preserves 0 on the right.

• smul : M → A → A
• smul_zero : ∀ (a : M), a • 0 = 0

  Multiplying 0 by a scalar gives 0

Instances

• AddMonoidHom.smulZeroClass
• Units.instSMulZeroClass

theorem SMulZeroClass.smul_zero {M : Type u_10} {A : Type u_11} [Zero A] [self : SMulZeroClass M A] (a : M) : a • 0 = 0

Multiplying 0 by a scalar gives 0

@[simp]

theorem smul_zero {M : Type u_1} {A : Type u_4} [Zero A] [SMulZeroClass M A] (a : M) : a • 0 = 0

theorem smul_ite_zero {M : Type u_1} {A : Type u_4} [Zero A] [SMulZeroClass M A] (p : Prop) [Decidable p] (a : M) (b : A) : (a • if p then b else 0) = if p then a • b else 0

theorem smul_eq_zero_of_right {M : Type u_1} {A : Type u_4} [Zero A] [SMulZeroClass M A] (a : M) {b : A} (h : b = 0) : a • b = 0

theorem right_ne_zero_of_smul {M : Type u_1} {A : Type u_4} [Zero A] [SMulZeroClass M A] {a : M} {b : A} : a • b ≠ 0 → b ≠ 0

@[reducible, inline]

abbrev Function.Injective.smulZeroClass {M : Type u_1} {A : Type u_4} {B : Type u_5} [Zero A] [SMulZeroClass M A] [Zero B] [SMul M B] (f : ZeroHom B A) (hf : Function.Injective ⇑f) (smul : ∀ (c : M) (x : B), f (c • x) = c • f x) : SMulZeroClass M B

Pullback a zero-preserving scalar multiplication along an injective zero-preserving map. See note [reducible non-instances].

Equations

• Function.Injective.smulZeroClass f hf smul = SMulZeroClass.mk ⋯

@[reducible, inline]

abbrev ZeroHom.smulZeroClass {M : Type u_1} {A : Type u_4} {B : Type u_5} [Zero A] [SMulZeroClass M A] [Zero B] [SMul M B] (f : ZeroHom A B) (smul : ∀ (c : M) (x : A), f (c • x) = c • f x) : SMulZeroClass M B

Pushforward a zero-preserving scalar multiplication along a zero-preserving map. See note [reducible non-instances].

Equations

• f.smulZeroClass smul = SMulZeroClass.mk ⋯

@[reducible, inline]

abbrev Function.Surjective.smulZeroClassLeft {R : Type u_10} {S : Type u_11} {M : Type u_12} [Zero M] [SMulZeroClass R M] [SMul S M] (f : R → S) (hf : Function.Surjective f) (hsmul : ∀ (c : R) (x : M), f c • x = c • x) : SMulZeroClass S M

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

See also Function.Surjective.distribMulActionLeft.

Equations

• Function.Surjective.smulZeroClassLeft f hf hsmul = SMulZeroClass.mk ⋯

@[reducible, inline]

abbrev SMulZeroClass.compFun {M : Type u_1} {N : Type u_2} (A : Type u_4) [Zero A] [SMulZeroClass M A] (f : N → M) : SMulZeroClass N A

Compose a SMulZeroClass with a function, with scalar multiplication f r' • m. See note [reducible non-instances].

Equations

• SMulZeroClass.compFun A f = SMulZeroClass.mk ⋯

@[simp]

theorem SMulZeroClass.toZeroHom_apply {M : Type u_1} (A : Type u_4) [Zero A] [SMulZeroClass M A] (x : M) : ∀ (x_1 : A), (SMulZeroClass.toZeroHom A x) x_1 = x • x_1

def SMulZeroClass.toZeroHom {M : Type u_1} (A : Type u_4) [Zero A] [SMulZeroClass M A] (x : M) : ZeroHom A A

Each element of the scalars defines a zero-preserving map.

Equations

• SMulZeroClass.toZeroHom A x = { toFun := fun (x_1 : A) => x • x_1, map_zero' := ⋯ }

theorem DistribSMul.ext {M : Type u_10} {A : Type u_11} : ∀ {inst : AddZeroClass A} (x y : DistribSMul M A), SMul.smul = SMul.smul → x = y

theorem DistribSMul.ext_iff {M : Type u_10} {A : Type u_11} : ∀ {inst : AddZeroClass A} (x y : DistribSMul M A), x = y ↔ SMul.smul = SMul.smul

class DistribSMul (M : Type u_10) (A : Type u_11) [AddZeroClass A] extends SMulZeroClass : Type (max u_10 u_11)

Typeclass for scalar multiplication that preserves 0 and + on the right.

This is exactly DistribMulAction without the MulAction part.

• smul : M → A → A
• smul_zero : ∀ (a : M), a • 0 = 0
• smul_add : ∀ (a : M) (x y : A), a • (x + y) = a • x + a • y

  Scalar multiplication distributes across addition

Instances

• DistribMulAction.toDistribSMul
• NNRat.instDistribSMul
• NonUnitalNonAssocSemiring.toDistribSMul
• Rat.instDistribSMul
• Units.instDistribSMulUnits

theorem DistribSMul.smul_add {M : Type u_10} {A : Type u_11} [AddZeroClass A] [self : DistribSMul M A] (a : M) (x : A) (y : A) : a • (x + y) = a • x + a • y

Scalar multiplication distributes across addition

theorem smul_add {M : Type u_1} {A : Type u_4} [AddZeroClass A] [DistribSMul M A] (a : M) (b₁ : A) (b₂ : A) : a • (b₁ + b₂) = a • b₁ + a • b₂

instance AddMonoidHom.smulZeroClass {M : Type u_1} {A : Type u_4} {B : Type u_5} [AddZeroClass A] [DistribSMul M A] [AddZeroClass B] : SMulZeroClass M (B →+ A)

Equations

• AddMonoidHom.smulZeroClass = SMulZeroClass.mk ⋯

@[reducible, inline]

abbrev Function.Injective.distribSMul {M : Type u_1} {A : Type u_4} {B : Type u_5} [AddZeroClass A] [DistribSMul M A] [AddZeroClass B] [SMul M B] (f : B →+ A) (hf : Function.Injective ⇑f) (smul : ∀ (c : M) (x : B), f (c • x) = c • f x) : DistribSMul M B

Pullback a distributive scalar multiplication along an injective additive monoid homomorphism. See note [reducible non-instances].

Equations

• Function.Injective.distribSMul f hf smul = let __src := Function.Injective.smulZeroClass (↑f) hf smul; DistribSMul.mk ⋯

@[reducible, inline]

abbrev Function.Surjective.distribSMul {M : Type u_1} {A : Type u_4} {B : Type u_5} [AddZeroClass A] [DistribSMul M A] [AddZeroClass B] [SMul M B] (f : A →+ B) (hf : Function.Surjective ⇑f) (smul : ∀ (c : M) (x : A), f (c • x) = c • f x) : DistribSMul M B

Pushforward a distributive scalar multiplication along a surjective additive monoid homomorphism. See note [reducible non-instances].

Equations

• Function.Surjective.distribSMul f hf smul = let __src := (↑f).smulZeroClass smul; DistribSMul.mk ⋯

@[reducible, inline]

abbrev Function.Surjective.distribSMulLeft {R : Type u_10} {S : Type u_11} {M : Type u_12} [AddZeroClass M] [DistribSMul R M] [SMul S M] (f : R → S) (hf : Function.Surjective f) (hsmul : ∀ (c : R) (x : M), f c • x = c • x) : DistribSMul S M

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

See also Function.Surjective.distribMulActionLeft.

Equations

• Function.Surjective.distribSMulLeft f hf hsmul = let __src := Function.Surjective.smulZeroClassLeft f hf hsmul; DistribSMul.mk ⋯

@[reducible, inline]

abbrev DistribSMul.compFun {M : Type u_1} {N : Type u_2} (A : Type u_4) [AddZeroClass A] [DistribSMul M A] (f : N → M) : DistribSMul N A

Compose a DistribSMul with a function, with scalar multiplication f r' • m. See note [reducible non-instances].

Equations

• DistribSMul.compFun A f = let __src := SMulZeroClass.compFun A f; DistribSMul.mk ⋯

@[simp]

theorem DistribSMul.toAddMonoidHom_apply {M : Type u_1} (A : Type u_4) [AddZeroClass A] [DistribSMul M A] (x : M) : ∀ (x_1 : A), (DistribSMul.toAddMonoidHom A x) x_1 = (fun (x : M) (x_2 : A) => x • x_2) x x_1

def DistribSMul.toAddMonoidHom {M : Type u_1} (A : Type u_4) [AddZeroClass A] [DistribSMul M A] (x : M) : A →+ A

Each element of the scalars defines an additive monoid homomorphism.

Equations

• DistribSMul.toAddMonoidHom A x = let __src := SMulZeroClass.toZeroHom A x; { toFun := (fun (x : M) (x_1 : A) => x • x_1) x, map_zero' := ⋯, map_add' := ⋯ }

theorem DistribMulAction.ext_iff {M : Type u_10} {A : Type u_11} : ∀ {inst : Monoid M} {inst_1 : AddMonoid A} (x y : DistribMulAction M A), x = y ↔ SMul.smul = SMul.smul

theorem DistribMulAction.ext {M : Type u_10} {A : Type u_11} : ∀ {inst : Monoid M} {inst_1 : AddMonoid A} (x y : DistribMulAction M A), SMul.smul = SMul.smul → x = y

class DistribMulAction (M : Type u_10) (A : Type u_11) [Monoid M] [AddMonoid A] extends MulAction : Type (max u_10 u_11)

Typeclass for multiplicative actions on additive structures. This generalizes group modules.

• smul : M → A → A
• one_smul : ∀ (b : A), 1 • b = b
• mul_smul : ∀ (x y : M) (b : A), (x * y) • b = x • y • b
• smul_zero : ∀ (a : M), a • 0 = 0

  Multiplying 0 by a scalar gives 0

• smul_add : ∀ (a : M) (x y : A), a • (x + y) = a • x + a • y

  Scalar multiplication distributes across addition

Instances

• AddAut.applyDistribMulAction
• AddMonoid.End.applyDistribMulAction
• Submonoid.distribMulAction
• Units.instDistribMulAction
• selfAdjoint.instDistribMulActionSubtypeMemAddSubgroupOfStarModule
• skewAdjoint.instDistribMulActionSubtypeMemAddSubgroupOfStarModule

theorem DistribMulAction.smul_zero {M : Type u_10} {A : Type u_11} [Monoid M] [AddMonoid A] [self : DistribMulAction M A] (a : M) : a • 0 = 0

Multiplying 0 by a scalar gives 0

theorem DistribMulAction.smul_add {M : Type u_10} {A : Type u_11} [Monoid M] [AddMonoid A] [self : DistribMulAction M A] (a : M) (x : A) (y : A) : a • (x + y) = a • x + a • y

Scalar multiplication distributes across addition

@[instance 100]

instance DistribMulAction.toDistribSMul {M : Type u_1} {A : Type u_4} [Monoid M] [AddMonoid A] [DistribMulAction M A] : DistribSMul M A

Equations

• DistribMulAction.toDistribSMul = let __src := inst; DistribSMul.mk ⋯

Since Lean 3 does not have definitional eta for structures, we have to make sure that the definition of DistribMulAction.toDistribSMul was done correctly, and the two paths from DistribMulAction to SMul are indeed definitionally equal.

@[reducible, inline]

abbrev Function.Injective.distribMulAction {M : Type u_1} {A : Type u_4} {B : Type u_5} [Monoid M] [AddMonoid A] [DistribMulAction M A] [AddMonoid B] [SMul M B] (f : B →+ A) (hf : Function.Injective ⇑f) (smul : ∀ (c : M) (x : B), f (c • x) = c • f x) : DistribMulAction M B

Pullback a distributive multiplicative action along an injective additive monoid homomorphism. See note [reducible non-instances].

Equations

• Function.Injective.distribMulAction f hf smul = let __src := Function.Injective.distribSMul f hf smul; let __src_1 := Function.Injective.mulAction (⇑f) hf smul; DistribMulAction.mk ⋯ ⋯

@[reducible, inline]

abbrev Function.Surjective.distribMulAction {M : Type u_1} {A : Type u_4} {B : Type u_5} [Monoid M] [AddMonoid A] [DistribMulAction M A] [AddMonoid B] [SMul M B] (f : A →+ B) (hf : Function.Surjective ⇑f) (smul : ∀ (c : M) (x : A), f (c • x) = c • f x) : DistribMulAction M B

Pushforward a distributive multiplicative action along a surjective additive monoid homomorphism. See note [reducible non-instances].

Equations

• Function.Surjective.distribMulAction f hf smul = let __src := Function.Surjective.distribSMul f hf smul; let __src_1 := Function.Surjective.mulAction (⇑f) hf smul; DistribMulAction.mk ⋯ ⋯

@[reducible, inline]

abbrev Function.Surjective.distribMulActionLeft {R : Type u_10} {S : Type u_11} {M : Type u_12} [Monoid R] [AddMonoid M] [DistribMulAction R M] [Monoid S] [SMul S M] (f : R →* S) (hf : Function.Surjective ⇑f) (hsmul : ∀ (c : R) (x : M), f c • x = c • x) : DistribMulAction S M

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

See also Function.Surjective.mulActionLeft and Function.Surjective.moduleLeft.

Equations

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

@[reducible, inline]

abbrev DistribMulAction.compHom {M : Type u_1} {N : Type u_2} (A : Type u_4) [Monoid M] [AddMonoid A] [DistribMulAction M A] [Monoid N] (f : N →* M) : DistribMulAction N A

Compose a DistribMulAction with a MonoidHom, with action f r' • m. See note [reducible non-instances].

Equations

• DistribMulAction.compHom A f = let __src := DistribSMul.compFun A ⇑f; let __src_1 := MulAction.compHom A f; DistribMulAction.mk ⋯ ⋯

@[simp]

theorem DistribMulAction.toAddMonoidHom_apply {M : Type u_1} (A : Type u_4) [Monoid M] [AddMonoid A] [DistribMulAction M A] (x : M) : ∀ (x_1 : A), (DistribMulAction.toAddMonoidHom A x) x_1 = x • x_1

def DistribMulAction.toAddMonoidHom {M : Type u_1} (A : Type u_4) [Monoid M] [AddMonoid A] [DistribMulAction M A] (x : M) : A →+ A

Each element of the monoid defines an additive monoid homomorphism.

Equations

• DistribMulAction.toAddMonoidHom A x = DistribSMul.toAddMonoidHom A x

@[simp]

theorem DistribMulAction.toAddMonoidEnd_apply (M : Type u_1) (A : Type u_4) [Monoid M] [AddMonoid A] [DistribMulAction M A] (x : M) : (DistribMulAction.toAddMonoidEnd M A) x = DistribMulAction.toAddMonoidHom A x

def DistribMulAction.toAddMonoidEnd (M : Type u_1) (A : Type u_4) [Monoid M] [AddMonoid A] [DistribMulAction M A] : M →* AddMonoid.End A

Each element of the monoid defines an additive monoid homomorphism.

Equations

• DistribMulAction.toAddMonoidEnd M A = { toFun := DistribMulAction.toAddMonoidHom A, map_one' := ⋯, map_mul' := ⋯ }

instance AddMonoid.nat_smulCommClass (M : Type u_1) (A : Type u_4) [Monoid M] [AddMonoid A] [DistribMulAction M A] : SMulCommClass ℕ M A

Equations

• ⋯ = ⋯

instance AddMonoid.nat_smulCommClass' (M : Type u_1) (A : Type u_4) [Monoid M] [AddMonoid A] [DistribMulAction M A] : SMulCommClass M ℕ A

Equations

• ⋯ = ⋯

instance AddGroup.int_smulCommClass {M : Type u_1} {A : Type u_4} [Monoid M] [AddGroup A] [DistribMulAction M A] : SMulCommClass ℤ M A

Equations

• ⋯ = ⋯

instance AddGroup.int_smulCommClass' {M : Type u_1} {A : Type u_4} [Monoid M] [AddGroup A] [DistribMulAction M A] : SMulCommClass M ℤ A

Equations

• ⋯ = ⋯

@[simp]

theorem smul_neg {M : Type u_1} {A : Type u_4} [Monoid M] [AddGroup A] [DistribMulAction M A] (r : M) (x : A) : r • -x = -(r • x)

theorem smul_sub {M : Type u_1} {A : Type u_4} [Monoid M] [AddGroup A] [DistribMulAction M A] (r : M) (x : A) (y : A) : r • (x - y) = r • x - r • y

theorem MulDistribMulAction.ext {M : Type u_10} {A : Type u_11} : ∀ {inst : Monoid M} {inst_1 : Monoid A} (x y : MulDistribMulAction M A), SMul.smul = SMul.smul → x = y

theorem MulDistribMulAction.ext_iff {M : Type u_10} {A : Type u_11} : ∀ {inst : Monoid M} {inst_1 : Monoid A} (x y : MulDistribMulAction M A), x = y ↔ SMul.smul = SMul.smul

class MulDistribMulAction (M : Type u_10) (A : Type u_11) [Monoid M] [Monoid A] extends MulAction : Type (max u_10 u_11)

Typeclass for multiplicative actions on multiplicative structures. This generalizes conjugation actions.

• smul : M → A → A
• one_smul : ∀ (b : A), 1 • b = b
• mul_smul : ∀ (x y : M) (b : A), (x * y) • b = x • y • b
• smul_mul : ∀ (r : M) (x y : A), r • (x * y) = r • x * r • y

  Distributivity of • across *

• smul_one : ∀ (r : M), r • 1 = 1

  Multiplying 1 by a scalar gives 1

Instances

• MulAut.applyMulDistribMulAction
• MulSemiringAction.toMulDistribMulAction
• Submonoid.mulDistribMulAction
• Units.instMulDistribMulAction
• Units.mulDistribMulAction'

theorem MulDistribMulAction.smul_mul {M : Type u_10} {A : Type u_11} [Monoid M] [Monoid A] [self : MulDistribMulAction M A] (r : M) (x : A) (y : A) : r • (x * y) = r • x * r • y

Distributivity of • across *

theorem MulDistribMulAction.smul_one {M : Type u_10} {A : Type u_11} [Monoid M] [Monoid A] [self : MulDistribMulAction M A] (r : M) : r • 1 = 1

Multiplying 1 by a scalar gives 1

theorem smul_mul' {M : Type u_1} {A : Type u_4} [Monoid M] [Monoid A] [MulDistribMulAction M A] (a : M) (b₁ : A) (b₂ : A) : a • (b₁ * b₂) = a • b₁ * a • b₂

@[reducible, inline]

abbrev Function.Injective.mulDistribMulAction {M : Type u_1} {A : Type u_4} {B : Type u_5} [Monoid M] [Monoid A] [MulDistribMulAction M A] [Monoid B] [SMul M B] (f : B →* A) (hf : Function.Injective ⇑f) (smul : ∀ (c : M) (x : B), f (c • x) = c • f x) : MulDistribMulAction M B

Pullback a multiplicative distributive multiplicative action along an injective monoid homomorphism. See note [reducible non-instances].

Equations

• Function.Injective.mulDistribMulAction f hf smul = let __src := Function.Injective.mulAction (⇑f) hf smul; MulDistribMulAction.mk ⋯ ⋯

@[reducible, inline]

abbrev Function.Surjective.mulDistribMulAction {M : Type u_1} {A : Type u_4} {B : Type u_5} [Monoid M] [Monoid A] [MulDistribMulAction M A] [Monoid B] [SMul M B] (f : A →* B) (hf : Function.Surjective ⇑f) (smul : ∀ (c : M) (x : A), f (c • x) = c • f x) : MulDistribMulAction M B

Pushforward a multiplicative distributive multiplicative action along a surjective monoid homomorphism. See note [reducible non-instances].

Equations

• Function.Surjective.mulDistribMulAction f hf smul = let __src := Function.Surjective.mulAction (⇑f) hf smul; MulDistribMulAction.mk ⋯ ⋯

@[reducible, inline]

abbrev MulDistribMulAction.compHom {M : Type u_1} {N : Type u_2} (A : Type u_4) [Monoid M] [Monoid A] [MulDistribMulAction M A] [Monoid N] (f : N →* M) : MulDistribMulAction N A

Compose a MulDistribMulAction with a MonoidHom, with action f r' • m. See note [reducible non-instances].

Equations

• MulDistribMulAction.compHom A f = let __src := MulAction.compHom A f; MulDistribMulAction.mk ⋯ ⋯

def MulDistribMulAction.toMonoidHom {M : Type u_1} (A : Type u_4) [Monoid M] [Monoid A] [MulDistribMulAction M A] (r : M) : A →* A

Scalar multiplication by r as a MonoidHom.

Equations

• MulDistribMulAction.toMonoidHom A r = { toFun := fun (x : A) => r • x, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem MulDistribMulAction.toMonoidHom_apply {M : Type u_1} {A : Type u_4} [Monoid M] [Monoid A] [MulDistribMulAction M A] (r : M) (x : A) : (MulDistribMulAction.toMonoidHom A r) x = r • x

@[simp]

theorem smul_pow' {M : Type u_1} {A : Type u_4} [Monoid M] [Monoid A] [MulDistribMulAction M A] (r : M) (x : A) (n : ℕ) : r • x ^ n = (r • x) ^ n

@[simp]

theorem MulDistribMulAction.toMonoidEnd_apply (M : Type u_1) (A : Type u_4) [Monoid M] [Monoid A] [MulDistribMulAction M A] (r : M) : (MulDistribMulAction.toMonoidEnd M A) r = MulDistribMulAction.toMonoidHom A r

def MulDistribMulAction.toMonoidEnd (M : Type u_1) (A : Type u_4) [Monoid M] [Monoid A] [MulDistribMulAction M A] : M →* Monoid.End A

Each element of the monoid defines a monoid homomorphism.

Equations

• MulDistribMulAction.toMonoidEnd M A = { toFun := MulDistribMulAction.toMonoidHom A, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem smul_inv' {M : Type u_1} {A : Type u_4} [Monoid M] [Group A] [MulDistribMulAction M A] (r : M) (x : A) : r • x⁻¹ = (r • x)⁻¹

theorem smul_div' {M : Type u_1} {A : Type u_4} [Monoid M] [Group A] [MulDistribMulAction M A] (r : M) (x : A) (y : A) : r • (x / y) = r • x / r • y

instance AddMonoid.End.applyDistribMulAction {α : Type u_6} [AddMonoid α] : DistribMulAction (AddMonoid.End α) α

The tautological action by AddMonoid.End α on α.

This generalizes Function.End.applyMulAction.

Equations

• AddMonoid.End.applyDistribMulAction = DistribMulAction.mk ⋯ ⋯

@[simp]

theorem AddMonoid.End.smul_def {α : Type u_6} [AddMonoid α] (f : AddMonoid.End α) (a : α) : f • a = f a

instance AddMonoid.End.applyFaithfulSMul {α : Type u_6} [AddMonoid α] : FaithfulSMul (AddMonoid.End α) α

AddMonoid.End.applyDistribMulAction is faithful.

Equations

• ⋯ = ⋯