Prod instances for additive and multiplicative actions

This file defines instances for binary product of additive and multiplicative actions and provides scalar multiplication as a homomorphism from α × β to β.

Main declarations

• smulMulHom /smulMonoidHom : Scalar multiplication bundled as a multiplicative/monoid homomorphism.

See also

• Mathlib.GroupTheory.GroupAction.Option
• Mathlib.GroupTheory.GroupAction.Pi
• Mathlib.GroupTheory.GroupAction.Sigma
• Mathlib.GroupTheory.GroupAction.Sum

Porting notes

The to_additive attribute can be used to generate both the smul and vadd lemmas from the corresponding pow lemmas, as explained on zulip here: https://leanprover.zulipchat.com/#narrow/near/316087838

This was not done as part of the port in order to stay as close as possible to the mathlib3 code.

instance Prod.vadd {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] : VAdd M (α × β)

instance Prod.smul {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] : SMul M (α × β)

@[simp]

theorem Prod.vadd_fst {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] (a : M) (x : α × β) : (a +ᵥ x).fst = a +ᵥ x.fst

@[simp]

theorem Prod.smul_fst {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] (a : M) (x : α × β) : (a • x).fst = a • x.fst

@[simp]

theorem Prod.vadd_snd {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] (a : M) (x : α × β) : (a +ᵥ x).snd = a +ᵥ x.snd

@[simp]

theorem Prod.smul_snd {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] (a : M) (x : α × β) : (a • x).snd = a • x.snd

@[simp]

theorem Prod.vadd_mk {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] (a : M) (b : α) (c : β) : a +ᵥ (b, c) = (a +ᵥ b, a +ᵥ c)

@[simp]

theorem Prod.smul_mk {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] (a : M) (b : α) (c : β) : a • (b, c) = (a • b, a • c)

theorem Prod.vadd_def {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] (a : M) (x : α × β) : a +ᵥ x = (a +ᵥ x.fst, a +ᵥ x.snd)

theorem Prod.smul_def {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] (a : M) (x : α × β) : a • x = (a • x.fst, a • x.snd)

@[simp]

theorem Prod.vadd_swap {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] (a : M) (x : α × β) : Prod.swap (a +ᵥ x) = a +ᵥ Prod.swap x

@[simp]

theorem Prod.smul_swap {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] (a : M) (x : α × β) : Prod.swap (a • x) = a • Prod.swap x

theorem Prod.smul_zero_mk {M : Type u_1} {β : Type u_6} [SMul M β] {α : Type u_7} [Monoid M] [AddMonoid α] [DistribMulAction M α] (a : M) (c : β) : a • (0, c) = (0, a • c)

theorem Prod.smul_mk_zero {M : Type u_1} {α : Type u_5} [SMul M α] {β : Type u_7} [Monoid M] [AddMonoid β] [DistribMulAction M β] (a : M) (b : α) : a • (b, 0) = (a • b, 0)

instance Prod.pow {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] : Pow (α × β) E

@[simp]

theorem Prod.pow_fst {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] (p : α × β) (c : E) : (p ^ c).fst = p.fst ^ c

@[simp]

theorem Prod.pow_snd {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] (p : α × β) (c : E) : (p ^ c).snd = p.snd ^ c

@[simp]

theorem Prod.pow_mk {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] (c : E) (a : α) (b : β) : (a, b) ^ c = (a ^ c, b ^ c)

theorem Prod.pow_def {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] (p : α × β) (c : E) : p ^ c = (p.fst ^ c, p.snd ^ c)

@[simp]

theorem Prod.pow_swap {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] (p : α × β) (c : E) : Prod.swap (p ^ c) = Prod.swap p ^ c

instance Prod.vaddAssocClass {M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] [VAdd N α] [VAdd N β] [VAdd M N] [VAddAssocClass M N α] [VAddAssocClass M N β] : VAddAssocClass M N (α × β)

theorem Prod.vaddAssocClass.proof_1 {M : Type u_1} {N : Type u_2} {α : Type u_3} {β : Type u_4} [VAdd M α] [VAdd M β] [VAdd N α] [VAdd N β] [VAdd M N] [VAddAssocClass M N α] [VAddAssocClass M N β] : VAddAssocClass M N (α × β)

instance Prod.isScalarTower {M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [SMul N α] [SMul N β] [SMul M N] [IsScalarTower M N α] [IsScalarTower M N β] : IsScalarTower M N (α × β)

instance Prod.vaddCommClass {M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] [VAdd N α] [VAdd N β] [VAddCommClass M N α] [VAddCommClass M N β] : VAddCommClass M N (α × β)

theorem Prod.vaddCommClass.proof_1 {M : Type u_1} {N : Type u_2} {α : Type u_3} {β : Type u_4} [VAdd M α] [VAdd M β] [VAdd N α] [VAdd N β] [VAddCommClass M N α] [VAddCommClass M N β] : VAddCommClass M N (α × β)

instance Prod.smulCommClass {M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [SMul N α] [SMul N β] [SMulCommClass M N α] [SMulCommClass M N β] : SMulCommClass M N (α × β)

theorem Prod.isCentralVAdd.proof_1 {M : Type u_1} {α : Type u_2} {β : Type u_3} [VAdd M α] [VAdd M β] [VAdd Mᵃᵒᵖ α] [VAdd Mᵃᵒᵖ β] [IsCentralVAdd M α] [IsCentralVAdd M β] : IsCentralVAdd M (α × β)

instance Prod.isCentralVAdd {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] [VAdd Mᵃᵒᵖ α] [VAdd Mᵃᵒᵖ β] [IsCentralVAdd M α] [IsCentralVAdd M β] : IsCentralVAdd M (α × β)

instance Prod.isCentralScalar {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [SMul Mᵐᵒᵖ α] [SMul Mᵐᵒᵖ β] [IsCentralScalar M α] [IsCentralScalar M β] : IsCentralScalar M (α × β)

instance Prod.faithfulVAddLeft {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] [FaithfulVAdd M α] [Nonempty β] : FaithfulVAdd M (α × β)

theorem Prod.faithfulVAddLeft.proof_1 {M : Type u_1} {α : Type u_2} {β : Type u_3} [VAdd M α] [VAdd M β] [FaithfulVAdd M α] [Nonempty β] : FaithfulVAdd M (α × β)

abbrev Prod.faithfulVAddLeft.match_1 {β : Type u_1} (motive : Nonempty β → Prop) : (x : Nonempty β) → ((b : β) → motive (_ : Nonempty β)) → motive x

instance Prod.faithfulSMulLeft {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [FaithfulSMul M α] [Nonempty β] : FaithfulSMul M (α × β)

theorem Prod.faithfulVAddRight.proof_1 {M : Type u_1} {α : Type u_2} {β : Type u_3} [VAdd M α] [VAdd M β] [Nonempty α] [FaithfulVAdd M β] : FaithfulVAdd M (α × β)

instance Prod.faithfulVAddRight {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] [Nonempty α] [FaithfulVAdd M β] : FaithfulVAdd M (α × β)

instance Prod.faithfulSMulRight {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [Nonempty α] [FaithfulSMul M β] : FaithfulSMul M (α × β)

theorem Prod.vaddCommClassBoth.proof_1 {M : Type u_1} {N : Type u_2} {P : Type u_3} [Add N] [Add P] [VAdd M N] [VAdd M P] [VAddCommClass M N N] [VAddCommClass M P P] : VAddCommClass M (N × P) (N × P)

instance Prod.vaddCommClassBoth {M : Type u_1} {N : Type u_2} {P : Type u_3} [Add N] [Add P] [VAdd M N] [VAdd M P] [VAddCommClass M N N] [VAddCommClass M P P] : VAddCommClass M (N × P) (N × P)

instance Prod.smulCommClassBoth {M : Type u_1} {N : Type u_2} {P : Type u_3} [Mul N] [Mul P] [SMul M N] [SMul M P] [SMulCommClass M N N] [SMulCommClass M P P] : SMulCommClass M (N × P) (N × P)

instance Prod.isScalarTowerBoth {M : Type u_1} {N : Type u_2} {P : Type u_3} [Mul N] [Mul P] [SMul M N] [SMul M P] [IsScalarTower M N N] [IsScalarTower M P P] : IsScalarTower M (N × P) (N × P)

instance Prod.addAction {M : Type u_1} {α : Type u_5} {β : Type u_6} [AddMonoid M] [AddAction M α] [AddAction M β] : AddAction M (α × β)

theorem Prod.addAction.proof_2 {M : Type u_3} {α : Type u_1} {β : Type u_2} [AddMonoid M] [AddAction M α] [AddAction M β] : ∀ (x x_1 : M) (x_2 : α × β), (x + x_1 +ᵥ x_2.fst, x + x_1 +ᵥ x_2.snd) = (x +ᵥ (x_1 +ᵥ x_2).fst, x +ᵥ (x_1 +ᵥ x_2).snd)

abbrev Prod.addAction.match_1 {α : Type u_1} {β : Type u_2} (motive : α × β → Prop) : (x : α × β) → ((fst : α) → (snd : β) → motive (fst, snd)) → motive x

theorem Prod.addAction.proof_1 {M : Type u_3} {α : Type u_1} {β : Type u_2} [AddMonoid M] [AddAction M α] [AddAction M β] : ∀ (x : α × β), 0 +ᵥ x = x

instance Prod.mulAction {M : Type u_1} {α : Type u_5} {β : Type u_6} [Monoid M] [MulAction M α] [MulAction M β] : MulAction M (α × β)

instance Prod.smulZeroClass {R : Type u_7} {M : Type u_8} {N : Type u_9} [Zero M] [Zero N] [SMulZeroClass R M] [SMulZeroClass R N] : SMulZeroClass R (M × N)

instance Prod.distribSMul {R : Type u_7} {M : Type u_8} {N : Type u_9} [AddZeroClass M] [AddZeroClass N] [DistribSMul R M] [DistribSMul R N] : DistribSMul R (M × N)

instance Prod.distribMulAction {M : Type u_1} {N : Type u_2} {R : Type u_7} [Monoid R] [AddMonoid M] [AddMonoid N] [DistribMulAction R M] [DistribMulAction R N] : DistribMulAction R (M × N)

instance Prod.mulDistribMulAction {M : Type u_1} {N : Type u_2} {R : Type u_7} [Monoid R] [Monoid M] [Monoid N] [MulDistribMulAction R M] [MulDistribMulAction R N] : MulDistribMulAction R (M × N)

Scalar multiplication as a homomorphism

@[simp]

theorem smulMulHom_apply {α : Type u_5} {β : Type u_6} [Monoid α] [Mul β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] (a : α × β) : ↑smulMulHom a = a.fst • a.snd

def smulMulHom {α : Type u_5} {β : Type u_6} [Monoid α] [Mul β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] : α × β →ₙ* β

Scalar multiplication as a multiplicative homomorphism.

@[simp]

theorem smulMonoidHom_apply {α : Type u_5} {β : Type u_6} [Monoid α] [MulOneClass β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] : ∀ (a : α × β), ↑smulMonoidHom a = MulHom.toFun smulMulHom a

def smulMonoidHom {α : Type u_5} {β : Type u_6} [Monoid α] [MulOneClass β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] : α × β →* β

Scalar multiplication as a monoid homomorphism.