Sigma instances for additive and multiplicative actions

This file defines instances for arbitrary sum of additive and multiplicative actions.

See also

• Mathlib/Algebra/Group/Action/Option.lean
• Mathlib/Algebra/Group/Action/Pi.lean
• Mathlib/Algebra/Group/Action/Prod.lean
• Mathlib/Algebra/Group/Action/Sum.lean

@[instance_reducible]

instance Sigma.instSMul_mathlib {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → SMul M (α i)] : SMul M ((i : ι) × α i)

Equations

• Sigma.instSMul_mathlib = { smul := fun (a : M) => Sigma.map id fun (x : ι) (x_1 : α x) => a • x_1 }

@[instance_reducible]

instance Sigma.VAdd {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → _root_.VAdd M (α i)] : _root_.VAdd M ((i : ι) × α i)

Equations

• Sigma.VAdd = { vadd := fun (a : M) => Sigma.map id fun (x : ι) (x_1 : α x) => a +ᵥ x_1 }

theorem Sigma.smul_def {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → SMul M (α i)] (a : M) (x : (i : ι) × α i) : a • x = map id (fun (x : ι) (x_1 : α x) => a • x_1) x

theorem Sigma.vadd_def {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → _root_.VAdd M (α i)] (a : M) (x : (i : ι) × α i) : a +ᵥ x = map id (fun (x : ι) (x_1 : α x) => a +ᵥ x_1) x

@[simp]

theorem Sigma.smul_mk {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → SMul M (α i)] (a : M) (i : ι) (b : α i) : a • ⟨i, b⟩ = ⟨i, a • b⟩

@[simp]

theorem Sigma.vadd_mk {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → _root_.VAdd M (α i)] (a : M) (i : ι) (b : α i) : a +ᵥ ⟨i, b⟩ = ⟨i, a +ᵥ b⟩

instance Sigma.instIsScalarTowerOfSMul {ι : Type u_1} {M : Type u_2} {N : Type u_3} {α : ι → Type u_4} [(i : ι) → SMul M (α i)] [(i : ι) → SMul N (α i)] [SMul M N] [∀ (i : ι), IsScalarTower M N (α i)] : IsScalarTower M N ((i : ι) × α i)

instance Sigma.instVAddAssocClassOfVAdd {ι : Type u_1} {M : Type u_2} {N : Type u_3} {α : ι → Type u_4} [(i : ι) → _root_.VAdd M (α i)] [(i : ι) → _root_.VAdd N (α i)] [_root_.VAdd M N] [∀ (i : ι), VAddAssocClass M N (α i)] : VAddAssocClass M N ((i : ι) × α i)

instance Sigma.instSMulCommClass {ι : Type u_1} {M : Type u_2} {N : Type u_3} {α : ι → Type u_4} [(i : ι) → SMul M (α i)] [(i : ι) → SMul N (α i)] [∀ (i : ι), SMulCommClass M N (α i)] : SMulCommClass M N ((i : ι) × α i)

instance Sigma.instVAddCommClass {ι : Type u_1} {M : Type u_2} {N : Type u_3} {α : ι → Type u_4} [(i : ι) → _root_.VAdd M (α i)] [(i : ι) → _root_.VAdd N (α i)] [∀ (i : ι), VAddCommClass M N (α i)] : VAddCommClass M N ((i : ι) × α i)

instance Sigma.instIsCentralScalar {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → SMul M (α i)] [(i : ι) → SMul Mᵐᵒᵖ (α i)] [∀ (i : ι), IsCentralScalar M (α i)] : IsCentralScalar M ((i : ι) × α i)

instance Sigma.instIsCentralVAdd {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → _root_.VAdd M (α i)] [(i : ι) → _root_.VAdd Mᵃᵒᵖ (α i)] [∀ (i : ι), IsCentralVAdd M (α i)] : IsCentralVAdd M ((i : ι) × α i)

theorem Sigma.FaithfulSMul' {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → SMul M (α i)] (i : ι) [FaithfulSMul M (α i)] : FaithfulSMul M ((i : ι) × α i)

This is not an instance because i becomes a metavariable.

theorem Sigma.FaithfulVAdd' {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → _root_.VAdd M (α i)] (i : ι) [FaithfulVAdd M (α i)] : FaithfulVAdd M ((i : ι) × α i)

This is not an instance because i becomes a metavariable.

instance Sigma.instFaithfulSMulOfNonempty {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → SMul M (α i)] [Nonempty ι] [∀ (i : ι), FaithfulSMul M (α i)] : FaithfulSMul M ((i : ι) × α i)

instance Sigma.instFaithfulVAddOfNonempty {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → _root_.VAdd M (α i)] [Nonempty ι] [∀ (i : ι), FaithfulVAdd M (α i)] : FaithfulVAdd M ((i : ι) × α i)

@[instance_reducible]

instance Sigma.instMulAction {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} {m : Monoid M} [(i : ι) → MulAction M (α i)] : MulAction M ((i : ι) × α i)

Equations

• Sigma.instMulAction = { toSMul := Sigma.instSMul_mathlib, mul_smul := ⋯, one_smul := ⋯ }

@[instance_reducible]

instance Sigma.instAddAction {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} {m : AddMonoid M} [(i : ι) → AddAction M (α i)] : AddAction M ((i : ι) × α i)

Equations

• Sigma.instAddAction = { toVAdd := Sigma.VAdd, add_vadd := ⋯, zero_vadd := ⋯ }