Documentation

Mathlib.GroupTheory.GroupAction.Sigma

Sigma instances for additive and multiplicative actions #

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

See also #

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instance Sigma.VAdd {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → VAdd M (α i)] :
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 }
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instance Sigma.instSMulSigma {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.instSMulSigma = { smul := fun (a : M) => Sigma.map id fun (x : ι) (x_1 : α x) => a • x_1 }
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theorem Sigma.vadd_def {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → VAdd M (α i)] (a : M) (x : (i : ι) × α i) :
a +ᵥ x = Sigma.map id (fun (x : ι) (x_1 : α x) => a +ᵥ x_1) x
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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 = Sigma.map id (fun (x : ι) (x_1 : α x) => a • x_1) x
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@[simp]
theorem Sigma.vadd_mk {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → VAdd M (α i)] (a : M) (i : ι) (b : α i) :
a +ᵥ { fst := i, snd := b } = { fst := i, snd := a +ᵥ b }
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@[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 • { fst := i, snd := b } = { fst := i, snd := a • b }
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instance Sigma.instIsScalarTowerSigmaInstVAddSigmaInstVAddSigma {ι : Type u_1} {M : Type u_2} {N : Type u_3} {α : ι → Type u_4} [(i : ι) → VAdd M (α i)] [(i : ι) → VAdd N (α i)] [VAdd M N] [∀ (i : ι), VAddAssocClass M N (α i)] :
VAddAssocClass M N ((i : ι) × α i)
Equations
  • ⋯ = ⋯
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instance Sigma.instIsScalarTowerSigmaInstSMulSigmaInstSMulSigma {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)
Equations
  • ⋯ = ⋯
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instance Sigma.instVAddCommClassSigmaInstVAddSigmaInstVAddSigma {ι : Type u_1} {M : Type u_2} {N : Type u_3} {α : ι → Type u_4} [(i : ι) → VAdd M (α i)] [(i : ι) → VAdd N (α i)] [∀ (i : ι), VAddCommClass M N (α i)] :
VAddCommClass M N ((i : ι) × α i)
Equations
  • ⋯ = ⋯
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instance Sigma.instSMulCommClassSigmaInstSMulSigmaInstSMulSigma {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)
Equations
  • ⋯ = ⋯
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instance Sigma.instIsCentralVAddSigmaInstVAddSigmaAddOpposite {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → VAdd M (α i)] [(i : ι) → VAdd Mᵃᵒᵖ (α i)] [∀ (i : ι), IsCentralVAdd M (α i)] :
IsCentralVAdd M ((i : ι) × α i)
Equations
  • ⋯ = ⋯
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instance Sigma.instIsCentralScalarSigmaInstSMulSigmaMulOpposite {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)
Equations
  • ⋯ = ⋯
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theorem Sigma.FaithfulVAdd' {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → 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.

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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.

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instance Sigma.instFaithfulVAddSigmaInstVAddSigma {ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} [(i : ι) → VAdd M (α i)] [Nonempty ι] [∀ (i : ι), FaithfulVAdd M (α i)] :
FaithfulVAdd M ((i : ι) × α i)
Equations
  • ⋯ = ⋯
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instance Sigma.instFaithfulSMulSigmaInstSMulSigma {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)
Equations
  • ⋯ = ⋯
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instance Sigma.instAddActionSigma {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
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theorem Sigma.instAddActionSigma.proof_2 {ι : Type u_1} {M : Type u_3} {α : ι → Type u_2} {m : AddMonoid M} [(i : ι) → AddAction M (α i)] (a : M) (b : M) (x : (i : ι) × α i) :
a + b +ᵥ x = a +ᵥ (b +ᵥ x)
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theorem Sigma.instAddActionSigma.proof_1 {ι : Type u_1} {M : Type u_3} {α : ι → Type u_2} {m : AddMonoid M} [(i : ι) → AddAction M (α i)] (x : (i : ι) × α i) :
0 +ᵥ x = x
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instance Sigma.instMulActionSigma {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