File: Mathlib/Algebra/Group/Action/Sigma.lean

full_name: Sigma.instSMul
code: @[to_additive Sigma.VAdd]
instance : SMul M (Σi, α i) 
goal: SMul M ((i : ι) × α i)

full_name: Sigma.VAdd
code: @[to_additive Sigma.VAdd]
instance : SMul M (Σi, α i) 
goal: VAdd M ((i : ι) × α i)

full_name: Sigma.vadd_def
code: @[to_additive]
theorem smul_def : a • x = x.map id fun _ => (a • ·) 
goal: a +ᵥ x = Sigma.map id (fun x x_1 => a +ᵥ x_1) x

full_name: Sigma.smul_def
code: @[to_additive]
theorem smul_def : a • x = x.map id fun _ => (a • ·) 
goal: a • x = Sigma.map id (fun x x_1 => a • x_1) x

full_name: Sigma.smul_mk
code: @[to_additive (attr := simp)]
theorem smul_mk : a • mk i b = ⟨i, a • b⟩ 
goal: a • ⟨i, b⟩ = ⟨i, a • b⟩

full_name: Sigma.vadd_mk
code: @[to_additive (attr := simp)]
theorem smul_mk : a • mk i b = ⟨i, a • b⟩ 
goal: a +ᵥ ⟨i, b⟩ = ⟨i, a +ᵥ b⟩

full_name: Sigma.instIsScalarTowerOfSMul
code: @[to_additive]
instance instIsScalarTowerOfSMul [SMul M N] [∀ i, IsScalarTower M N (α i)] :
    IsScalarTower M N (Σi, α i) 
goal: IsScalarTower M N ((i : ι) × α i)

full_name: Sigma.instIsScalarTowerOfVAdd
code: @[to_additive]
instance instIsScalarTowerOfSMul [SMul M N] [∀ i, IsScalarTower M N (α i)] :
    IsScalarTower M N (Σi, α i) 
goal: VAddAssocClass M N ((i : ι) × α i)

full_name: Sigma.instSMulCommClass
code: @[to_additive]
instance [∀ i, SMulCommClass M N (α i)] : SMulCommClass M N (Σi, α i) 
goal: SMulCommClass M N ((i : ι) × α i)

full_name: Sigma.instVAddCommClass
code: @[to_additive]
instance [∀ i, SMulCommClass M N (α i)] : SMulCommClass M N (Σi, α i) 
goal: VAddCommClass M N ((i : ι) × α i)

full_name: Sigma.instIsCentralVAdd
code: @[to_additive]
instance [∀ i, SMul Mᵐᵒᵖ (α i)] [∀ i, IsCentralScalar M (α i)] : IsCentralScalar M (Σi, α i) 
goal: IsCentralVAdd M ((i : ι) × α i)

full_name: Sigma.instIsCentralScalar
code: @[to_additive]
instance [∀ i, SMul Mᵐᵒᵖ (α i)] [∀ i, IsCentralScalar M (α i)] : IsCentralScalar M (Σi, α i) 
goal: IsCentralScalar M ((i : ι) × α i)

full_name: Sigma.FaithfulSMul'
code: /-- This is not an instance because `i` becomes a metavariable. -/
@[to_additive "This is not an instance because `i` becomes a metavariable."]
protected theorem FaithfulSMul' [FaithfulSMul M (α i)] : FaithfulSMul M (Σi, α i) 
goal: FaithfulSMul M ((i : ι) × α i)

full_name: Sigma.FaithfulVAdd'
code: /-- This is not an instance because `i` becomes a metavariable. -/
@[to_additive "This is not an instance because `i` becomes a metavariable."]
protected theorem FaithfulSMul' [FaithfulSMul M (α i)] : FaithfulSMul M (Σi, α i) 
goal: FaithfulVAdd M ((i : ι) × α i)

full_name: Sigma.instFaithfulVAddOfNonempty
code: @[to_additive]
instance [Nonempty ι] [∀ i, FaithfulSMul M (α i)] : FaithfulSMul M (Σi, α i) 
goal: FaithfulVAdd M ((i : ι) × α i)

full_name: Sigma.instFaithfulSMulOfNonempty
code: @[to_additive]
instance [Nonempty ι] [∀ i, FaithfulSMul M (α i)] : FaithfulSMul M (Σi, α i) 
goal: FaithfulSMul M ((i : ι) × α i)

full_name: Sigma.instAddAction
code: @[to_additive]
instance {m : Monoid M} [∀ i, MulAction M (α i)] :
    MulAction M (Σi, α i) 
goal: AddAction M ((i : ι) × α i)

full_name: Sigma.instMulAction
code: @[to_additive]
instance {m : Monoid M} [∀ i, MulAction M (α i)] :
    MulAction M (Σi, α i) 
goal: MulAction M ((i : ι) × α i)