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

full_name: Sum.instSMul
code: @[to_additive Sum.hasVAdd]
instance : SMul M (α ⊕ β) 
goal: SMul M (α ⊕ β)

full_name: Sum.hasVAdd
code: @[to_additive Sum.hasVAdd]
instance : SMul M (α ⊕ β) 
goal: VAdd M (α ⊕ β)

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

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

full_name: Sum.vadd_inl
code: @[to_additive (attr := simp)]
theorem smul_inl : a • (inl b : α ⊕ β) = inl (a • b) 
goal: a +ᵥ Sum.inl b = Sum.inl (a +ᵥ b)

full_name: Sum.smul_inl
code: @[to_additive (attr := simp)]
theorem smul_inl : a • (inl b : α ⊕ β) = inl (a • b) 
goal: a • Sum.inl b = Sum.inl (a • b)

full_name: Sum.vadd_inr
code: @[to_additive (attr := simp)]
theorem smul_inr : a • (inr c : α ⊕ β) = inr (a • c) 
goal: a +ᵥ Sum.inr c = Sum.inr (a +ᵥ c)

full_name: Sum.smul_inr
code: @[to_additive (attr := simp)]
theorem smul_inr : a • (inr c : α ⊕ β) = inr (a • c) 
goal: a • Sum.inr c = Sum.inr (a • c)

full_name: Sum.smul_swap
code: @[to_additive (attr := simp)]
theorem smul_swap : (a • x).swap = a • x.swap 
goal: (a • x).swap = a • x.swap

full_name: Sum.vadd_swap
code: @[to_additive (attr := simp)]
theorem smul_swap : (a • x).swap = a • x.swap 
goal: (a +ᵥ x).swap = a +ᵥ x.swap

full_name: Sum.instIsScalarTower
code: instance [SMul M N] [IsScalarTower M N α] [IsScalarTower M N β] : IsScalarTower M N (α ⊕ β) 
goal: IsScalarTower M N (α ⊕ β)

full_name: Sum.instVAddCommClass
code: @[to_additive]
instance [SMulCommClass M N α] [SMulCommClass M N β] : SMulCommClass M N (α ⊕ β) 
goal: VAddCommClass M N (α ⊕ β)

full_name: Sum.instSMulCommClass
code: @[to_additive]
instance [SMulCommClass M N α] [SMulCommClass M N β] : SMulCommClass M N (α ⊕ β) 
goal: SMulCommClass M N (α ⊕ β)

full_name: Sum.instIsCentralVAdd
code: @[to_additive]
instance [SMul Mᵐᵒᵖ α] [SMul Mᵐᵒᵖ β] [IsCentralScalar M α] [IsCentralScalar M β] :
    IsCentralScalar M (α ⊕ β) 
goal: IsCentralVAdd M (α ⊕ β)

full_name: Sum.instIsCentralScalar
code: @[to_additive]
instance [SMul Mᵐᵒᵖ α] [SMul Mᵐᵒᵖ β] [IsCentralScalar M α] [IsCentralScalar M β] :
    IsCentralScalar M (α ⊕ β) 
goal: IsCentralScalar M (α ⊕ β)

full_name: Sum.FaithfulVAddLeft
code: @[to_additive]
instance FaithfulSMulLeft [FaithfulSMul M α] : FaithfulSMul M (α ⊕ β) 
goal: FaithfulVAdd M (α ⊕ β)

full_name: Sum.FaithfulSMulLeft
code: @[to_additive]
instance FaithfulSMulLeft [FaithfulSMul M α] : FaithfulSMul M (α ⊕ β) 
goal: FaithfulSMul M (α ⊕ β)

full_name: Sum.FaithfulSMulRight
code: @[to_additive]
instance FaithfulSMulRight [FaithfulSMul M β] : FaithfulSMul M (α ⊕ β) 
goal: FaithfulSMul M (α ⊕ β)

full_name: Sum.FaithfulVAddRight
code: @[to_additive]
instance FaithfulSMulRight [FaithfulSMul M β] : FaithfulSMul M (α ⊕ β) 
goal: FaithfulVAdd M (α ⊕ β)

full_name: Sum.instMulAction
code: @[to_additive]
instance {m : Monoid M} [MulAction M α] [MulAction M β] :
    MulAction M (α ⊕ β) 
goal: MulAction M (α ⊕ β)

full_name: Sum.instAddAction
code: @[to_additive]
instance {m : Monoid M} [MulAction M α] [MulAction M β] :
    MulAction M (α ⊕ β) 
goal: AddAction M (α ⊕ β)