File: Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean

full_name: AddMonoidAlgebra.toDirectSum
code: /-- Interpret an `AddMonoidAlgebra` as a homogenous `DirectSum`. -/
def AddMonoidAlgebra.toDirectSum [Semiring M] (f : AddMonoidAlgebra M ι) : ⨁ _ : ι, M 
goal: ⨁ (x : ι), M

full_name: AddMonoidAlgebra.toDirectSum_single
code: @[simp]
theorem AddMonoidAlgebra.toDirectSum_single (i : ι) (m : M) :
    AddMonoidAlgebra.toDirectSum (Finsupp.single i m) = DirectSum.of _ i m 
goal: AddMonoidAlgebra.toDirectSum (Finsupp.single i m) = (DirectSum.of (fun i => M) i) m

full_name: DirectSum.toAddMonoidAlgebra
code: /-- Interpret a homogenous `DirectSum` as an `AddMonoidAlgebra`. -/
def DirectSum.toAddMonoidAlgebra (f : ⨁ _ : ι, M) : AddMonoidAlgebra M ι 
goal: AddMonoidAlgebra M ι

full_name: DirectSum.toAddMonoidAlgebra_of
code: @[simp]
theorem DirectSum.toAddMonoidAlgebra_of (i : ι) (m : M) :
    (DirectSum.of _ i m : ⨁ _ : ι, M).toAddMonoidAlgebra = Finsupp.single i m 
goal: ((DirectSum.of (fun x => M) i) m).toAddMonoidAlgebra = Finsupp.single i m

full_name: AddMonoidAlgebra.toDirectSum_toAddMonoidAlgebra
code: @[simp]
theorem AddMonoidAlgebra.toDirectSum_toAddMonoidAlgebra (f : AddMonoidAlgebra M ι) :
    f.toDirectSum.toAddMonoidAlgebra = f 
goal: f.toDirectSum.toAddMonoidAlgebra = f

full_name: DirectSum.toAddMonoidAlgebra_toDirectSum
code: @[simp]
theorem DirectSum.toAddMonoidAlgebra_toDirectSum (f : ⨁ _ : ι, M) :
    f.toAddMonoidAlgebra.toDirectSum = f 
goal: f.toAddMonoidAlgebra.toDirectSum = f

full_name: AddMonoidAlgebra.toDirectSum_zero
code: @[simp]
theorem toDirectSum_zero [Semiring M] : (0 : AddMonoidAlgebra M ι).toDirectSum = 0 
goal: AddMonoidAlgebra.toDirectSum 0 = 0

full_name: AddMonoidAlgebra.toDirectSum_add
code: @[simp]
theorem toDirectSum_add [Semiring M] (f g : AddMonoidAlgebra M ι) :
    (f + g).toDirectSum = f.toDirectSum + g.toDirectSum 
goal: (f + g).toDirectSum = f.toDirectSum + g.toDirectSum

full_name: AddMonoidAlgebra.toDirectSum_mul
code: @[simp]
theorem toDirectSum_mul [DecidableEq ι] [AddMonoid ι] [Semiring M] (f g : AddMonoidAlgebra M ι) :
    (f * g).toDirectSum = f.toDirectSum * g.toDirectSum 
goal: (f * g).toDirectSum = f.toDirectSum * g.toDirectSum

full_name: DirectSum.toAddMonoidAlgebra_zero
code: @[simp]
theorem toAddMonoidAlgebra_zero [Semiring M] [∀ m : M, Decidable (m ≠ 0)] :
    toAddMonoidAlgebra 0 = (0 : AddMonoidAlgebra M ι) 
goal: DirectSum.toAddMonoidAlgebra 0 = 0

full_name: DirectSum.toAddMonoidAlgebra_add
code: @[simp]
theorem toAddMonoidAlgebra_add [Semiring M] [∀ m : M, Decidable (m ≠ 0)] (f g : ⨁ _ : ι, M) :
    (f + g).toAddMonoidAlgebra = toAddMonoidAlgebra f + toAddMonoidAlgebra g 
goal: (f + g).toAddMonoidAlgebra = f.toAddMonoidAlgebra + g.toAddMonoidAlgebra

full_name: DirectSum.toAddMonoidAlgebra_mul
code: @[simp]
theorem toAddMonoidAlgebra_mul [AddMonoid ι] [Semiring M]
    [∀ m : M, Decidable (m ≠ 0)] (f g : ⨁ _ : ι, M) :
    (f * g).toAddMonoidAlgebra = toAddMonoidAlgebra f * toAddMonoidAlgebra g 
goal: (f * g).toAddMonoidAlgebra = f.toAddMonoidAlgebra * g.toAddMonoidAlgebra

full_name: addMonoidAlgebraEquivDirectSum_symm_apply
code: /-- `AddMonoidAlgebra.toDirectSum` and `DirectSum.toAddMonoidAlgebra` together form an
equiv. -/
@[simps (config := .asFn)]
def addMonoidAlgebraEquivDirectSum [DecidableEq ι] [Semiring M] [∀ m : M, Decidable (m ≠ 0)] :
    AddMonoidAlgebra M ι ≃ ⨁ _ : ι, M 
goal: ⇑addMonoidAlgebraEquivDirectSum.symm = DirectSum.toAddMonoidAlgebra

full_name: addMonoidAlgebraEquivDirectSum
code: /-- `AddMonoidAlgebra.toDirectSum` and `DirectSum.toAddMonoidAlgebra` together form an
equiv. -/
@[simps (config := .asFn)]
def addMonoidAlgebraEquivDirectSum [DecidableEq ι] [Semiring M] [∀ m : M, Decidable (m ≠ 0)] :
    AddMonoidAlgebra M ι ≃ ⨁ _ : ι, M 
goal: AddMonoidAlgebra M ι ≃ ⨁ (x : ι), M

full_name: addMonoidAlgebraEquivDirectSum_apply
code: /-- `AddMonoidAlgebra.toDirectSum` and `DirectSum.toAddMonoidAlgebra` together form an
equiv. -/
@[simps (config := .asFn)]
def addMonoidAlgebraEquivDirectSum [DecidableEq ι] [Semiring M] [∀ m : M, Decidable (m ≠ 0)] :
    AddMonoidAlgebra M ι ≃ ⨁ _ : ι, M 
goal: ⇑addMonoidAlgebraEquivDirectSum = AddMonoidAlgebra.toDirectSum

full_name: addMonoidAlgebraAddEquivDirectSum_apply
code: /-- The additive version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`.  -/
@[simps (config := .asFn)]
def addMonoidAlgebraAddEquivDirectSum [DecidableEq ι] [Semiring M] [∀ m : M, Decidable (m ≠ 0)] :
    AddMonoidAlgebra M ι ≃+ ⨁ _ : ι, M 
goal: ⇑addMonoidAlgebraAddEquivDirectSum = AddMonoidAlgebra.toDirectSum

full_name: addMonoidAlgebraAddEquivDirectSum
code: /-- The additive version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`.  -/
@[simps (config := .asFn)]
def addMonoidAlgebraAddEquivDirectSum [DecidableEq ι] [Semiring M] [∀ m : M, Decidable (m ≠ 0)] :
    AddMonoidAlgebra M ι ≃+ ⨁ _ : ι, M 
goal: AddMonoidAlgebra M ι ≃+ ⨁ (x : ι), M

full_name: addMonoidAlgebraAddEquivDirectSum_symm_apply
code: /-- The additive version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`.  -/
@[simps (config := .asFn)]
def addMonoidAlgebraAddEquivDirectSum [DecidableEq ι] [Semiring M] [∀ m : M, Decidable (m ≠ 0)] :
    AddMonoidAlgebra M ι ≃+ ⨁ _ : ι, M 
goal: ⇑addMonoidAlgebraAddEquivDirectSum.symm = DirectSum.toAddMonoidAlgebra

full_name: addMonoidAlgebraRingEquivDirectSum
code: /-- The ring version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`.  -/
@[simps (config := .asFn)]
def addMonoidAlgebraRingEquivDirectSum [DecidableEq ι] [AddMonoid ι] [Semiring M]
    [∀ m : M, Decidable (m ≠ 0)] : AddMonoidAlgebra M ι ≃+* ⨁ _ : ι, M 
goal: AddMonoidAlgebra M ι ≃+* ⨁ (x : ι), M

full_name: addMonoidAlgebraRingEquivDirectSum_apply
code: /-- The ring version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`.  -/
@[simps (config := .asFn)]
def addMonoidAlgebraRingEquivDirectSum [DecidableEq ι] [AddMonoid ι] [Semiring M]
    [∀ m : M, Decidable (m ≠ 0)] : AddMonoidAlgebra M ι ≃+* ⨁ _ : ι, M 
goal: ⇑addMonoidAlgebraRingEquivDirectSum = AddMonoidAlgebra.toDirectSum

full_name: addMonoidAlgebraRingEquivDirectSum_symm_apply
code: /-- The ring version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`.  -/
@[simps (config := .asFn)]
def addMonoidAlgebraRingEquivDirectSum [DecidableEq ι] [AddMonoid ι] [Semiring M]
    [∀ m : M, Decidable (m ≠ 0)] : AddMonoidAlgebra M ι ≃+* ⨁ _ : ι, M 
goal: ⇑addMonoidAlgebraRingEquivDirectSum.symm = DirectSum.toAddMonoidAlgebra

full_name: addMonoidAlgebraAlgEquivDirectSum_symm_apply
code: /-- The algebra version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`. -/
@[simps (config := .asFn)]
def addMonoidAlgebraAlgEquivDirectSum [DecidableEq ι] [AddMonoid ι] [CommSemiring R] [Semiring A]
    [Algebra R A] [∀ m : A, Decidable (m ≠ 0)] : AddMonoidAlgebra A ι ≃ₐ[R] ⨁ _ : ι, A 
goal: ⇑addMonoidAlgebraAlgEquivDirectSum.symm = DirectSum.toAddMonoidAlgebra

full_name: addMonoidAlgebraAlgEquivDirectSum_apply
code: /-- The algebra version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`. -/
@[simps (config := .asFn)]
def addMonoidAlgebraAlgEquivDirectSum [DecidableEq ι] [AddMonoid ι] [CommSemiring R] [Semiring A]
    [Algebra R A] [∀ m : A, Decidable (m ≠ 0)] : AddMonoidAlgebra A ι ≃ₐ[R] ⨁ _ : ι, A 
goal: ⇑addMonoidAlgebraAlgEquivDirectSum = AddMonoidAlgebra.toDirectSum

full_name: addMonoidAlgebraAlgEquivDirectSum
code: /-- The algebra version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`. -/
@[simps (config := .asFn)]
def addMonoidAlgebraAlgEquivDirectSum [DecidableEq ι] [AddMonoid ι] [CommSemiring R] [Semiring A]
    [Algebra R A] [∀ m : A, Decidable (m ≠ 0)] : AddMonoidAlgebra A ι ≃ₐ[R] ⨁ _ : ι, A 
goal: AddMonoidAlgebra A ι ≃ₐ[R] ⨁ (x : ι), A