ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CochainComplex.mappingCone.rotateHomotopyEquiv_comm₃
@[reassoc (attr := simp)]
lemma rotateHomotopyEquiv_comm₃ :
    (rotateHomotopyEquiv φ).hom ≫ (triangle (inr φ)).mor₃ = -φ⟦1⟧' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C ℤ} (φ : K ⟶ L)
Goal: (CochainComplex.mappingCone.rotateHomotopyEquiv φ).hom ≫
    (CochainComplex.mappingCone.triangle (CochainComplex.mappingCone.inr φ)).mor₃ =
  -(CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map φ
-- CochainComplex.mappingCone.map_δ
lemma map_δ :
    (G.mapHomologicalComplex (ComplexShape.up ℤ)).map (triangle φ).mor₃ ≫
      NatTrans.app ((Functor.mapHomologicalComplex G (ComplexShape.up ℤ)).commShiftIso  1).hom K =
    (mapHomologicalComplexIso φ G).hom ≫
      (triangle ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).mor₃ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] [CategoryTheory.Preadditive D] [CategoryTheory.Limits.HasBinaryBiproducts D] {K L : CochainComplex C ℤ} (φ : K ⟶ L) (G : C ⥤ D) [G.Additive]
Goal: (G.mapHomologicalComplex (ComplexShape.up ℤ)).map (CochainComplex.mappingCone.triangle φ).mor₃ ≫
    ((G.mapHomologicalComplex (ComplexShape.up ℤ)).commShiftIso 1).hom.app K =
  (CochainComplex.mappingCone.mapHomologicalComplexIso φ G).hom ≫
    (CochainComplex.mappingCone.triangle ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).mor₃
-- HomotopyCategory.Pretriangulated.isomorphic_distinguished
lemma isomorphic_distinguished (T₁ : Triangle (HomotopyCategory C (ComplexShape.up ℤ)))
    (hT₁ : T₁ ∈ distinguishedTriangles C) (T₂ : Triangle (HomotopyCategory C (ComplexShape.up ℤ)))
    (e : T₂ ≅ T₁) : T₂ ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] (T₁ : CategoryTheory.Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))) (hT₁ : T₁ ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C) (T₂ : CategoryTheory.Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))) (e : T₂ ≅ T₁)
Goal: T₂ ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C
-- HomotopyCategory.Pretriangulated.contractible_distinguished
lemma contractible_distinguished (X : HomotopyCategory C (ComplexShape.up ℤ)) :
    Pretriangulated.contractibleTriangle X ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasBinaryBiproducts C] (X : HomotopyCategory C (ComplexShape.up ℤ))
Goal: CategoryTheory.Pretriangulated.contractibleTriangle X ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C
-- HomotopyCategory.Pretriangulated.distinguished_cocone_triangle
lemma distinguished_cocone_triangle {X Y : HomotopyCategory C (ComplexShape.up ℤ)} (f : X ⟶ Y) :
    ∃ (Z : HomotopyCategory C (ComplexShape.up ℤ)) (g : Y ⟶ Z) (h : Z ⟶ X⟦1⟧),
      Triangle.mk f g h ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {X Y : HomotopyCategory C (ComplexShape.up ℤ)} (f : X ⟶ Y)
Goal: ∃ Z g h, CategoryTheory.Pretriangulated.Triangle.mk f g h ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C
-- HomotopyCategory.Pretriangulated.rotate_distinguished_triangle'
lemma rotate_distinguished_triangle' (T : Triangle (HomotopyCategory C (ComplexShape.up ℤ)))
    (hT : T ∈ distinguishedTriangles C) : T.rotate ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] (T : CategoryTheory.Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))) (hT : T ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C)
Goal: T.rotate ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C
-- HomotopyCategory.Pretriangulated.shift_distinguished_triangle
lemma shift_distinguished_triangle (T : Triangle (HomotopyCategory C (ComplexShape.up ℤ)))
    (hT : T ∈ distinguishedTriangles C) (n : ℤ) :
      (Triangle.shiftFunctor _ n).obj T ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] (T : CategoryTheory.Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))) (hT : T ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C) (n : ℤ)
Goal: (CategoryTheory.Pretriangulated.Triangle.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).obj T ∈
  HomotopyCategory.Pretriangulated.distinguishedTriangles C
-- HomotopyCategory.Pretriangulated.invRotate_distinguished_triangle'
lemma invRotate_distinguished_triangle' (T : Triangle (HomotopyCategory C (ComplexShape.up ℤ)))
    (hT : T ∈ distinguishedTriangles C) : T.invRotate ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] (T : CategoryTheory.Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))) (hT : T ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C)
Goal: T.invRotate ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C
-- HomotopyCategory.Pretriangulated.rotate_distinguished_triangle
lemma rotate_distinguished_triangle (T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))) :
    T ∈ distinguishedTriangles C ↔ T.rotate ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] (T : CategoryTheory.Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ)))
Goal: T ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C ↔
  T.rotate ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C
-- HomotopyCategory.Pretriangulated.complete_distinguished_triangle_morphism
lemma complete_distinguished_triangle_morphism
    (T₁ T₂ : Triangle (HomotopyCategory C (ComplexShape.up ℤ)))
    (hT₁ : T₁ ∈ distinguishedTriangles C) (hT₂ : T₂ ∈ distinguishedTriangles C)
    (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁) :
    ∃ (c : T₁.obj₃ ⟶ T₂.obj₃), T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧
      T₁.mor₃ ≫ a⟦(1 : ℤ)⟧' = c ≫ T₂.mor₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] (T₁ T₂ : CategoryTheory.Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))) (hT₁ : T₁ ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C) (hT₂ : T₂ ∈ HomotopyCategory.Pretriangulated.distinguishedTriangles C) (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁)
Goal: ∃ c,
  T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧
    T₁.mor₃ ≫ (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map a = c ≫ T₂.mor₃
-- HomotopyCategory.mappingCone_triangleh_distinguished
lemma mappingCone_triangleh_distinguished {X Y : CochainComplex C ℤ} (f : X ⟶ Y) :
    CochainComplex.mappingCone.triangleh f ∈ distTriang (HomotopyCategory _ _) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasBinaryBiproducts C] {X Y : CochainComplex C ℤ} (f : X ⟶ Y)
Goal: CochainComplex.mappingCone.triangleh f ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
-- DirectSum.smul_apply
theorem smul_apply (b : R) (v : ⨁ i, M i) (i : ι) : (b • v) i = b • v i 
Arguments: {R : Type u} [Semiring R] {ι : Type v} {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (b : R) (v : DirectSum ι fun i => M i) (i : ι)
Goal: (b • v) i = b • v i
-- DirectSum.lof_eq_of
theorem lof_eq_of (i : ι) (b : M i) : lof R ι M i b = of M i b 
Arguments: (R : Type u) [Semiring R] (ι : Type v) [dec_ι : DecidableEq ι] (M : ι → Type w) [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (i : ι) (b : M i)
Goal: (DirectSum.lof R ι M i) b = (DirectSum.of M i) b
-- DirectSum.single_eq_lof
theorem single_eq_lof (i : ι) (b : M i) : DFinsupp.single i b = lof R ι M i b 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (i : ι) (b : M i)
Goal: DFinsupp.single i b = (DirectSum.lof R ι M i) b
-- DirectSum.mk_smul
theorem mk_smul (s : Finset ι) (c : R) (x) : mk M s (c • x) = c • mk M s x 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (s : Finset ι) (c : R) (x : (i : ↑↑s) → M ↑i)
Goal: (DirectSum.mk M s) (c • x) = c • (DirectSum.mk M s) x
-- DirectSum.of_smul
theorem of_smul (i : ι) (c : R) (x) : of M i (c • x) = c • of M i x 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (i : ι) (c : R) (x : M i)
Goal: (DirectSum.of M i) (c • x) = c • (DirectSum.of M i) x
-- DirectSum.support_smul
theorem support_smul [∀ (i : ι) (x : M i), Decidable (x ≠ 0)] (c : R) (v : ⨁ i, M i) :
    (c • v).support ⊆ v.support 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] [(i : ι) → (x : M i) → Decidable (x ≠ 0)] (c : R) (v : DirectSum ι fun i => M i)
Goal: DFinsupp.support (c • v) ⊆ DFinsupp.support v
-- DirectSum.coe_toModule_eq_coe_toAddMonoid
theorem coe_toModule_eq_coe_toAddMonoid :
    (toModule R ι N φ : (⨁ i, M i) → N) = toAddMonoid fun i ↦ (φ i).toAddMonoidHom 
Arguments: (R : Type u) [Semiring R] (ι : Type v) [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (N : Type u₁) [AddCommMonoid N] [Module R N] (φ : (i : ι) → M i →ₗ[R] N)
Goal: ⇑(DirectSum.toModule R ι N φ) = ⇑(DirectSum.toAddMonoid fun i => (φ i).toAddMonoidHom)
-- DirectSum.toModule_lof
@[simp]
theorem toModule_lof (i) (x : M i) : toModule R ι N φ (lof R ι M i x) = φ i x 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] {N : Type u₁} [AddCommMonoid N] [Module R N] {φ : (i : ι) → M i →ₗ[R] N} (i : ι) (x : M i)
Goal: (DirectSum.toModule R ι N φ) ((DirectSum.lof R ι M i) x) = (φ i) x
-- DirectSum.toModule.unique
theorem toModule.unique (f : ⨁ i, M i) : ψ f = toModule R ι N (fun i ↦ ψ.comp <| lof R ι M i) f 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] {N : Type u₁} [AddCommMonoid N] [Module R N] (ψ : (DirectSum ι fun i => M i) →ₗ[R] N) (f : DirectSum ι fun i => M i)
Goal: ψ f = (DirectSum.toModule R ι N fun i => ψ ∘ₗ DirectSum.lof R ι M i) f
-- DirectSum.linearMap_ext
@[ext]
theorem linearMap_ext ⦃ψ ψ' : (⨁ i, M i) →ₗ[R] N⦄
    (H : ∀ i, ψ.comp (lof R ι M i) = ψ'.comp (lof R ι M i)) : ψ = ψ' 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] {N : Type u₁} [AddCommMonoid N] [Module R N] ⦃ψ ψ' : (DirectSum ι fun i => M i) →ₗ[R] N⦄ (H : ∀ (i : ι), ψ ∘ₗ DirectSum.lof R ι M i = ψ' ∘ₗ DirectSum.lof R ι M i)
Goal: ψ = ψ'
-- DirectSum.linearEquivFunOnFintype_apply
@[simps apply]
def linearEquivFunOnFintype [Fintype ι] : (⨁ i, M i) ≃ₗ[R] ∀ i, M i 
Arguments: (R : Type u) [Semiring R] (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] [Fintype ι]
Goal: ∀ (a : Π₀ (i : ι), (fun i => M i) i) (a_1 : ι), (DirectSum.linearEquivFunOnFintype R ι M) a a_1 = a a_1
-- DirectSum.linearEquivFunOnFintype_lof
@[simp]
theorem linearEquivFunOnFintype_lof [Fintype ι] [DecidableEq ι] (i : ι) (m : M i) :
    (linearEquivFunOnFintype R ι M) (lof R ι M i m) = Pi.single i m 
Arguments: (R : Type u) [Semiring R] {ι : Type v} {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] [Fintype ι] [DecidableEq ι] (i : ι) (m : M i)
Goal: (DirectSum.linearEquivFunOnFintype R ι M) ((DirectSum.lof R ι M i) m) = Pi.single i m
-- DirectSum.linearEquivFunOnFintype_symm_single
@[simp]
theorem linearEquivFunOnFintype_symm_single [Fintype ι] [DecidableEq ι] (i : ι) (m : M i) :
    (linearEquivFunOnFintype R ι M).symm (Pi.single i m) = lof R ι M i m 
Arguments: (R : Type u) [Semiring R] {ι : Type v} {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] [Fintype ι] [DecidableEq ι] (i : ι) (m : M i)
Goal: (DirectSum.linearEquivFunOnFintype R ι M).symm (Pi.single i m) = (DirectSum.lof R ι M i) m
-- DirectSum.linearEquivFunOnFintype_symm_coe
@[simp]
theorem linearEquivFunOnFintype_symm_coe [Fintype ι] (f : ⨁ i, M i) :
    (linearEquivFunOnFintype R ι M).symm f = f 
Arguments: (R : Type u) [Semiring R] {ι : Type v} {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] [Fintype ι] (f : DirectSum ι fun i => M i)
Goal: (DirectSum.linearEquivFunOnFintype R ι M).symm ⇑f = f
-- DirectSum.apply_eq_component
theorem apply_eq_component (f : ⨁ i, M i) (i : ι) : f i = component R ι M i f 
Arguments: (R : Type u) [Semiring R] {ι : Type v} {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (f : DirectSum ι fun i => M i) (i : ι)
Goal: f i = (DirectSum.component R ι M i) f
-- DirectSum.ext
@[ext]
theorem ext {f g : ⨁ i, M i} (h : ∀ i, component R ι M i f = component R ι M i g) : f = g 
Arguments: (R : Type u) [Semiring R] {ι : Type v} {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] {f g : DirectSum ι fun i => M i} (h : ∀ (i : ι), (DirectSum.component R ι M i) f = (DirectSum.component R ι M i) g)
Goal: f = g
-- DirectSum.ext_iff
theorem ext_iff {f g : ⨁ i, M i} : f = g ↔ ∀ i, component R ι M i f = component R ι M i g 
Arguments: (R : Type u) [Semiring R] {ι : Type v} {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] {f g : DirectSum ι fun i => M i}
Goal: f = g ↔ ∀ (i : ι), (DirectSum.component R ι M i) f = (DirectSum.component R ι M i) g
-- DirectSum.lof_apply
@[simp]
theorem lof_apply (i : ι) (b : M i) : ((lof R ι M i) b) i = b 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (i : ι) (b : M i)
Goal: ((DirectSum.lof R ι M i) b) i = b
-- DirectSum.component.lof_self
@[simp]
theorem component.lof_self (i : ι) (b : M i) : component R ι M i ((lof R ι M i) b) = b 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (i : ι) (b : M i)
Goal: (DirectSum.component R ι M i) ((DirectSum.lof R ι M i) b) = b
-- DirectSum.component.of
theorem component.of (i j : ι) (b : M j) :
    component R ι M i ((lof R ι M j) b) = if h : j = i then Eq.recOn h b else 0 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (i j : ι) (b : M j)
Goal: (DirectSum.component R ι M i) ((DirectSum.lof R ι M j) b) = if h : j = i then Eq.recOn h b else 0
-- DirectSum.lequivCongrLeft_apply
@[simp]
theorem lequivCongrLeft_apply (h : ι ≃ κ) (f : ⨁ i, M i) (k : κ) :
    lequivCongrLeft R h f k = f (h.symm k) 
Arguments: (R : Type u) [Semiring R] {ι : Type v} {M : ι → Type w} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] {κ : Type u_1} (h : ι ≃ κ) (f : DirectSum ι fun i => M i) (k : κ)
Goal: ((DirectSum.lequivCongrLeft R h) f) k = f (h.symm k)
-- DirectSum.sigmaLcurry_apply
@[simp]
theorem sigmaLcurry_apply (f : ⨁ i : Σ_, _, δ i.1 i.2) (i : ι) (j : α i) :
    sigmaLcurry R f i j = f ⟨i, j⟩ 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {α : ι → Type u_1} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] [(i : ι) → (j : α i) → Module R (δ i j)] (f : DirectSum ((x : ι) × α x) fun i => δ i.fst i.snd) (i : ι) (j : α i)
Goal: (((DirectSum.sigmaLcurry R) f) i) j = f ⟨i, j⟩
-- DirectSum.sigmaLuncurry_apply
@[simp]
theorem sigmaLuncurry_apply (f : ⨁ (i) (j), δ i j) (i : ι) (j : α i) :
    sigmaLuncurry R f ⟨i, j⟩ = f i j 
Arguments: (R : Type u) [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {α : ι → Type u_1} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] [(i : ι) → (j : α i) → Module R (δ i j)] (f : DirectSum ι fun i => DirectSum (α i) fun j => δ i j) (i : ι) (j : α i)
Goal: ((DirectSum.sigmaLuncurry R) f) ⟨i, j⟩ = (f i) j
-- DirectSum.lequivProdDirectSum_apply
@[simps]
noncomputable def lequivProdDirectSum : (⨁ i, α i) ≃ₗ[R] α none × ⨁ i, α (some i) 
Arguments: (R : Type u) [Semiring R] {ι : Type v} {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] [(i : Option ι) → Module R (α i)]
Goal: ∀ (a : DirectSum (Option ι) fun i => α i), (DirectSum.lequivProdDirectSum R) a = DirectSum.addEquivProdDirectSum.toFun a
-- DirectSum.lequivProdDirectSum_symm_apply
@[simps]
noncomputable def lequivProdDirectSum : (⨁ i, α i) ≃ₗ[R] α none × ⨁ i, α (some i) 
Arguments: (R : Type u) [Semiring R] {ι : Type v} {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] [(i : Option ι) → Module R (α i)]
Goal: ∀ (a : α none × DirectSum ι fun i => α (some i)),
  (DirectSum.lequivProdDirectSum R).symm a = DirectSum.addEquivProdDirectSum.invFun a
-- DirectSum.coeLinearMap_eq_dfinsupp_sum
theorem coeLinearMap_eq_dfinsupp_sum [DecidableEq M] (x : DirectSum ι fun i => A i) :
    coeLinearMap A x = DFinsupp.sum x fun i => (fun x : A i => ↑x) 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] (A : ι → Submodule R M) [DecidableEq M] (x : DirectSum ι fun i => ↥(A i))
Goal: (DirectSum.coeLinearMap A) x = DFinsupp.sum x fun i x => ↑x
-- DirectSum.coeLinearMap_of
@[simp]
theorem coeLinearMap_of (i : ι) (x : A i) : DirectSum.coeLinearMap A (of (fun i ↦ A i) i x) = x 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] (A : ι → Submodule R M) (i : ι) (x : ↥(A i))
Goal: (DirectSum.coeLinearMap A) ((DirectSum.of (fun i => ↥(A i)) i) x) = ↑x
-- DirectSum.IsInternal.ofBijective_coeLinearMap_same
@[simp]
theorem IsInternal.ofBijective_coeLinearMap_same (h : IsInternal A)
    {i : ι} (x : A i) :
    (LinearEquiv.ofBijective (coeLinearMap A) h).symm x i = x 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {i : ι} (x : ↥(A i))
Goal: ((LinearEquiv.ofBijective (DirectSum.coeLinearMap A) h).symm ↑x) i = x
-- DirectSum.IsInternal.ofBijective_coeLinearMap_of_ne
@[simp]
theorem IsInternal.ofBijective_coeLinearMap_of_ne (h : IsInternal A)
    {i j : ι} (hij : i ≠ j) (x : A i) :
    (LinearEquiv.ofBijective (coeLinearMap A) h).symm x j = 0 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {i j : ι} (hij : i ≠ j) (x : ↥(A i))
Goal: ((LinearEquiv.ofBijective (DirectSum.coeLinearMap A) h).symm ↑x) j = 0
-- DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem
theorem IsInternal.ofBijective_coeLinearMap_of_mem (h : IsInternal A)
    {i : ι} {x : M} (hx : x ∈ A i) :
    (LinearEquiv.ofBijective (coeLinearMap A) h).symm x i = ⟨x, hx⟩ 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {i : ι} {x : M} (hx : x ∈ A i)
Goal: ((LinearEquiv.ofBijective (DirectSum.coeLinearMap A) h).symm x) i = ⟨x, hx⟩
-- DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem_ne
theorem IsInternal.ofBijective_coeLinearMap_of_mem_ne (h : IsInternal A)
    {i j : ι} (hij : i ≠ j) {x : M} (hx : x ∈ A i) :
    (LinearEquiv.ofBijective (coeLinearMap A) h).symm x j = 0 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {i j : ι} (hij : i ≠ j) {x : M} (hx : x ∈ A i)
Goal: ((LinearEquiv.ofBijective (DirectSum.coeLinearMap A) h).symm x) j = 0
-- DirectSum.IsInternal.submodule_iSup_eq_top
theorem IsInternal.submodule_iSup_eq_top (h : IsInternal A) : iSup A = ⊤ 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A)
Goal: iSup A = ⊤
-- DirectSum.IsInternal.submodule_independent
theorem IsInternal.submodule_independent (h : IsInternal A) : CompleteLattice.Independent A 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A)
Goal: CompleteLattice.Independent A
-- DirectSum.IsInternal.collectedBasis_coe
@[simp]
theorem IsInternal.collectedBasis_coe (h : IsInternal A) {α : ι → Type*}
    (v : ∀ i, Basis (α i) R (A i)) : ⇑(h.collectedBasis v) = fun a : Σi, α i ↦ ↑(v a.1 a.2) 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {α : ι → Type u_2} (v : (i : ι) → Basis (α i) R ↥(A i))
Goal: ⇑(h.collectedBasis v) = fun a => ↑((v a.fst) a.snd)
-- DirectSum.IsInternal.collectedBasis_mem
theorem IsInternal.collectedBasis_mem (h : IsInternal A) {α : ι → Type*}
    (v : ∀ i, Basis (α i) R (A i)) (a : Σi, α i) : h.collectedBasis v a ∈ A a.1 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {α : ι → Type u_2} (v : (i : ι) → Basis (α i) R ↥(A i)) (a : (i : ι) × α i)
Goal: (h.collectedBasis v) a ∈ A a.fst
-- DirectSum.IsInternal.collectedBasis_repr_of_mem
theorem IsInternal.collectedBasis_repr_of_mem (h : IsInternal A) {α : ι → Type*}
    (v : ∀ i, Basis (α i) R (A i)) {x : M} {i : ι} {a : α i} (hx : x ∈ A i) :
    (h.collectedBasis v).repr x ⟨i, a⟩ = (v i).repr ⟨x, hx⟩ a 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {α : ι → Type u_2} (v : (i : ι) → Basis (α i) R ↥(A i)) {x : M} {i : ι} {a : α i} (hx : x ∈ A i)
Goal: ((h.collectedBasis v).repr x) ⟨i, a⟩ = ((v i).repr ⟨x, hx⟩) a
-- DirectSum.IsInternal.collectedBasis_repr_of_mem_ne
theorem IsInternal.collectedBasis_repr_of_mem_ne (h : IsInternal A) {α : ι → Type*}
    (v : ∀ i, Basis (α i) R (A i)) {x : M} {i j : ι} (hij : i ≠ j) {a : α j} (hx : x ∈ A i) :
    (h.collectedBasis v).repr x ⟨j, a⟩ = 0 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {α : ι → Type u_2} (v : (i : ι) → Basis (α i) R ↥(A i)) {x : M} {i j : ι} (hij : i ≠ j) {a : α j} (hx : x ∈ A i)
Goal: ((h.collectedBasis v).repr x) ⟨j, a⟩ = 0
-- DirectSum.IsInternal.isCompl
theorem IsInternal.isCompl {A : ι → Submodule R M} {i j : ι} (hij : i ≠ j)
    (h : (Set.univ : Set ι) = {i, j}) (hi : IsInternal A) : IsCompl (A i) (A j) 
Arguments: {R : Type u} [Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommMonoid M] [Module R M] {A : ι → Submodule R M} {i j : ι} (hij : i ≠ j) (h : Set.univ = {i, j}) (hi : DirectSum.IsInternal A)
Goal: IsCompl (A i) (A j)
-- DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top
theorem isInternal_submodule_of_independent_of_iSup_eq_top {A : ι → Submodule R M}
    (hi : CompleteLattice.Independent A) (hs : iSup A = ⊤) : IsInternal A 
Arguments: {R : Type u} [Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommGroup M] [Module R M] {A : ι → Submodule R M} (hi : CompleteLattice.Independent A) (hs : iSup A = ⊤)
Goal: DirectSum.IsInternal A