Theorem: AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand
File: Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean

Proof steps:
state_before: C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ ((splitting K).cofan Δ).inj A ≫ (obj K).map θ =
    Termwise.mapMono K i ≫ ((splitting K).cofan Δ').inj (Splitting.IndexSet.mk e)
state_after: C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ ((splitting K).ι (unop A.fst).len ≫ map K A.e.op) ≫ map K θ =
    Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
tactic: dsimp [Splitting.cofan]
state_before: C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ ((splitting K).ι (unop A.fst).len ≫ map K A.e.op) ≫ map K θ =
    Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
state_after: C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ ((splitting K).ι (unop A.fst).len ≫ (obj K).map A.e.op) ≫ (obj K).map θ =
    Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
tactic: change (_ ≫ (Γ₀.obj K).map A.e.op) ≫ (Γ₀.obj K).map θ = _
state_before: C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ ((splitting K).ι (unop A.fst).len ≫ (obj K).map A.e.op) ≫ (obj K).map θ =
    Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
state_after: C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ (splitting K).ι (unop A.fst).len ≫ (obj K).map (A.e.op ≫ θ) =
    Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
tactic: rw [assoc, ← Functor.map_comp]
state_before: C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ (splitting K).ι (unop A.fst).len ≫ (obj K).map (A.e.op ≫ θ) =
    Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
state_after: C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ Sigma.ι (summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ map K (A.e.op ≫ θ) =
    Termwise.mapMono K i ≫
      Sigma.ι (summand K (op Δ'')) (Splitting.IndexSet.id (op Δ'')) ≫ map K (Splitting.IndexSet.mk e).e.op
tactic: dsimp [splitting]
state_before: C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ Sigma.ι (summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ map K (A.e.op ≫ θ) =
    Termwise.mapMono K i ≫
      Sigma.ι (summand K (op Δ'')) (Splitting.IndexSet.id (op Δ'')) ≫ map K (Splitting.IndexSet.mk e).e.op
state_after: no goals
tactic: erw [Γ₀.Obj.map_on_summand₀ K (Splitting.IndexSet.id A.1)
    (show e ≫ i = ((Splitting.IndexSet.e A).op ≫ θ).unop ≫ 𝟙 _ by rw [comp_id, fac]; rfl),
    Γ₀.Obj.map_on_summand₀ K (Splitting.IndexSet.id (op Δ''))
      (show e ≫ 𝟙 Δ'' = e.op.unop ≫ 𝟙 _ by simp), Termwise.mapMono_id, id_comp]