-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq' pq'' : κ
i₀' i₀ i₁ i₂ i₃ : ι
hi₀' : i₀' = data.i₀ r' pq' ⋯
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq' pq''
pos :
  Eq (α := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ pageX X data r pq'' ⋯)
    ((X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃ ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ hn₁ hn₂ ≫
          (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).inv) ≫
        Iso.hom (pageXIso X data r ⋯ pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂) ≫
          X.d (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ (n₂ + 1) hn₁ hn₂ ⋯ ≫
            Iso.inv (pageXIso X data r ⋯ pq'' (data.i₀ r pq'' ⋯) i₀' i₀ i₁ ⋯ ⋯ ⋯ ⋯ n₁ n₂ (n₂ + 1) ⋯ hn₂ ⋯) :
      X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ pageX X data r pq'' ⋯)
    (0 : X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ (page X data r hr).X pq'')
pos₁ :
  Eq (α := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ pageX X data r pq'' ⋯)
    (X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃ ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ hn₁ hn₂ ≫
        (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).inv ≫
          Iso.hom (pageXIso X data r ⋯ pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂) ≫
            X.d (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ (n₂ + 1) hn₁ hn₂ ⋯ ≫
              Iso.inv (pageXIso X data r ⋯ pq'' (data.i₀ r pq'' ⋯) i₀' i₀ i₁ ⋯ ⋯ ⋯ ⋯ n₁ n₂ (n₂ + 1) ⋯ hn₂ ⋯) :
      X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ pageX X data r pq'' ⋯)
    (0 : X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ (page X data r hr).X pq'')
pos₂ :
  Eq (α := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ pageX X data r pq'' ⋯)
    (X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃ ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ hn₁ hn₂ ≫
        X.d (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ (n₂ + 1) hn₁ hn₂ ⋯ ≫
          Iso.inv (pageXIso X data r ⋯ pq'' (data.i₀ r pq'' ⋯) i₀' i₀ i₁ ⋯ ⋯ ⋯ ⋯ n₁ n₂ (n₂ + 1) ⋯ hn₂ ⋯) :
      X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ pageX X data r pq'' ⋯)
    (0 : X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ (page X data r hr).X pq'')
pos₃ :
  Eq (α := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ pageX X data r pq'' ⋯)
    (CategoryStruct.comp (Y := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₁ n₂ (n₂ + 1) ⋯ ⋯)
      (0 :
        X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ ⟶
          X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₁ n₂ (n₂ + 1) ⋯ ⋯)
      (Iso.inv (pageXIso X data r ⋯ pq'' (data.i₀ r pq'' ⋯) i₀' i₀ i₁ ⋯ ⋯ ⋯ ⋯ n₁ n₂ (n₂ + 1) ⋯ hn₂ ⋯) :
        X.E (i := data.i₀ r pq'' ⋯) (homOfLE ⋯ : data.i₀ r pq'' ⋯ ⟶ i₀') (homOfLE ⋯) (homOfLE ⋯) n₁ n₂ (n₂ + 1) hn₂ ⋯ ⟶
          pageX X data r pq'' ⋯))
    0
pos₄ pos₅ : Eq.{v_1 + 1} (α := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ pageX X data r pq'' ⋯) 0 0
⊢ Eq (α := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ pageX X data r pq'' ⋯)
    ((X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃ ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ hn₁ hn₂ ≫
          (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).inv) ≫
        pageD X data r pq' pq'' ⋯ :
      X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ pageX X data r pq'' ⋯)
    (0 : X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ hn₂ ⟶ (page X data r hr).X pq'')

Tactic:
rw [pageD_eq X data r hr pq' pq'' h
      (homOfLE (by simpa only [hi₀', data.i₀_prev r r' _ _ h] using data.le₀₁ r pq''))
      (homOfLE (data.i₀_le' hrr' hr pq' hi₀' hi₀)) (homOfLE (data.le₀₁' r hr pq' hi₀ hi₁))
      (homOfLE (data.le₁₂' pq' hi₁ hi₂)) (homOfLE (data.le₂₃' r hr pq' hi₂ hi₃)) rfl
      (by rw [hi₀', data.i₀_prev r r' pq' pq'' h]) hi₀ hi₁ hi₂ hi₃ _ _ _ _ hn₁' hn₁ hn₂ rfl,
      Category.assoc, Iso.inv_hom_id_assoc, map_fourδ₁Toδ₀_d_assoc .., zero_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₀₁', CategoryTheory.Abelian.SpectralObject.map_fourδ₁Toδ₀_d_assoc, CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.homOfLE, CategoryTheory.Abelian.SpectralObject.SpectralSequence.pageD_eq, CategoryTheory.Limits.zero_comp, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₂₃', CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₁₂', CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.i₀_le', rfl