-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic
Goal:
σ : Type u_1
A : Type u
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
X : Scheme
f : A →+* (↑Γ(X, ⊤) : Type u)
hf : Eq (α := Ideal (↑Γ(X, ⊤) : Type u)) (Ideal.map f (HomogeneousIdeal.toIdeal (HomogeneousIdeal.irrelevant 𝒜))) ⊤
x : (openCoverOfMapIrrelevantEqTop 𝒜 f hf).I₀
simp :
  Eq (α :=
    (↑(X.toSpecΓ ⁻¹ᵁ
            PrimeSpectrum.basicOpen
              ((f : (a : A) → (↑Γ(X, ⊤) : Type u))
                (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                  0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                    r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))) :
        Scheme) ⟶
      Spec (CommRingCat.of ↥(𝒜 0)))
    (X.toSpecΓ ∣_
        PrimeSpectrum.basicOpen
          ((f : (a : A) → (↑Γ(X, ⊤) : Type u))
            (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
              0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i))))) ≫
      Iso.hom
          (basicOpenIsoSpecAway
            ((f : (a : A) → (↑Γ(X, ⊤) : Type u))
              (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                  r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))) ≫
        CategoryStruct.comp (Y :=
          Spec
            (CommRingCat.of
              (Away 𝒜
                (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                  0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                    r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))))
          (Spec.map
            (CategoryStruct.comp (Y :=
              CommRingCat.of
                (Localization.Away
                  (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                    0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                      r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i))))))
              (CommRingCat.ofHom
                (algebraMap
                  (Away 𝒜
                    (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                      0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                        r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))
                  (Localization.Away
                    (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                      0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                        r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))))
              (CommRingCat.ofHom
                (IsLocalization.map
                  (Localization.Away (M := (↑Γ(X, ⊤) : Type u))
                    ((f : (a : A) → (↑Γ(X, ⊤) : Type u))
                        (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                          0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                            r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))) :
                      (↑Γ(X, ⊤) : Type u)))
                  f ⋯))))
          (Spec.map
            (CommRingCat.ofHom
              (fromZeroRingHom 𝒜
                (Submonoid.powers
                  (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                    0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                      r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))))))
    (Iso.hom (X.isoOfEq ⋯) ≫
      (openCoverOfMapIrrelevantEqTop 𝒜 f hf).f x ≫
        X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom (algebraMap (↥(𝒜 0)) A) ≫ CommRingCat.ofHom f))
⊢ Eq (α := (openCoverOfMapIrrelevantEqTop 𝒜 f hf).X x ⟶ Spec (CommRingCat.of ↥(𝒜 0)))
    ((openCoverOfMapIrrelevantEqTop 𝒜 f hf).f x ≫ fromOfGlobalSections 𝒜 f hf ≫ toSpecZero 𝒜)
    ((openCoverOfMapIrrelevantEqTop 𝒜 f hf).f x ≫
      X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom (f.comp (algebraMap (↥(𝒜 0)) A))))

Tactic:
simp only [fromOfGlobalSections, toBasicOpenOfGlobalSections, CommRingCat.ofHom_comp,
    Category.assoc, Scheme.Cover.ι_glueMorphisms_assoc, basicOpenIsoSpec_inv_ι_assoc,
    awayι_toSpecZero, Iso.inv_comp_eq]

Premises used: CategoryTheory.Category.assoc, AlgebraicGeometry.Proj.basicOpenIsoSpec_inv_ι_assoc, CategoryTheory.Iso.inv_comp_eq, AlgebraicGeometry.Proj.awayι_toSpecZero, AlgebraicGeometry.Scheme.Cover.ι_glueMorphisms_assoc, AlgebraicGeometry.Proj.toBasicOpenOfGlobalSections, CommRingCat.ofHom_comp, AlgebraicGeometry.Proj.fromOfGlobalSections