-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic (goal)
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
⊢ 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))))
have 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 sorry))))
        (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 sorry) ≫
    (openCoverOfMapIrrelevantEqTop 𝒜 f hf).f x ≫
      X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom (algebraMap (↥(𝒜 0)) A) ≫ CommRingCat.ofHom f))
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]