-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x :
  (↑(↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).toPresheafedSpace :
        TopCat) :
    Type u_1)
rw :
  Eq (α :=
    CommRingCat.of (A⁰_ f) ⟶
      CommRingCat.of (↑(Presheaf.stalk (Sheaf.presheaf (structureSheaf 𝒜)) (↑x : (↑Proj.T : Type u_1))) : Type u_1))
    (awayToSection 𝒜 f ≫
      CategoryStruct.comp (Y :=
        LocallyRingedSpace.Γ.obj
          (op (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) ⋯)))
        (CategoryTheory.Functor.map
            (ObjectProperty.FullSubcategory.obj (P :=
              (CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
              (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)))
            (Quiver.Hom.op (homOfLE ⋯)) :
          Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
              (ObjectProperty.FullSubcategory.obj (P :=
                  (CategoryTheory.Presheaf.IsSheaf
                      (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                    (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                  (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
                (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat)
              (op (pbo f) : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ) ⟶
            Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
              (ObjectProperty.FullSubcategory.obj (P :=
                  (CategoryTheory.Presheaf.IsSheaf
                      (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                    (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                  (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
                (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat)
              (Functor.obj (C :=
                  (Opens
                      (↑(Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) : Type u_1))ᵒᵖ)
                  (Functor.op (Topology.IsOpenEmbedding.functor ⋯)) (op ⊤) :
                (Opens (↑Proj.T : Type u_1))ᵒᵖ))
        ((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    ⋯).presheaf.germ
              ⊤ x True.intro ≫
            Iso.hom (Proj.restrictStalkIso ⋯ x) :
          LocallyRingedSpace.Γ.obj
              (op
                (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  ⋯)) ⟶
            CommRingCat.of
              (↑(Presheaf.stalk (Sheaf.presheaf (structureSheaf 𝒜)) (↑x : (↑Proj.T : Type u_1))) : Type u_1)))
    (CategoryStruct.comp (Y :=
      CommRingCat.of (HomogeneousLocalization 𝒜 (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl))
      (CommRingCat.ofHom (HomogeneousLocalization.mapId 𝒜 ⋯))
      (Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : (↑Proj.T : Type u_1))))))
⊢ awayToΓ 𝒜 f ≫
      (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).presheaf.Γgerm
        x =
    CategoryStruct.comp (Y :=
      CommRingCat.of (HomogeneousLocalization 𝒜 (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl))
      (CommRingCat.ofHom (HomogeneousLocalization.mapId 𝒜 ⋯))
      (Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : (↑Proj.T : Type u_1)))) ≫
        Iso.inv (Proj.restrictStalkIso ⋯ x))

Tactic:
rw [awayToΓ, Category.assoc, ← Category.assoc _ (Iso.inv _),
    Iso.eq_comp_inv, Category.assoc, Category.assoc, Presheaf.Γgerm]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.eq_comp_inv, TopCat.Presheaf.Γgerm, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ, CategoryTheory.Iso.inv