-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
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
x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)
y : at x
f g : NumDenSameDeg 𝒜 x.asHomogeneousIdeal.toIdeal.primeCompl
c : A
hc : c ∈ x.asHomogeneousIdeal.toIdeal.primeCompl
hc' :
  (↑(Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ x.asHomogeneousIdeal.toIdeal.primeCompl) c hc) : A) *
      ((↑g.den : A) * (↑f.num : A)) =
    (↑(Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ x.asHomogeneousIdeal.toIdeal.primeCompl) c hc) : A) *
      ((↑f.den : A) * (↑g.num : A))
t : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)
ht' : t ∈ (↑(ProjectiveSpectrum.basicOpen 𝒜 c) : Set (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))
htf : t ∈ (↑(ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A)) : Set (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))
htg : t ∈ (↑(ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A)) : Set (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))
a :
  Eq (α :=
    Localization (M := A)
      (↑(Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦
                  Membership.mem (γ := Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))
                    (unop
                      (op
                        (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                          ProjectiveSpectrum.basicOpen 𝒜 c)))
                    x_1)
                  t ⋯) :
              (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl)
    (val.{0, u_1, u_2} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑(sectionInBasicOpen 𝒜 x f) :
          (x_1 : ↥(unop (op (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A))))) →
            (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ at x)
              (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
        ((Quiver.Hom.unop (Quiver.Hom.op (homOfLE ⋯ ≫ homOfLE ⋯)) :
            (a :
                ↥(unop
                    (op
                      (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                        ProjectiveSpectrum.basicOpen 𝒜 c)))) →
              ↥(unop (op (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A)))))
          (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦
            Membership.mem (γ := Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))
              (unop
                (op
                  (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                    ProjectiveSpectrum.basicOpen 𝒜 c)))
              x_1)
            t ⋯))))
    (val.{0, u_1, u_2} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑(sectionInBasicOpen 𝒜 x g) :
          (x_1 : ↥(unop (op (ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A))))) →
            (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ at x)
              (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
        ((Quiver.Hom.unop (Quiver.Hom.op (homOfLE ⋯ ≫ homOfLE ⋯)) :
            (a :
                ↥(unop
                    (op
                      (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                        ProjectiveSpectrum.basicOpen 𝒜 c)))) →
              ↥(unop (op (ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A)))))
          (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦
            Membership.mem (γ := Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))
              (unop
                (op
                  (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                    ProjectiveSpectrum.basicOpen 𝒜 c)))
              x_1)
            t ⋯))))
⊢ Eq (α :=
    Localization (M := A)
      (↑(Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦
                  Membership.mem (γ := Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))
                    (unop
                      (op
                        (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                          ProjectiveSpectrum.basicOpen 𝒜 c)))
                    x_1)
                  t ⋯) :
              (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl)
    (val.{0, u_1, u_2} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑(((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
                    CommRingCat.carrier) (X :=
                    Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                      (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A)) :
                        (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ))
                    (Y :=
                    Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                      (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op
                          (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                            ProjectiveSpectrum.basicOpen 𝒜 c) :
                        (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ))
                    (CategoryTheory.Functor.map (Sheaf.presheaf (Proj.structureSheaf 𝒜))
                        (Quiver.Hom.op (homOfLE ⋯ ≫ homOfLE ⋯)) :
                      Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A)) :
                            (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ) ⟶
                        Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op
                              (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓
                                  ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                                ProjectiveSpectrum.basicOpen 𝒜 c) :
                            (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)) :
                  (↑(Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A)) :
                            (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)) :
                      Type u_1) →+*
                    (↑(Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op
                              (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓
                                  ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                                ProjectiveSpectrum.basicOpen 𝒜 c) :
                            (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)) :
                      Type u_1)) :
                (a :
                    (↑((Proj.structureSheaf 𝒜).presheaf.obj (op (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A)))) :
                      Type u_1)) →
                  (↑((Proj.structureSheaf 𝒜).presheaf.obj
                        (op
                          (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                            ProjectiveSpectrum.basicOpen 𝒜 c))) :
                    Type u_1))
              (sectionInBasicOpen 𝒜 x f)) :
          (x_1 :
              ↥(unop
                  (op
                    (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                      ProjectiveSpectrum.basicOpen 𝒜 c)))) →
            (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ at x)
              (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
        (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦
          Membership.mem (γ := Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))
            (unop
              (op
                (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                  ProjectiveSpectrum.basicOpen 𝒜 c)))
            x_1)
          t ⋯)))
    (val.{0, u_1, u_2} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑(((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
                    CommRingCat.carrier) (X :=
                    Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                      (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op (ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A)) :
                        (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ))
                    (Y :=
                    Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                      (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op
                          (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                            ProjectiveSpectrum.basicOpen 𝒜 c) :
                        (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ))
                    (CategoryTheory.Functor.map (Sheaf.presheaf (Proj.structureSheaf 𝒜))
                        (Quiver.Hom.op (homOfLE ⋯ ≫ homOfLE ⋯)) :
                      Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op (ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A)) :
                            (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ) ⟶
                        Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op
                              (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓
                                  ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                                ProjectiveSpectrum.basicOpen 𝒜 c) :
                            (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)) :
                  (↑(Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op (ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A)) :
                            (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)) :
                      Type u_1) →+*
                    (↑(Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op
                              (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓
                                  ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                                ProjectiveSpectrum.basicOpen 𝒜 c) :
                            (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)) :
                      Type u_1)) :
                (a :
                    (↑((Proj.structureSheaf 𝒜).presheaf.obj (op (ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A)))) :
                      Type u_1)) →
                  (↑((Proj.structureSheaf 𝒜).presheaf.obj
                        (op
                          (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                            ProjectiveSpectrum.basicOpen 𝒜 c))) :
                    Type u_1))
              (sectionInBasicOpen 𝒜 x g)) :
          (x_1 :
              ↥(unop
                  (op
                    (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                      ProjectiveSpectrum.basicOpen 𝒜 c)))) →
            (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ at x)
              (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
        (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦
          Membership.mem (γ := Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))
            (unop
              (op
                (ProjectiveSpectrum.basicOpen 𝒜 (↑f.den : A) ⊓ ProjectiveSpectrum.basicOpen 𝒜 (↑g.den : A) ⊓
                  ProjectiveSpectrum.basicOpen 𝒜 c)))
            x_1)
          t ⋯)))

Tactic:
rw [Proj.res_apply, Proj.res_apply]

Premises used: AlgebraicGeometry.Proj.res_apply