-- 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)
U : Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)
hxU : x ∈ U
s :
  ToType (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC := CommRingCat.carrier)
    (Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) (op U))
V : Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)
hxV :
  (↑(Subtype.mk (p := fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ x ∈ unop (op U)) x hxU) :
      (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ∈
    V
i : V ⟶ unop (op U)
n : ℕ
a b : ↥(𝒜 n)
h : ∀ (x : ↥V), (↑b : A) ∉ (↑x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)).asHomogeneousIdeal
e :
  ∀ (a_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) (b_1 : a_1 ∈ V),
    (↑s : (x : ↥U) → at (↑x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
        (Subtype.mk (p := fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ x ∈ U) a_1 ⋯) =
      HomogeneousLocalization.mk { deg := n, num := a, den := b, den_mem := ⋯ }
rw :
  Eq (α := (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1))
    (((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
                      (↑(HomogeneousLocalization.NumDenSameDeg.den { deg := n, num := a, den := b, den_mem := ⋯ }) :
                        A)) :
                  (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ))
              (Y := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
              (Presheaf.germ (Sheaf.presheaf (Proj.structureSheaf 𝒜))
                  (ProjectiveSpectrum.basicOpen 𝒜
                    (↑(HomogeneousLocalization.NumDenSameDeg.den { deg := n, num := a, den := b, den_mem := ⋯ }) : A))
                  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
                          (↑(HomogeneousLocalization.NumDenSameDeg.den
                                { deg := n, num := a, den := b, den_mem := ⋯ :
                                  HomogeneousLocalization.NumDenSameDeg 𝒜 x.asHomogeneousIdeal.toIdeal.primeCompl }) :
                            A)) :
                      (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ) ⟶
                  Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) 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
                          (↑(HomogeneousLocalization.NumDenSameDeg.den { deg := n, num := a, den := b, den_mem := ⋯ }) :
                            A)) :
                      (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)) :
                Type u_1) →+*
              (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)) :
          (a :
              (↑((Proj.structureSheaf 𝒜).presheaf.obj
                    (op (ProjectiveSpectrum.basicOpen 𝒜 (↑{ deg := n, num := a, den := b, den_mem := ⋯ }.den : A)))) :
                Type u_1)) →
            (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1))
        (sectionInBasicOpen 𝒜 x { deg := n, num := a, den := b, den_mem := ⋯ }) :
      (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1))
    (((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
              CommRingCat.carrier) (X :=
              Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                (op U))
              (Y := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
              (Presheaf.germ (Sheaf.presheaf (Proj.structureSheaf 𝒜)) U x hxU :
                Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                    (op U) ⟶
                  Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) :
            (↑(Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                    (op U)) :
                Type u_1) →+*
              (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)) :
          (a : (↑((Proj.structureSheaf 𝒜).presheaf.obj (op U)) : Type u_1)) →
            (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1))
        s :
      (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1))
⊢ Eq (α := (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1))
    (homogeneousLocalizationToStalk 𝒜 x
      ((CommRingCat.Hom.hom (stalkToFiberRingHom 𝒜 x) :
          (a : (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1)) → (↑(CommRingCat.of (at x)) : Type u_1))
        (((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
                CommRingCat.carrier) (X :=
                Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                  (op U))
                (Y := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
                (Presheaf.germ (Sheaf.presheaf (Proj.structureSheaf 𝒜)) U x hxU :
                  Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op U) ⟶
                    Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) :
              (↑(Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op U)) :
                  Type u_1) →+*
                (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)) :
            (a : (↑((Proj.structureSheaf 𝒜).presheaf.obj (op U)) : Type u_1)) →
              (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1))
          s)))
    (((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
            CommRingCat.carrier) (X :=
            Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
              (op U))
            (Y := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
            (Presheaf.germ (Sheaf.presheaf (Proj.structureSheaf 𝒜)) U x hxU :
              Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                  (op U) ⟶
                Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) :
          (↑(Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                  (op U)) :
              Type u_1) →+*
            (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)) :
        (a : (↑((Proj.structureSheaf 𝒜).presheaf.obj (op U)) : Type u_1)) →
          (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1))
      s)

Tactic:
rw [stalkToFiberRingHom_germ, homogeneousLocalizationToStalk, e x hxV, Quotient.liftOn'_mk'']

Premises used: Quotient.liftOn'_mk'', AlgebraicGeometry.homogeneousLocalizationToStalk, AlgebraicGeometry.stalkToFiberRingHom_germ