-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor
Goal:
A B σ τ : Type u
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
inst_3 : CommRing B
inst_4 : SetLike τ B
inst_5 : AddSubgroupClass τ B
𝒜 : ℕ → σ
ℬ : ℕ → τ
inst_6 : GradedRing 𝒜
inst_7 : GradedRing ℬ
f : 𝒜 →+*ᵍ ℬ
hf : LE.le (α := HomogeneousIdeal ℬ) (ℬ₊) (HomogeneousIdeal.map f (𝒜₊))
p : ProjectiveSpectrum ℬ
c :
  NumDenSameDeg 𝒜
    (Ideal.primeCompl
      (toIdeal (σ := σ) (𝒜 := 𝒜)
        (asHomogeneousIdeal
          (((ProjectiveSpectrum.comap f hf : C(ProjectiveSpectrum ℬ, ProjectiveSpectrum 𝒜)) :
              (a : ProjectiveSpectrum ℬ) → ProjectiveSpectrum 𝒜)
            p))))
hf₁ :
  Eq (α := Localization (M := B) p.asHomogeneousIdeal.toIdeal.primeCompl)
    (Localization.mk ((f : (a : A) → B) (↑c.num : A))
        (Subtype.mk (p := fun (x : B) ↦ x ∈ p.asHomogeneousIdeal.toIdeal.primeCompl) ((f : (a : A) → B) (↑c.den : A))
          ⋯) :
      Localization (M := B) p.asHomogeneousIdeal.toIdeal.primeCompl)
    (val (ι := ℕ) (σ := τ) (𝒜 := ℬ)
        ((↑(sectionInBasicOpen ℬ p (NumDenSameDeg.map f ⋯ c)) :
            (x : ↥(Opposite.unop (Opposite.op (ProjectiveSpectrum.basicOpen ℬ ((f : (a : A) → B) (↑c.den : A)))))) →
              (fun (x : (↑(top ℬ) : Type u)) ↦ AtPrime ℬ x.asHomogeneousIdeal.toIdeal) (↑x : (↑(top ℬ) : Type u)))
          (Subtype.mk (p := fun (x : (↑(top ℬ) : Type u)) ↦
            Membership.mem (γ := Opens (↑(top ℬ) : Type u))
              (Opposite.unop (Opposite.op (ProjectiveSpectrum.basicOpen ℬ ((f : (a : A) → B) (↑c.den : A))))) x)
            p ⋯)) :
      Localization (M := B) p.asHomogeneousIdeal.toIdeal.primeCompl)
⊢ Eq (α := Localization (M := B) p.asHomogeneousIdeal.toIdeal.primeCompl)
    (Localization.mk ((f : (a : A) → B) (↑c.num : A))
        (Subtype.mk (p := fun (x : B) ↦ x ∈ p.asHomogeneousIdeal.toIdeal.primeCompl) ((f : (a : A) → B) (↑c.den : A))
          ⋯) :
      Localization (M := B) p.asHomogeneousIdeal.toIdeal.primeCompl)
    (val (ι := ℕ) (σ := τ) (𝒜 := ℬ)
        ((CommRingCat.Hom.hom
              (CommRingCat.ofHom
                (↑(stalkIso' ℬ p) :
                  (↑((structureSheaf ℬ).presheaf.stalk p) : Type u) →+* AtPrime ℬ p.asHomogeneousIdeal.toIdeal)) :
            (a : (↑((Proj ℬ).presheaf.stalk p) : Type u)) → AtPrime ℬ p.asHomogeneousIdeal.toIdeal)
          ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X :=
                (toSheafedSpace ℬ).presheaf.obj
                  (Opposite.op (ProjectiveSpectrum.basicOpen ℬ ((f : (a : A) → B) (↑c.den : A)))))
                (Y := (toSheafedSpace ℬ).presheaf.stalk p)
                ((toSheafedSpace ℬ).presheaf.germ (ProjectiveSpectrum.basicOpen ℬ ((f : (a : A) → B) (↑c.den : A))) p
                  ⋯) :
              (a :
                  (↑((toSheafedSpace ℬ).presheaf.obj
                        (Opposite.op (ProjectiveSpectrum.basicOpen ℬ ((f : (a : A) → B) (↑c.den : A))))) :
                    Type u)) →
                (↑((toSheafedSpace ℬ).presheaf.stalk p) : Type u))
            (sectionInBasicOpen ℬ p (NumDenSameDeg.map f ⋯ c) :
              (↑(Functor.obj (C := (Opens (↑(top ℬ) : Type u))ᵒᵖ)
                    (ObjectProperty.FullSubcategory.obj (P :=
                        (Presheaf.IsSheaf (Opens.grothendieckTopology (↑(top ℬ) : Type u)) :
                          (P : (Opens (↑(top ℬ) : Type u))ᵒᵖ ⥤ CommRingCat) → Prop))
                        (structureSheaf ℬ : TopCat.Sheaf CommRingCat (top ℬ)) :
                      (Opens (↑(top ℬ) : Type u))ᵒᵖ ⥤ CommRingCat)
                    (Opposite.op (ProjectiveSpectrum.basicOpen ℬ (↑(NumDenSameDeg.den (NumDenSameDeg.map f ⋯ c)) : B)) :
                      (Opens (↑(top ℬ) : Type u))ᵒᵖ)) :
                Type u)))) :
      Localization (M := B) p.asHomogeneousIdeal.toIdeal.primeCompl)

Tactic:
erw [stalkIso'_germ]

Premises used: AlgebraicGeometry.Proj.stalkIso'_germ