-- 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 : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
a : A⁰_ f
a₁ :
  Eq (α :=
    (↑((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                ⋯).presheaf.stalk
          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 := CommRingCat.of (A⁰_ f)) (Y :=
            (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    ⋯).presheaf.stalk
              x)
            (StructureSheaf.toStalk (A⁰_ f)
                ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                      carrier) (X :=
                      (↑(Proj.restrict (U :=
                              Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                              ⋯).toPresheafedSpace :
                        TopCat))
                      (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
                    (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
                      (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
                  x) ≫
              Iso.hom (specStalkEquiv 𝒜 f x f_deg hm) ≫
                Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : (↑Proj.T : Type u_1)))) ≫
                  Iso.inv (Proj.restrictStalkIso ⋯ x)) :
          (a : (↑(CommRingCat.of (A⁰_ f)) : Type u_1)) → (↑((Proj.restrict ⋯).presheaf.stalk x) : Type u_1))
        a :
      (↑((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  ⋯).presheaf.stalk
            x) :
        Type u_1))
    ((RingHom.comp (β :=
            (↑(Presheaf.stalk (X := PrimeSpectrum.Top (A⁰_ f)) (Sheaf.presheaf («Spec».structureSheaf (A⁰_ f)))
                  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                        carrier) (X :=
                        (↑(Proj.restrict (U :=
                                Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                                ⋯).toPresheafedSpace :
                          TopCat))
                        (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
                      (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
                        (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
                    x)) :
              Type u_1))
            (CommRingCat.Hom.hom
              (Iso.hom (specStalkEquiv 𝒜 f x f_deg hm) ≫
                Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : ProjectiveSpectrum 𝒜))) ≫
                  Iso.inv (Proj.restrictStalkIso ⋯ x)))
            (algebraMap (A⁰_ f)
              (↑(Presheaf.stalk (X := PrimeSpectrum.Top (A⁰_ f)) (Sheaf.presheaf («Spec».structureSheaf (A⁰_ f)))
                    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                          carrier) (X :=
                          (↑(Proj.restrict (U :=
                                  Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                                  ⋯).toPresheafedSpace :
                            TopCat))
                          (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
                        (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
                          (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
                      x)) :
                Type u_1)) :
          (a : A⁰_ f) → (↑((Proj.restrict ⋯).presheaf.stalk x) : Type u_1))
        a :
      (↑((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  ⋯).presheaf.stalk
            x) :
        Type u_1))
a₂ : ℕ
a₃ : f ∈ 𝒜 m
a₄ : 0 < m
⊢ Eq (α :=
    (↑((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                ⋯).presheaf.stalk
          x) :
      Type u_1))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
            CommRingCat.carrier) (Y :=
            (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    ⋯).presheaf.stalk
              x)
            (awayToΓ 𝒜 f ≫
              (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                      ⋯).presheaf.Γgerm
                x) :
          (a : (↑(CommRingCat.of (A⁰_ f)) : Type u_1)) → (↑((Proj.restrict ⋯).presheaf.stalk x) : Type u_1))
        a :
      (↑((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  ⋯).presheaf.stalk
            x) :
        Type u_1))
    ((RingHom.comp (β :=
            (↑(Presheaf.stalk (X := PrimeSpectrum.Top (A⁰_ f)) (Sheaf.presheaf («Spec».structureSheaf (A⁰_ f)))
                  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                        carrier) (X :=
                        (↑(Proj.restrict (U :=
                                Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                                ⋯).toPresheafedSpace :
                          TopCat))
                        (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
                      (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
                        (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
                    x)) :
              Type u_1))
            (CommRingCat.Hom.hom
              (Iso.hom (specStalkEquiv 𝒜 f x f_deg hm) ≫
                Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : ProjectiveSpectrum 𝒜))) ≫
                  Iso.inv (Proj.restrictStalkIso ⋯ x)))
            (algebraMap (A⁰_ f)
              (↑(Presheaf.stalk (X := PrimeSpectrum.Top (A⁰_ f)) (Sheaf.presheaf («Spec».structureSheaf (A⁰_ f)))
                    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                          carrier) (X :=
                          (↑(Proj.restrict (U :=
                                  Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                                  ⋯).toPresheafedSpace :
                            TopCat))
                          (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
                        (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
                          (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
                      x)) :
                Type u_1)) :
          (a : A⁰_ f) → (↑((Proj.restrict ⋯).presheaf.stalk x) : Type u_1))
        a :
      (↑((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  ⋯).presheaf.stalk
            x) :
        Type u_1))

Tactic:
rw [awayToΓ_ΓToStalk, ← toStalk_specStalkEquiv, Category.assoc]

Premises used: CategoryTheory.Category.assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_specStalkEquiv, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk