-- 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
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :=
  RingHom.toAlgebra (mapId 𝒜 ⋯)
z : AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal
toIsLocalizationMap :
  ∀ (i : ℕ) (a : ↥(𝒜 i)) (b : A) (hb : b ∈ 𝒜 i) (hb' : b ∉ (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal),
    ∃ (x_1 :
      (A⁰_ f) ×
        ↥(Ideal.primeCompl
            (PrimeSpectrum.asIdeal
              ((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)))),
      HMul.hMul (α := AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) (β :=
          AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
          (HomogeneousLocalization.mk
            { deg := i, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 i) b hb, den_mem := hb' :
              NumDenSameDeg 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal.primeCompl })
          ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
                (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
              (↑x_1.2 : A⁰_ f) :
            AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) =
        (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
            (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
          x_1.1
⊢ ∃ (x_1 :
    (A⁰_ f) ×
      ↥(Ideal.primeCompl
          (PrimeSpectrum.asIdeal
            ((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)))),
    HMul.hMul (β := AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) z
        ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
              (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
            (↑x_1.2 : A⁰_ f) :
          AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) =
      (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
          (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
        x_1.1

Tactic:
obtain ⟨⟨i, a, ⟨b, hb⟩, (hb' : b ∉ x.1.1)⟩, rfl⟩ := z.mk_surjective

Premises used: HomogeneousLocalization.mk_surjective, Subtype.val, ProjectiveSpectrum.asHomogeneousIdeal, rfl