-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
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 ⋯)
y : NumDenSameDeg 𝒜 (Submonoid.powers f)
hy :
  Membership.mem (γ := Submonoid (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)))
    (HomogeneousLocalization.mk y)
⊢ IsUnit (M := 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)
        (↑(Subtype.mk (p := fun (x_1 : A⁰_ f) ↦
              Membership.mem (γ := Submonoid (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)))
                x_1)
              (HomogeneousLocalization.mk y) hy) :
          A⁰_ f) :
      AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
have toIsLocalizationMap : HomogeneousLocalization.val
    (HMul.hMul (α := AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) (β :=
      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)
          (↑(Subtype.mk (p := fun (x_1 : A⁰_ f) ↦
                Membership.mem (γ := Submonoid (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))
                                    sorry).toPresheafedSpace :
                              TopCat))
                            (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
                          (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
                            (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
                        x)))
                  x_1)
                (HomogeneousLocalization.mk y) hy) :
            A⁰_ f) :
        AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
      (HomogeneousLocalization.mk
        { deg := y.deg, num := y.den, den := y.num, den_mem := sorry :
          NumDenSameDeg 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal.primeCompl })) =
  HomogeneousLocalization.val.{0, u_1, u_2} (ι := ℕ) (σ := σ) (𝒜 := 𝒜) 1
Tactic: refine .of_mul_eq_one
      (.mk ⟨y.deg, y.den, y.num, (mk_mem_toSpec_base_apply _ _ _).not.mp hy⟩) <| val_injective _ ?_