-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic
Goal:
σ : Type u_1
A : Type u
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
x : A
hx : x = f * g
i_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)
hi :
  Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
    (Opposite.unop (Opposite.op (basicOpen 𝒜 x)) : TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u)) i_1
n : ℕ
a : ↥(𝒜 n)
i : ℕ
hb' : (fun (x : ℕ) ↦ f ^ x) i ∈ 𝒜 n
hf :
  Eq (α := Localization (M := A) i_1.asHomogeneousIdeal.toIdeal.primeCompl)
    (Localization.mk ((RingHom.id A : (a : A) → A) ((↑a : A) * g ^ i))
        (Subtype.mk (p := fun (x_1 : A) ↦
          x_1 ∈
            (↑(Subtype.mk (p := fun (x_2 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
                        Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
                          (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_2)
                        i_1 hi) :
                    (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl)
          ((RingHom.id A : (a : A) → A)
            (↑(Subtype.mk (p := fun (x_1 : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers x) x_1) (x ^ i) ⋯) :
              A))
          ⋯) :
      Localization (M := A)
        (↑(Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
                    Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
                      (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
                    i_1 hi) :
                (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl)
    (val.{0, u, u_1} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
        ((↑((CommRingCat.Hom.hom ((Proj 𝒜).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
                  (a : (↑Γ(Proj 𝒜, basicOpen 𝒜 f) : Type u)) → (↑Γ(Proj 𝒜, basicOpen 𝒜 x) : Type u))
                ((CommRingCat.Hom.hom (awayToSection 𝒜 f) : (a : Away 𝒜 f) → (↑Γ(Proj 𝒜, basicOpen 𝒜 f) : Type u))
                  (mk
                    { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb',
                      den_mem := ⋯ }))) :
            (x_1 : ↥(basicOpen 𝒜 x)) →
              AtPrime 𝒜 (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal)
          (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
            Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
              (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
            i_1 hi)) :
      Localization (M := A) i_1.asHomogeneousIdeal.toIdeal.primeCompl)
⊢ Eq (α := Localization (M := A) i_1.asHomogeneousIdeal.toIdeal.primeCompl)
    ((IsLocalization.map (M := Submonoid.powers x) (T :=
            (↑(Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
                        Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
                          (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
                        i_1 hi) :
                    (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl)
            (Localization (M := A)
              (↑(Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
                          Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
                            (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
                          i_1 hi) :
                      (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl)
            (RingHom.id A) ⋯ :
          (a : Localization (Submonoid.powers x)) →
            Localization (↑⟨i_1, hi⟩ : (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl)
        (val.{0, u, u_1} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
          ((CommRingCat.Hom.hom (CommRingCat.ofHom (awayMap 𝒜 g_deg hx)) : (a : Away 𝒜 f) → Away 𝒜 x)
            (mk { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb', den_mem := ⋯ }))) :
      Localization (M := A)
        (↑(Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
                    Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
                      (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
                    i_1 hi) :
                (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl)
    (val.{0, u, u_1} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
        ((↑((CommRingCat.Hom.hom ((Proj 𝒜).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
                  (a : (↑Γ(Proj 𝒜, basicOpen 𝒜 f) : Type u)) → (↑Γ(Proj 𝒜, basicOpen 𝒜 x) : Type u))
                ((CommRingCat.Hom.hom (awayToSection 𝒜 f) : (a : Away 𝒜 f) → (↑Γ(Proj 𝒜, basicOpen 𝒜 f) : Type u))
                  (mk
                    { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb',
                      den_mem := ⋯ }))) :
            (x_1 : ↥(basicOpen 𝒜 x)) →
              AtPrime 𝒜 (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal)
          (Subtype.mk (p := fun (x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u)) ↦
            Membership.mem (γ := TopologicalSpace.Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u))
              (basicOpen 𝒜 x : (Proj 𝒜).Opens) x_1)
            i_1 hi)) :
      Localization (M := A) i_1.asHomogeneousIdeal.toIdeal.primeCompl)

Tactic:
rw [CommRingCat.hom_ofHom, val_awayMap_mk, Localization.mk_eq_mk', IsLocalization.map_mk',
    ← Localization.mk_eq_mk']

Premises used: IsLocalization.map_mk', Localization.mk_eq_mk', CommRingCat.hom_ofHom, HomogeneousLocalization.val_awayMap_mk