-- 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 :
  ∃ (c :
    ↥(↑(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),
    (↑c : A) *
        HMul.hMul (β := A)
          (↑((↑(NumDenSameDeg.num
                        ((fun (x_1 : NumDenSameDeg 𝒜 (Submonoid.powers f)) ↦
                            ({ deg := x_1.deg,
                                num :=
                                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg)
                                    ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.num : A)) ⋯,
                                den :=
                                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg)
                                    ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.den : A)) ⋯,
                                den_mem := ⋯ } :
                              NumDenSameDeg 𝒜
                                (Ideal.primeCompl
                                  (HomogeneousIdeal.toIdeal.{0, u_1, u} (σ := σ) (𝒜 := 𝒜)
                                    (ProjectiveSpectrum.asHomogeneousIdeal
                                      (↑((Quiver.Hom.unop (Quiver.Hom.op (homOfLE ⋯)) :
                                              (a : ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)))) →
                                                ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 f))))
                                            (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)))))))
                          { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb',
                            den_mem := ⋯ : NumDenSameDeg 𝒜 (Submonoid.powers f) })) :
                    A),
                  Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ i_1.asHomogeneousIdeal.toIdeal.primeCompl)
                    (↑(NumDenSameDeg.den
                          ((fun (x_1 : NumDenSameDeg 𝒜 (Submonoid.powers f)) ↦
                              ({ deg := x_1.deg,
                                  num :=
                                    Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg)
                                      ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.num : A)) ⋯,
                                  den :=
                                    Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg)
                                      ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.den : A)) ⋯,
                                  den_mem := ⋯ } :
                                NumDenSameDeg 𝒜
                                  (Ideal.primeCompl
                                    (HomogeneousIdeal.toIdeal.{0, u_1, u} (σ := σ) (𝒜 := 𝒜)
                                      (ProjectiveSpectrum.asHomogeneousIdeal
                                        (↑((Quiver.Hom.unop (Quiver.Hom.op (homOfLE ⋯)) :
                                                (a : ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)))) →
                                                  ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 f))))
                                              (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)))))))
                            { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb',
                              den_mem := ⋯ : NumDenSameDeg 𝒜 (Submonoid.powers f) })) :
                      A)
                    ⋯).2 :
            A)
          (((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))
                  ⋯).1 :
            A) =
      (↑c : A) *
        HMul.hMul (α := A)
          ((↑((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))
                      ⋯).2 :
              A) :
            A)
          ((↑(NumDenSameDeg.num
                    ((fun (x_1 : NumDenSameDeg 𝒜 (Submonoid.powers f)) ↦
                        ({ deg := x_1.deg,
                            num :=
                              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg)
                                ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.num : A)) ⋯,
                            den :=
                              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg)
                                ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.den : A)) ⋯,
                            den_mem := ⋯ } :
                          NumDenSameDeg 𝒜
                            (Ideal.primeCompl
                              (HomogeneousIdeal.toIdeal.{0, u_1, u} (σ := σ) (𝒜 := 𝒜)
                                (ProjectiveSpectrum.asHomogeneousIdeal
                                  (↑((Quiver.Hom.unop (Quiver.Hom.op (homOfLE ⋯)) :
                                          (a : ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)))) →
                                            ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 f))))
                                        (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)))))))
                      { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb', den_mem := ⋯ :
                        NumDenSameDeg 𝒜 (Submonoid.powers f) })) :
                A),
              Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ i_1.asHomogeneousIdeal.toIdeal.primeCompl)
                (↑(NumDenSameDeg.den
                      ((fun (x_1 : NumDenSameDeg 𝒜 (Submonoid.powers f)) ↦
                          ({ deg := x_1.deg,
                              num :=
                                Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg)
                                  ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.num : A)) ⋯,
                              den :=
                                Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg)
                                  ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.den : A)) ⋯,
                              den_mem := ⋯ } :
                            NumDenSameDeg 𝒜
                              (Ideal.primeCompl
                                (HomogeneousIdeal.toIdeal.{0, u_1, u} (σ := σ) (𝒜 := 𝒜)
                                  (ProjectiveSpectrum.asHomogeneousIdeal
                                    (↑((Quiver.Hom.unop (Quiver.Hom.op (homOfLE ⋯)) :
                                            (a : ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)))) →
                                              ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 f))))
                                          (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)))))))
                        { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb', den_mem := ⋯ :
                          NumDenSameDeg 𝒜 (Submonoid.powers f) })) :
                  A)
                ⋯).1
⊢ ((Localization.r.{u} (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 :
        Con
          (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)) :
      (a : A × ↥(↑⟨i_1, hi⟩ : (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl) →
        A × ↥(↑⟨i_1, hi⟩ : (↑(ProjectiveSpectrum.top 𝒜) : Type u)).asHomogeneousIdeal.toIdeal.primeCompl → Prop)
    (((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))
          ⋯) :
      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)
    ((↑(NumDenSameDeg.num
            ((fun (x_1 : NumDenSameDeg 𝒜 (Submonoid.powers f)) ↦
                ({ deg := x_1.deg,
                    num :=
                      Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg) ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.num : A))
                        ⋯,
                    den :=
                      Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg) ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.den : A))
                        ⋯,
                    den_mem := ⋯ } :
                  NumDenSameDeg 𝒜
                    (Ideal.primeCompl
                      (HomogeneousIdeal.toIdeal.{0, u_1, u} (σ := σ) (𝒜 := 𝒜)
                        (ProjectiveSpectrum.asHomogeneousIdeal
                          (↑((Quiver.Hom.unop (Quiver.Hom.op (homOfLE ⋯)) :
                                  (a : ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)))) →
                                    ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 f))))
                                (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)))))))
              { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb', den_mem := ⋯ })) :
        A),
      Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ i_1.asHomogeneousIdeal.toIdeal.primeCompl)
        (↑(NumDenSameDeg.den
              ((fun (x_1 : NumDenSameDeg 𝒜 (Submonoid.powers f)) ↦
                  ({ deg := x_1.deg,
                      num :=
                        Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg)
                          ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.num : A)) ⋯,
                      den :=
                        Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 x_1.deg)
                          ((GradedRingHom.id 𝒜 : (a : A) → A) (↑x_1.den : A)) ⋯,
                      den_mem := ⋯ } :
                    NumDenSameDeg 𝒜
                      (Ideal.primeCompl
                        (HomogeneousIdeal.toIdeal.{0, u_1, u} (σ := σ) (𝒜 := 𝒜)
                          (ProjectiveSpectrum.asHomogeneousIdeal
                            (↑((Quiver.Hom.unop (Quiver.Hom.op (homOfLE ⋯)) :
                                    (a : ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 x)))) →
                                      ↥(Opposite.unop (Opposite.op (basicOpen 𝒜 f))))
                                  (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)))))))
                { deg := n, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 n) (f ^ i) hb', den_mem := ⋯ })) :
          A)
        ⋯)

Tactic:
rw [Localization.r_iff_exists]

Premises used: Localization.r_iff_exists