Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper

Goal state before:
pp: case e_6.e_6
σ : Type u_1
A : Type u_2
inst✝¹⁰ : CommRing A
inst✝⁹ : SetLike σ A
inst✝⁸ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝⁷ : GradedRing 𝒜
O : Type u_3
inst✝⁶ : CommRing O
inst✝⁵ : IsDomain O
inst✝⁴ : ValuationRing O
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra O K
inst✝¹ : IsFractionRing O K
φ₀ : ↥(𝒜 0) →+* O
ι : Type u_5
inst✝ : Finite ι
x : ι → A
h2 : Algebra.adjoin (↥(𝒜 0)) (Set.range x) = ⊤
j : ι
φ : Away 𝒜 (x j) →+* K
hcomm : (algebraMap O K).comp φ₀ = φ.comp (fromZeroRingHom 𝒜 (Submonoid.powers (x j)))
d : ι → ℕ
hdi : ∀ (i : ι), 0 < d i
hxdi : ∀ (i : ι), x i ∈ 𝒜 (d i)
val✝ : Fintype ι
ψ : ι → ValueGroup O K := fun i ↦ (valuation O K) (φ (Away.isLocalizationElem ⋯ ⋯) ^ ∏ k ∈ Finset.univ.erase i, d k)
this✝¹ : Nonempty ι
Kmax : ValueGroup O K := (Finset.image ψ Finset.univ).max' ⋯
i₀ : ι
hi1 : ψ i₀ = Kmax
hi₀ : ∀ (j : ι), ψ j ≤ ψ i₀
hKmax : 0 < Kmax
this✝ : Algebra (Away 𝒜 (x j)) (Away 𝒜 (x j * x i₀)) := (awayMap 𝒜 ⋯ ⋯).toAlgebra
this : IsLocalization.Away (Away.isLocalizationElem ⋯ ⋯) (Away 𝒜 (x j * x i₀))
hunit : IsUnit (φ (Away.isLocalizationElem ⋯ ⋯))
φ' : Away 𝒜 (x j * x i₀) →+* K := IsLocalization.Away.lift (Away.isLocalizationElem ⋯ ⋯) hunit
hφ'1 : ∀ (s : Away 𝒜 (x j)), φ' ((awayMap 𝒜 ⋯ ⋯) s) = φ s
sx : Away 𝒜 (x i₀)
a : ℕ
ai : ι → ℕ
hai✝ : ∑ i, ai i • d i = a • d i₀
hai : ∑ x, ai x * d x = a * d i₀
H : (∏ i, x i ^ ai i) * x i₀ ^ (a * (d j - 1)) ∈ 𝒜 ((a * d i₀) • d j)
⊢ (Localization.r (Submonoid.powers (x j)))
    (((∏ i, x i ^ ai i) * x i₀ ^ (a * (d j - 1))) ^ (d j * ∏ i, d i), ⟨x j ^ (a * d i₀), ⋯⟩ ^ (d j * ∏ i, d i))
    ((∏ i, ((x i ^ d j) ^ ∏ i ∈ Finset.univ.erase i, d i) ^ (d i * ai i)) *
        ((x i₀ ^ d j) ^ ∏ i ∈ Finset.univ.erase i₀, d i) ^ (d i₀ * a * (d j - 1)),
      (∏ i, (⟨x j ^ d i, ⋯⟩ ^ ∏ i ∈ Finset.univ.erase i, d i) ^ (d i * ai i)) *
        (⟨x j ^ d i₀, ⋯⟩ ^ ∏ i ∈ Finset.univ.erase i₀, d i) ^ (d i₀ * a * (d j - 1)))

Tactic:
Localization.r_iff_exists

Goal state after:
case e_6.e_6
σ : Type u_1
A : Type u_2
inst✝¹⁰ : CommRing A
inst✝⁹ : SetLike σ A
inst✝⁸ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝⁷ : GradedRing 𝒜
O : Type u_3
inst✝⁶ : CommRing O
inst✝⁵ : IsDomain O
inst✝⁴ : ValuationRing O
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra O K
inst✝¹ : IsFractionRing O K
φ₀ : ↥(𝒜 0) →+* O
ι : Type u_5
inst✝ : Finite ι
x : ι → A
h2 : Algebra.adjoin (↥(𝒜 0)) (Set.range x) = ⊤
j : ι
φ : Away 𝒜 (x j) →+* K
hcomm : (algebraMap O K).comp φ₀ = φ.comp (fromZeroRingHom 𝒜 (Submonoid.powers (x j)))
d : ι → ℕ
hdi : ∀ (i : ι), 0 < d i
hxdi : ∀ (i : ι), x i ∈ 𝒜 (d i)
val✝ : Fintype ι
ψ : ι → ValueGroup O K := fun i ↦ (valuation O K) (φ (Away.isLocalizationElem ⋯ ⋯) ^ ∏ k ∈ Finset.univ.erase i, d k)
this✝¹ : Nonempty ι
Kmax : ValueGroup O K := (Finset.image ψ Finset.univ).max' ⋯
i₀ : ι
hi1 : ψ i₀ = Kmax
hi₀ : ∀ (j : ι), ψ j ≤ ψ i₀
hKmax : 0 < Kmax
this✝ : Algebra (Away 𝒜 (x j)) (Away 𝒜 (x j * x i₀)) := (awayMap 𝒜 ⋯ ⋯).toAlgebra
this : IsLocalization.Away (Away.isLocalizationElem ⋯ ⋯) (Away 𝒜 (x j * x i₀))
hunit : IsUnit (φ (Away.isLocalizationElem ⋯ ⋯))
φ' : Away 𝒜 (x j * x i₀) →+* K := IsLocalization.Away.lift (Away.isLocalizationElem ⋯ ⋯) hunit
hφ'1 : ∀ (s : Away 𝒜 (x j)), φ' ((awayMap 𝒜 ⋯ ⋯) s) = φ s
sx : Away 𝒜 (x i₀)
a : ℕ
ai : ι → ℕ
hai✝ : ∑ i, ai i • d i = a • d i₀
hai : ∑ x, ai x * d x = a * d i₀
H : (∏ i, x i ^ ai i) * x i₀ ^ (a * (d j - 1)) ∈ 𝒜 ((a * d i₀) • d j)
⊢ ∃ c,
    ↑c *
        (↑((∏ i, ((x i ^ d j) ^ ∏ i ∈ Finset.univ.erase i, d i) ^ (d i * ai i)) *
                  ((x i₀ ^ d j) ^ ∏ i ∈ Finset.univ.erase i₀, d i) ^ (d i₀ * a * (d j - 1)),
                (∏ i, (⟨x j ^ d i, ⋯⟩ ^ ∏ i ∈ Finset.univ.erase i, d i) ^ (d i * ai i)) *
                  (⟨x j ^ d i₀, ⋯⟩ ^ ∏ i ∈ Finset.univ.erase i₀, d i) ^ (d i₀ * a * (d j - 1))).2 *
          (((∏ i, x i ^ ai i) * x i₀ ^ (a * (d j - 1))) ^ (d j * ∏ i, d i),
              ⟨x j ^ (a * d i₀), ⋯⟩ ^ (d j * ∏ i, d i)).1) =
      ↑c *
        (↑(((∏ i, x i ^ ai i) * x i₀ ^ (a * (d j - 1))) ^ (d j * ∏ i, d i),
                ⟨x j ^ (a * d i₀), ⋯⟩ ^ (d j * ∏ i, d i)).2 *
          ((∏ i, ((x i ^ d j) ^ ∏ i ∈ Finset.univ.erase i, d i) ^ (d i * ai i)) *
                ((x i₀ ^ d j) ^ ∏ i ∈ Finset.univ.erase i₀, d i) ^ (d i₀ * a * (d j - 1)),
              (∏ i, (⟨x j ^ d i, ⋯⟩ ^ ∏ i ∈ Finset.univ.erase i, d i) ^ (d i * ai i)) *
                (⟨x j ^ d i₀, ⋯⟩ ^ ∏ i ∈ Finset.univ.erase i₀, d i) ^ (d i₀ * a * (d j - 1))).1)