Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper

Goal state before:
pp: case h
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 := ⋯
this✝¹ : Nonempty ι
Kmax : ValueGroup O K := ⋯
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)) := ⋯
this : IsLocalization.Away (Away.isLocalizationElem ⋯ ⋯) (Away 𝒜 (x j * x i₀))
hunit : IsUnit (φ (Away.isLocalizationElem ⋯ ⋯))
φ' : Away 𝒜 (x j * x i₀) →+* K := ⋯
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)
⊢ ↑1 *
      (↑((∏ 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) =
    ↑1 *
      (↑(((∏ 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)

Tactic:
simp only [OneMemClass.coe_one, ← pow_mul, Submonoid.coe_mul,
            SubmonoidClass.coe_finsetProd, one_mul]

Goal state after:
case h
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)
⊢ (∏ 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)))) *
      ((∏ 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)) *
      ((∏ x_1, x x_1 ^ ((d j * ∏ i ∈ Finset.univ.erase x_1, d i) * (d x_1 * ai x_1))) *
        x i₀ ^ ((d j * ∏ i ∈ Finset.univ.erase i₀, d i) * (d i₀ * a * (d j - 1))))