Theorem: FractionalIdeal.finprod_heightOneSpectrum_factorization
File: Mathlib/RingTheory/DedekindDomain/Factorization.lean

Proof steps:
state_before: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    I
state_after: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    I
tactic: have hJ_ne_zero : J ≠ 0 := ideal_factor_ne_zero hI haJ
state_before: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    I
state_after: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    I
tactic: have hJ := Ideal.finprod_heightOneSpectrum_factorization_coe K hJ_ne_zero
state_before: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    I
state_after: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    I
tactic: have ha_ne_zero : Ideal.span {a} ≠ 0 := constant_factor_ne_zero hI haJ
state_before: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    I
state_after: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    I
tactic: have ha := Ideal.finprod_heightOneSpectrum_factorization_coe K ha_ne_zero
state_before: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    I
state_after: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    (∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors)) *
      ∏ᶠ (x : HeightOneSpectrum R),
        (↑x.asIdeal ^ ↑((Associates.mk x.asIdeal).count (Associates.mk (Ideal.span {a})).factors))⁻¹
tactic: rw [haJ, ← div_spanSingleton, div_eq_mul_inv, ← coeIdeal_span_singleton, ← hJ, ← ha,
    ← finprod_inv_distrib]
state_before: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    (∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors)) *
      ∏ᶠ (x : HeightOneSpectrum R),
        (↑x.asIdeal ^ ↑((Associates.mk x.asIdeal).count (Associates.mk (Ideal.span {a})).factors))⁻¹
state_after: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    (∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors)) *
      ∏ᶠ (x : HeightOneSpectrum R),
        ↑x.asIdeal ^ (-↑((Associates.mk x.asIdeal).count (Associates.mk (Ideal.span {a})).factors))
tactic: simp_rw [← zpow_neg]
state_before: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    (∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors)) *
      ∏ᶠ (x : HeightOneSpectrum R),
        ↑x.asIdeal ^ (-↑((Associates.mk x.asIdeal).count (Associates.mk (Ideal.span {a})).factors))
state_after: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    ∏ᶠ (i : HeightOneSpectrum R),
      ↑i.asIdeal ^ ↑((Associates.mk i.asIdeal).count (Associates.mk J).factors) *
        ↑i.asIdeal ^ (-↑((Associates.mk i.asIdeal).count (Associates.mk (Ideal.span {a})).factors))
tactic: rw [← finprod_mul_distrib (Ideal.finite_mulSupport_coe hJ_ne_zero)
    (Ideal.finite_mulSupport_inv ha_ne_zero)]
state_before: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    ∏ᶠ (i : HeightOneSpectrum R),
      ↑i.asIdeal ^ ↑((Associates.mk i.asIdeal).count (Associates.mk J).factors) *
        ↑i.asIdeal ^ (-↑((Associates.mk i.asIdeal).count (Associates.mk (Ideal.span {a})).factors))
state_after: case h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
⊢ ∀ (x : HeightOneSpectrum R),
    ↑x.asIdeal ^
        (↑((Associates.mk x.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk x.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
      ↑x.asIdeal ^ ↑((Associates.mk x.asIdeal).count (Associates.mk J).factors) *
        ↑x.asIdeal ^ (-↑((Associates.mk x.asIdeal).count (Associates.mk (Ideal.span {a})).factors))
tactic: apply finprod_congr
state_before: case h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
⊢ ∀ (x : HeightOneSpectrum R),
    ↑x.asIdeal ^
        (↑((Associates.mk x.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk x.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
      ↑x.asIdeal ^ ↑((Associates.mk x.asIdeal).count (Associates.mk J).factors) *
        ↑x.asIdeal ^ (-↑((Associates.mk x.asIdeal).count (Associates.mk (Ideal.span {a})).factors))
state_after: case h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v✝ : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
v : HeightOneSpectrum R
⊢ ↑v.asIdeal ^
      (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
        ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) *
      ↑v.asIdeal ^ (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors))
tactic: intro v
state_before: case h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v✝ : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑(Ideal.span {a})
v : HeightOneSpectrum R
⊢ ↑v.asIdeal ^
      (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
        ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) *
      ↑v.asIdeal ^ (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors))
state_after: no goals
tactic: rw [← zpow_add₀ ((@coeIdeal_ne_zero R _ K _ _ _ _).mpr v.ne_bot), sub_eq_add_neg]