Theorem: FractionalIdeal.finprod_heightOneSpectrum_factorization_principal_fraction
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
n : R
hn : n ≠ 0
d : ↥R⁰
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) =
    spanSingleton R⁰ (mk' K n d)
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
n : R
hn : n ≠ 0
d : ↥R⁰
hd_ne_zero : (algebraMap R K) ↑d ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) =
    spanSingleton R⁰ (mk' K n d)
tactic: have hd_ne_zero : (algebraMap R K) (d : R) ≠ 0 :=
    map_ne_zero_of_mem_nonZeroDivisors _ (IsFractionRing.injective R K) d.property
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
n : R
hn : n ≠ 0
d : ↥R⁰
hd_ne_zero : (algebraMap R K) ↑d ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) =
    spanSingleton R⁰ (mk' K n d)
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
n : R
hn : n ≠ 0
d : ↥R⁰
hd_ne_zero : (algebraMap R K) ↑d ≠ 0
h0 : spanSingleton R⁰ (mk' K n d) ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) =
    spanSingleton R⁰ (mk' K n d)
tactic: have h0 : spanSingleton R⁰ (mk' K n d) ≠ 0 := by
    rw [spanSingleton_ne_zero_iff, IsFractionRing.mk'_eq_div, ne_eq, div_eq_zero_iff, not_or]
    exact ⟨(map_ne_zero_iff (algebraMap R K) (IsFractionRing.injective R K)).mpr hn, hd_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
n : R
hn : n ≠ 0
d : ↥R⁰
hd_ne_zero : (algebraMap R K) ↑d ≠ 0
h0 : spanSingleton R⁰ (mk' K n d) ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) =
    spanSingleton R⁰ (mk' K n d)
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
n : R
hn : n ≠ 0
d : ↥R⁰
hd_ne_zero : (algebraMap R K) ↑d ≠ 0
h0 : spanSingleton R⁰ (mk' K n d) ≠ 0
hI : spanSingleton R⁰ (mk' K n d) = spanSingleton R⁰ ((algebraMap R K) ↑d)⁻¹ * ↑(Ideal.span {n})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) =
    spanSingleton R⁰ (mk' K n d)
tactic: have hI : spanSingleton R⁰ (mk' K n d) =
      spanSingleton R⁰ ((algebraMap R K) d)⁻¹ * ↑(Ideal.span {n} : Ideal R) := by
    rw [coeIdeal_span_singleton, spanSingleton_mul_spanSingleton]
    apply congr_arg
    rw [IsFractionRing.mk'_eq_div, div_eq_mul_inv, mul_comm]
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
n : R
hn : n ≠ 0
d : ↥R⁰
hd_ne_zero : (algebraMap R K) ↑d ≠ 0
h0 : spanSingleton R⁰ (mk' K n d) ≠ 0
hI : spanSingleton R⁰ (mk' K n d) = spanSingleton R⁰ ((algebraMap R K) ↑d)⁻¹ * ↑(Ideal.span {n})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) =
    spanSingleton R⁰ (mk' K n d)
state_after: no goals
tactic: exact finprod_heightOneSpectrum_factorization h0 hI