-- Mathlib/RingTheory/DedekindDomain/Factorization.lean
theorem count_finprod_coprime (exps : HeightOneSpectrum R → ℤ) :
    count K v (∏ᶠ (w : HeightOneSpectrum R) (_ : w ≠ v),
      (w.asIdeal : (FractionalIdeal R⁰ K)) ^ exps w) = 0

Proof state:
case hp₂
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
exps : HeightOneSpectrum R → ℤ
⊢ ∀ x ∈ fun i => i = v → False, count K v (↑x.asIdeal ^ exps x) = 0

Tactic:
intro w hw

Result:
case hp₂
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
exps : HeightOneSpectrum R → ℤ
w : HeightOneSpectrum R
hw : w ∈ fun i => i = v → False
⊢ count K v (↑w.asIdeal ^ exps w) = 0

Proof state:
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
exps : HeightOneSpectrum R → ℤ
⊢ count K v (∏ᶠ (w : HeightOneSpectrum R) (_ : w ≠ v), ↑w.asIdeal ^ exps w) = 0

Tactic:
apply finprod_mem_induction fun I => count K v I = 0

Result:
case hp₀
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
exps : HeightOneSpectrum R → ℤ
⊢ count K v 1 = 0

case hp₁
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
exps : HeightOneSpectrum R → ℤ
⊢ ∀ (x y : FractionalIdeal R⁰ K), count K v x = 0 → count K v y = 0 → count K v (x * y) = 0

case hp₂
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
exps : HeightOneSpectrum R → ℤ
⊢ ∀ x ∈ fun i => i = v → False, count K v (↑x.asIdeal ^ exps x) = 0

Proof state:
case hp₀
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
exps : HeightOneSpectrum R → ℤ
⊢ count K v 1 = 0

Tactic:
exact count_one K v

Result:
no goals

Proof state:
case hp₁
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
exps : HeightOneSpectrum R → ℤ
I I' : FractionalIdeal R⁰ K
hI : count K v I = 0
hI' : count K v I' = 0
⊢ count K v (I * I') = 0

Tactic:
by_cases h : I ≠ 0 ∧ I' ≠ 0

Result:
case pos
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
exps : HeightOneSpectrum R → ℤ
I I' : FractionalIdeal R⁰ K
hI : count K v I = 0
hI' : count K v I' = 0
h : I ≠ 0 ∧ I' ≠ 0
⊢ count K v (I * I') = 0

case neg
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
exps : HeightOneSpectrum R → ℤ
I I' : FractionalIdeal R⁰ K
hI : count K v I = 0
hI' : count K v I' = 0
h : ¬(I ≠ 0 ∧ I' ≠ 0)
⊢ count K v (I * I') = 0

Proof state:
case hp₁
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
exps : HeightOneSpectrum R → ℤ
⊢ ∀ (x y : FractionalIdeal R⁰ K), count K v x = 0 → count K v y = 0 → count K v (x * y) = 0

Tactic:
intro I I' hI hI'

Result:
case hp₁
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
exps : HeightOneSpectrum R → ℤ
I I' : FractionalIdeal R⁰ K
hI : count K v I = 0
hI' : count K v I' = 0
⊢ count K v (I * I') = 0

Proof state:
case pos
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
exps : HeightOneSpectrum R → ℤ
I I' : FractionalIdeal R⁰ K
hI : count K v I = 0
hI' : count K v I' = 0
h : I ≠ 0 ∧ I' ≠ 0
⊢ count K v (I * I') = 0

Tactic:
rw [count_mul' K v, if_pos h, hI, hI', add_zero]

Result:
no goals

Proof state:
case neg
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
exps : HeightOneSpectrum R → ℤ
I I' : FractionalIdeal R⁰ K
hI : count K v I = 0
hI' : count K v I' = 0
h : ¬(I ≠ 0 ∧ I' ≠ 0)
⊢ count K v (I * I') = 0

Tactic:
rw [count_mul' K v, if_neg h]

Result:
no goals

Proof state:
case hp₂
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
exps : HeightOneSpectrum R → ℤ
w : HeightOneSpectrum R
hw : w ∈ fun i => i = v → False
⊢ count K v (↑w.asIdeal ^ exps w) = 0

Tactic:
rw [count_zpow, count_maximal_coprime K v hw, MulZeroClass.mul_zero]

Result:
no goals