Theorem: sup_eq_prod_inf_factors
File: Mathlib/RingTheory/DedekindDomain/Ideal.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod
state_after: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
⊢ I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod
tactic: have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    rw [mem_inter] at hp
    exact prime_of_normalized_factor p hp.left
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
⊢ I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod
state_after: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod
tactic: have := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>
      prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod
state_after: case a
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ I ⊔ J ≤ (normalizedFactors I ∩ normalizedFactors J).prod

case a
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ≤ I ⊔ J
tactic: apply le_antisymm