-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
theorem sup_eq_prod_inf_factors [DecidableEq (Ideal T)] (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod

Proof state:
case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
p : Ideal T
hp : p ∈ normalizedFactors I ∧ p ∈ normalizedFactors J
⊢ Prime p

Tactic:
exact prime_of_normalized_factor p hp.left

Result:
no goals

Proof state:
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : 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

Result:
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
⊢ I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod

Proof state:
case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (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:
rw [sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]

Result:
case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (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 ∧ (normalizedFactors I ∩ normalizedFactors J).prod ∣ J

Proof state:
case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
p : Ideal T
hp : p ∈ normalizedFactors I ∩ normalizedFactors J
⊢ Prime p

Tactic:
rw [mem_inter] at hp

Result:
case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
p : Ideal T
hp : p ∈ normalizedFactors I ∧ p ∈ normalizedFactors J
⊢ Prime p

Proof state:
case a.hx
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (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 ≠ 0

Tactic:
exact ne_bot_of_le_ne_bot hI le_sup_left

Result:
no goals

Proof state:
case a.left
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (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 ≤ normalizedFactors I

Tactic:
exact inf_le_left

Result:
no goals

Proof state:
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (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 := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>
    prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1

Result:
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (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

Proof state:
case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
a : Ideal T
⊢ count a (normalizedFactors (I ⊔ J)) ≤ count a (normalizedFactors I) ⊓ count a (normalizedFactors J)

Tactic:
exact le_min (count_le_of_ideal_ge le_sup_left hI a) (count_le_of_ideal_ge le_sup_right hJ a)

Result:
no goals

Proof state:
case a.right
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (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 ≤ normalizedFactors J

Tactic:
exact inf_le_right

Result:
no goals

Proof state:
case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (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:
rw [← dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors,
  normalizedFactors_prod_of_prime, le_iff_count]

Result:
case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ ∀ (a : Ideal T), count a (normalizedFactors (I ⊔ J)) ≤ count a (normalizedFactors I ∩ normalizedFactors J)

case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ ∀ p ∈ normalizedFactors I ∩ normalizedFactors J, Prime p

case a.hx
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (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 ≠ 0

case a.hy
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (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 ≠ 0