-- 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 ∈ normalizedFactors I ∩ normalizedFactors J, Prime p

Tactic:
intro p 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 ∩ normalizedFactors J
⊢ Prime p

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 ≠ ⊥
⊢ normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J

Tactic:
apply normalizedFactors_prod_of_prime

Result:
case h
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ ∀ p ∈ normalizedFactors I ∩ normalizedFactors J, Prime p

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
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod

Tactic:
apply le_antisymm

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
⊢ I ⊔ J ≤ (normalizedFactors I ∩ normalizedFactors J).prod

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

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

Tactic:
rw [dvd_iff_normalizedFactors_le_normalizedFactors this hJ, H]

Result:
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

Proof state:
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

Tactic:
exact this

Result:
no goals

Proof state:
case h
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
p : Ideal T
hp : p ∈ normalizedFactors I ∩ normalizedFactors J
⊢ Prime p

Tactic:
rw [mem_inter] at hp

Result:
case h
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
p : Ideal T
hp : p ∈ normalizedFactors I ∧ p ∈ normalizedFactors J
⊢ Prime p

Proof state:
case h
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ ∀ p ∈ normalizedFactors I ∩ normalizedFactors J, Prime p

Tactic:
intro p hp

Result:
case h
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
p : Ideal T
hp : p ∈ normalizedFactors I ∩ normalizedFactors J
⊢ Prime p

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 ∩ normalizedFactors J)

Tactic:
intro a

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)

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).prod ∣ I

Tactic:
rw [dvd_iff_normalizedFactors_le_normalizedFactors this hI, H]

Result:
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

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 ∩ normalizedFactors J)

Tactic:
rw [Multiset.count_inter]

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) ⊓ count a (normalizedFactors J)