l3lab/ntp-mathlib huggingface dataset:
theorem idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors (hJ : J ≠ ⊥)
    {L : Ideal R} (hL : L ∈ normalizedFactors I) :
    ↑(idealFactorsEquivOfQuotEquiv f ⟨L, dvd_of_mem_normalizedFactors hL⟩)
      ∈ normalizedFactors J := by
  have hI : I ≠ ⊥
  · intro hI
    rw [hI, bot_eq_zero, normalizedFactors_zero, ← Multiset.empty_eq_zero] at hL
    exact Finset.not_mem_empty _ hL
  refine mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors hI hJ hL
    (d := (idealFactorsEquivOfQuotEquiv f).toEquiv) ?_
  rintro ⟨l, hl⟩ ⟨l', hl'⟩
  rw [Subtype.coe_mk, Subtype.coe_mk]
  

Proof state:
case mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing A
inst✝⁹ : Field K
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
I : Ideal R
J : Ideal A
B : Type u_4
inst✝⁶ : CommRing B
inst✝⁵ : IsDomain B
inst✝⁴ : IsDedekindDomain B
L✝ : Ideal B
inst✝³ : IsDomain R
inst✝² : IsDedekindDomain R
f : R ⧸ I ≃+* A ⧸ J
inst✝¹ : DecidableEq (Ideal R)
inst✝ : DecidableEq (Ideal A)
hJ : J ≠ ⊥
L : Ideal R
hL : L ∈ normalizedFactors I
hI : I ≠ ⊥
l : Ideal R
hl : l ∣ I
l' : Ideal R
hl' : l' ∣ I
⊢ ↑((idealFactorsEquivOfQuotEquiv f).toEquiv { val := l, property := hl }) ∣
      ↑((idealFactorsEquivOfQuotEquiv f).toEquiv { val := l', property := hl' }) ↔
    ↑{ val := l, property := hl } ∣ ↑{ val := l', property := hl' }

Next tactic:
apply idealFactorsEquivOfQuotEquiv_is_dvd_iso f
/-- The bijection between the sets of normalized factors of I and J induced by a ring
    isomorphism `f : R/I ≅ A/J`. -/
-- @[simps apply] -- Porting note: simpNF complains about the lemmas generated by simps
def normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    { L : Ideal R | L ∈ normalizedFactors I } ≃ { M : Ideal A | M ∈ normalizedFactors J } where
  toFun j :=
    ⟨idealFactorsEquivOfQuotEquiv f ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩,
      idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f hJ j.prop⟩
  invFun j :=
    ⟨(idealFactorsEquivOfQuotEquiv f).symm ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩, by
      

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing A
inst✝⁹ : Field K
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
I : Ideal R
J : Ideal A
B : Type u_4
inst✝⁶ : CommRing B
inst✝⁵ : IsDomain B
inst✝⁴ : IsDedekindDomain B
L : Ideal B
inst✝³ : IsDomain R
inst✝² : IsDedekindDomain R
f : R ⧸ I ≃+* A ⧸ J
inst✝¹ : DecidableEq (Ideal R)
inst✝ : DecidableEq (Ideal A)
hI : I ≠ ⊥
hJ : J ≠ ⊥
j : ↑{M | M ∈ normalizedFactors J}
⊢ ↑((OrderIso.symm (idealFactorsEquivOfQuotEquiv f)) { val := ↑j, property := (_ : ↑j ∣ J) }) ∈
    {L | L ∈ normalizedFactors I}

Next tactic:
rw [idealFactorsEquivOfQuotEquiv_symm]
/-- The bijection between the sets of normalized factors of I and J induced by a ring
    isomorphism `f : R/I ≅ A/J`. -/
-- @[simps apply] -- Porting note: simpNF complains about the lemmas generated by simps
def normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    { L : Ideal R | L ∈ normalizedFactors I } ≃ { M : Ideal A | M ∈ normalizedFactors J } where
  toFun j :=
    ⟨idealFactorsEquivOfQuotEquiv f ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩,
      idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f hJ j.prop⟩
  invFun j :=
    ⟨(idealFactorsEquivOfQuotEquiv f).symm ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩, by
      rw [idealFactorsEquivOfQuotEquiv_symm]
      

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing A
inst✝⁹ : Field K
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
I : Ideal R
J : Ideal A
B : Type u_4
inst✝⁶ : CommRing B
inst✝⁵ : IsDomain B
inst✝⁴ : IsDedekindDomain B
L : Ideal B
inst✝³ : IsDomain R
inst✝² : IsDedekindDomain R
f : R ⧸ I ≃+* A ⧸ J
inst✝¹ : DecidableEq (Ideal R)
inst✝ : DecidableEq (Ideal A)
hI : I ≠ ⊥
hJ : J ≠ ⊥
j : ↑{M | M ∈ normalizedFactors J}
⊢ ↑((idealFactorsEquivOfQuotEquiv (RingEquiv.symm f)) { val := ↑j, property := (_ : ↑j ∣ J) }) ∈
    {L | L ∈ normalizedFactors I}

Next tactic:
exact
        idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f.symm hI
          j.prop
/-- The bijection between the sets of normalized factors of I and J induced by a ring
    isomorphism `f : R/I ≅ A/J`. -/
-- @[simps apply] -- Porting note: simpNF complains about the lemmas generated by simps
def normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    { L : Ideal R | L ∈ normalizedFactors I } ≃ { M : Ideal A | M ∈ normalizedFactors J } where
  toFun j :=
    ⟨idealFactorsEquivOfQuotEquiv f ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩,
      idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f hJ j.prop⟩
  invFun j :=
    ⟨(idealFactorsEquivOfQuotEquiv f).symm ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩, by
      rw [idealFactorsEquivOfQuotEquiv_symm]
      exact
        idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f.symm hI
          j.prop⟩
  left_inv := fun ⟨j, hj⟩ => by 

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing A
inst✝⁹ : Field K
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
I : Ideal R
J : Ideal A
B : Type u_4
inst✝⁶ : CommRing B
inst✝⁵ : IsDomain B
inst✝⁴ : IsDedekindDomain B
L : Ideal B
inst✝³ : IsDomain R
inst✝² : IsDedekindDomain R
f : R ⧸ I ≃+* A ⧸ J
inst✝¹ : DecidableEq (Ideal R)
inst✝ : DecidableEq (Ideal A)
hI : I ≠ ⊥
hJ : J ≠ ⊥
x✝ : ↑{L | L ∈ normalizedFactors I}
j : Ideal R
hj : j ∈ {L | L ∈ normalizedFactors I}
⊢ (fun j =>
        { val := ↑((OrderIso.symm (idealFactorsEquivOfQuotEquiv f)) { val := ↑j, property := (_ : ↑j ∣ J) }),
          property :=
            (_ :
              ↑((OrderIso.symm (idealFactorsEquivOfQuotEquiv f)) { val := ↑j, property := (_ : ↑j ∣ J) }) ∈
                {L | L ∈ normalizedFactors I}) })
      ((fun j =>
          { val := ↑((idealFactorsEquivOfQuotEquiv f) { val := ↑j, property := (_ : ↑j ∣ I) }),
            property :=
              (_ : ↑((idealFactorsEquivOfQuotEquiv f) { val := ↑j, property := (_ : ↑j ∣ I) }) ∈ normalizedFactors J) })
        { val := j, property := hj }) =
    { val := j, property := hj }

Next tactic:
simp
/-- The bijection between the sets of normalized factors of I and J induced by a ring
    isomorphism `f : R/I ≅ A/J`. -/
-- @[simps apply] -- Porting note: simpNF complains about the lemmas generated by simps
def normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    { L : Ideal R | L ∈ normalizedFactors I } ≃ { M : Ideal A | M ∈ normalizedFactors J } where
  toFun j :=
    ⟨idealFactorsEquivOfQuotEquiv f ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩,
      idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f hJ j.prop⟩
  invFun j :=
    ⟨(idealFactorsEquivOfQuotEquiv f).symm ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩, by
      rw [idealFactorsEquivOfQuotEquiv_symm]
      exact
        idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f.symm hI
          j.prop⟩
  left_inv := fun ⟨j, hj⟩ => by simp
  right_inv := fun ⟨j, hj⟩ => by
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing A
inst✝⁹ : Field K
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
I : Ideal R
J : Ideal A
B : Type u_4
inst✝⁶ : CommRing B
inst✝⁵ : IsDomain B
inst✝⁴ : IsDedekindDomain B
L : Ideal B
inst✝³ : IsDomain R
inst✝² : IsDedekindDomain R
f : R ⧸ I ≃+* A ⧸ J
inst✝¹ : DecidableEq (Ideal R)
inst✝ : DecidableEq (Ideal A)
hI : I ≠ ⊥
hJ : J ≠ ⊥
x✝ : ↑{M | M ∈ normalizedFactors J}
j : Ideal A
hj : j ∈ {M | M ∈ normalizedFactors J}
⊢ (fun j =>
        { val := ↑((idealFactorsEquivOfQuotEquiv f) { val := ↑j, property := (_ : ↑j ∣ I) }),
          property :=
            (_ : ↑((idealFactorsEquivOfQuotEquiv f) { val := ↑j, property := (_ : ↑j ∣ I) }) ∈ normalizedFactors J) })
      ((fun j =>
          { val := ↑((OrderIso.symm (idealFactorsEquivOfQuotEquiv f)) { val := ↑j, property := (_ : ↑j ∣ J) }),
            property :=
              (_ :
                ↑((OrderIso.symm (idealFactorsEquivOfQuotEquiv f)) { val := ↑j, property := (_ : ↑j ∣ J) }) ∈
                  {L | L ∈ normalizedFactors I}) })
        { val := j, property := hj }) =
    { val := j, property := hj }

Next tactic:
simp
/-- The bijection between the sets of normalized factors of I and J induced by a ring
    isomorphism `f : R/I ≅ A/J`. -/
-- @[simps apply] -- Porting note: simpNF complains about the lemmas generated by simps
def normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    { L : Ideal R | L ∈ normalizedFactors I } ≃ { M : Ideal A | M ∈ normalizedFactors J } where
  toFun j :=
    ⟨idealFactorsEquivOfQuotEquiv f ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩,
      idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f hJ j.prop⟩
  invFun j :=
    ⟨(idealFactorsEquivOfQuotEquiv f).symm ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩, by
      rw [idealFactorsEquivOfQuotEquiv_symm]
      exact
        idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f.symm hI
          j.prop⟩
  left_inv := fun ⟨j, hj⟩ => by simp
  right_inv := fun ⟨j, hj⟩ => by
    simp
    -- This used to be the end of the proof before leanprover/lean4#2644
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing A
inst✝⁹ : Field K
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
I : Ideal R
J : Ideal A
B : Type u_4
inst✝⁶ : CommRing B
inst✝⁵ : IsDomain B
inst✝⁴ : IsDedekindDomain B
L : Ideal B
inst✝³ : IsDomain R
inst✝² : IsDedekindDomain R
f : R ⧸ I ≃+* A ⧸ J
inst✝¹ : DecidableEq (Ideal R)
inst✝ : DecidableEq (Ideal A)
hI : I ≠ ⊥
hJ : J ≠ ⊥
x✝ : ↑{M | M ∈ normalizedFactors J}
j : Ideal A
hj : j ∈ {M | M ∈ normalizedFactors J}
⊢ ↑((idealFactorsEquivOfQuotEquiv f)
        ((OrderIso.symm (idealFactorsEquivOfQuotEquiv f))
          { val := j, property := (_ : ↑{ val := j, property := hj } ∣ J) })) =
    j

Next tactic:
erw [OrderIso.apply_symm_apply]
/-- The map `normalizedFactorsEquivOfQuotEquiv` preserves multiplicities. -/
theorem normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity (hI : I ≠ ⊥) (hJ : J ≠ ⊥)
    (L : Ideal R) (hL : L ∈ normalizedFactors I) :
    multiplicity (↑(normalizedFactorsEquivOfQuotEquiv f hI hJ ⟨L, hL⟩)) J = multiplicity L I := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹³ : CommRing R
inst✝¹² : CommRing A
inst✝¹¹ : Field K
inst✝¹⁰ : IsDomain A
inst✝⁹ : IsDedekindDomain A
I : Ideal R
J : Ideal A
B : Type u_4
inst✝⁸ : CommRing B
inst✝⁷ : IsDomain B
inst✝⁶ : IsDedekindDomain B
L✝ : Ideal B
inst✝⁵ : IsDomain R
inst✝⁴ : IsDedekindDomain R
f : R ⧸ I ≃+* A ⧸ J
inst✝³ : DecidableEq (Ideal R)
inst✝² : DecidableEq (Ideal A)
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hI : I ≠ ⊥
hJ : J ≠ ⊥
L : Ideal R
hL : L ∈ normalizedFactors I
⊢ multiplicity (↑((normalizedFactorsEquivOfQuotEquiv f hI hJ) { val := L, property := hL })) J = multiplicity L I

Next tactic:
rw [normalizedFactorsEquivOfQuotEquiv, Equiv.coe_fn_mk, Subtype.coe_mk]
/-- The map `normalizedFactorsEquivOfQuotEquiv` preserves multiplicities. -/
theorem normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity (hI : I ≠ ⊥) (hJ : J ≠ ⊥)
    (L : Ideal R) (hL : L ∈ normalizedFactors I) :
    multiplicity (↑(normalizedFactorsEquivOfQuotEquiv f hI hJ ⟨L, hL⟩)) J = multiplicity L I := by
  rw [normalizedFactorsEquivOfQuotEquiv, Equiv.coe_fn_mk, Subtype.coe_mk]
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹³ : CommRing R
inst✝¹² : CommRing A
inst✝¹¹ : Field K
inst✝¹⁰ : IsDomain A
inst✝⁹ : IsDedekindDomain A
I : Ideal R
J : Ideal A
B : Type u_4
inst✝⁸ : CommRing B
inst✝⁷ : IsDomain B
inst✝⁶ : IsDedekindDomain B
L✝ : Ideal B
inst✝⁵ : IsDomain R
inst✝⁴ : IsDedekindDomain R
f : R ⧸ I ≃+* A ⧸ J
inst✝³ : DecidableEq (Ideal R)
inst✝² : DecidableEq (Ideal A)
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hI : I ≠ ⊥
hJ : J ≠ ⊥
L : Ideal R
hL : L ∈ normalizedFactors I
⊢ multiplicity
      (↑((idealFactorsEquivOfQuotEquiv f)
          { val := ↑{ val := L, property := hL }, property := (_ : ↑{ val := L, property := hL } ∣ I) }))
      J =
    multiplicity L I

Next tactic:
refine multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalizedFactors hI hJ hL
    (d := (idealFactorsEquivOfQuotEquiv f).toEquiv) ?_
/-- The map `normalizedFactorsEquivOfQuotEquiv` preserves multiplicities. -/
theorem normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity (hI : I ≠ ⊥) (hJ : J ≠ ⊥)
    (L : Ideal R) (hL : L ∈ normalizedFactors I) :
    multiplicity (↑(normalizedFactorsEquivOfQuotEquiv f hI hJ ⟨L, hL⟩)) J = multiplicity L I := by
  rw [normalizedFactorsEquivOfQuotEquiv, Equiv.coe_fn_mk, Subtype.coe_mk]
  refine multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalizedFactors hI hJ hL
    (d := (idealFactorsEquivOfQuotEquiv f).toEquiv) ?_
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹³ : CommRing R
inst✝¹² : CommRing A
inst✝¹¹ : Field K
inst✝¹⁰ : IsDomain A
inst✝⁹ : IsDedekindDomain A
I : Ideal R
J : Ideal A
B : Type u_4
inst✝⁸ : CommRing B
inst✝⁷ : IsDomain B
inst✝⁶ : IsDedekindDomain B
L✝ : Ideal B
inst✝⁵ : IsDomain R
inst✝⁴ : IsDedekindDomain R
f : R ⧸ I ≃+* A ⧸ J
inst✝³ : DecidableEq (Ideal R)
inst✝² : DecidableEq (Ideal A)
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hI : I ≠ ⊥
hJ : J ≠ ⊥
L : Ideal R
hL : L ∈ normalizedFactors I
⊢ ∀ (l l' : { l // l ∣ I }),
    ↑((idealFactorsEquivOfQuotEquiv f).toEquiv l) ∣ ↑((idealFactorsEquivOfQuotEquiv f).toEquiv l') ↔ ↑l ∣ ↑l'

Next tactic:
exact fun ⟨l, hl⟩ ⟨l', hl'⟩ => idealFactorsEquivOfQuotEquiv_is_dvd_iso f hl hl'
theorem Ideal.coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬IsPrime P) :
    IsCoprime I J := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDomain A
I J : Ideal R
h : ∀ (P : Ideal R), I ≤ P → J ≤ P → ¬IsPrime P
⊢ IsCoprime I J

Next tactic:
rw [isCoprime_iff_sup_eq]
theorem Ideal.coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬IsPrime P) :
    IsCoprime I J := by
  rw [isCoprime_iff_sup_eq]
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDomain A
I J : Ideal R
h : ∀ (P : Ideal R), I ≤ P → J ≤ P → ¬IsPrime P
⊢ I ⊔ J = ⊤

Next tactic:
by_contra hIJ
theorem Ideal.coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬IsPrime P) :
    IsCoprime I J := by
  rw [isCoprime_iff_sup_eq]
  by_contra hIJ
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDomain A
I J : Ideal R
h : ∀ (P : Ideal R), I ≤ P → J ≤ P → ¬IsPrime P
hIJ : ¬I ⊔ J = ⊤
⊢ False

Next tactic:
obtain ⟨P, hP, hIJ⟩ := Ideal.exists_le_maximal _ hIJ
theorem Ideal.coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬IsPrime P) :
    IsCoprime I J := by
  rw [isCoprime_iff_sup_eq]
  by_contra hIJ
  obtain ⟨P, hP, hIJ⟩ := Ideal.exists_le_maximal _ hIJ
  

Proof state:
case intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDomain A
I J : Ideal R
h : ∀ (P : Ideal R), I ≤ P → J ≤ P → ¬IsPrime P
hIJ✝ : ¬I ⊔ J = ⊤
P : Ideal R
hP : IsMaximal P
hIJ : I ⊔ J ≤ P
⊢ False

Next tactic:
exact h P (le_trans le_sup_left hIJ) (le_trans le_sup_right hIJ) hP.isPrime
theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
    (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : IsPrime I
a b : R
n : ℕ
h : a * b ∈ I ^ n
⊢ a ∈ I ∨ b ∈ I ^ n

Next tactic:
cases n
theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
    (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by
  cases n; · 

Proof state:
case zero
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : IsPrime I
a b : R
h : a * b ∈ I ^ Nat.zero
⊢ a ∈ I ∨ b ∈ I ^ Nat.zero

Next tactic:
simp
theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
    (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by
  cases n; · simp
  

Proof state:
case succ
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : IsPrime I
a b : R
n✝ : ℕ
h : a * b ∈ I ^ Nat.succ n✝
⊢ a ∈ I ∨ b ∈ I ^ Nat.succ n✝

Next tactic:
by_cases hI0 : I = ⊥
theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
    (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by
  cases n; · simp
  by_cases hI0 : I = ⊥; · 

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : IsPrime I
a b : R
n✝ : ℕ
h : a * b ∈ I ^ Nat.succ n✝
hI0 : I = ⊥
⊢ a ∈ I ∨ b ∈ I ^ Nat.succ n✝

Next tactic:
simpa [pow_succ, hI0] using h
theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
    (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by
  cases n; · simp
  by_cases hI0 : I = ⊥; · simpa [pow_succ, hI0] using h
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : IsPrime I
a b : R
n✝ : ℕ
h : a * b ∈ I ^ Nat.succ n✝
hI0 : ¬I = ⊥
⊢ a ∈ I ∨ b ∈ I ^ Nat.succ n✝

Next tactic:
simp only [← Submodule.span_singleton_le_iff_mem, Ideal.submodule_span_eq, ← Ideal.dvd_iff_le, ←
    Ideal.span_singleton_mul_span_singleton] at h ⊢
theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
    (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by
  cases n; · simp
  by_cases hI0 : I = ⊥; · simpa [pow_succ, hI0] using h
  simp only [← Submodule.span_singleton_le_iff_mem, Ideal.submodule_span_eq, ← Ideal.dvd_iff_le, ←
    Ideal.span_singleton_mul_span_singleton] at h ⊢
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : IsPrime I
a b : R
n✝ : ℕ
hI0 : ¬I = ⊥
h : I ^ Nat.succ n✝ ∣ span {a} * span {b}
⊢ I ∣ span {a} ∨ I ^ Nat.succ n✝ ∣ span {b}

Next tactic:
by_cases ha : I ∣ span {a}
theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
    (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by
  cases n; · simp
  by_cases hI0 : I = ⊥; · simpa [pow_succ, hI0] using h
  simp only [← Submodule.span_singleton_le_iff_mem, Ideal.submodule_span_eq, ← Ideal.dvd_iff_le, ←
    Ideal.span_singleton_mul_span_singleton] at h ⊢
  by_cases ha : I ∣ span {a}
  · 

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : IsPrime I
a b : R
n✝ : ℕ
hI0 : ¬I = ⊥
h : I ^ Nat.succ n✝ ∣ span {a} * span {b}
ha : I ∣ span {a}
⊢ I ∣ span {a} ∨ I ^ Nat.succ n✝ ∣ span {b}

Next tactic:
exact Or.inl ha
theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
    (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by
  cases n; · simp
  by_cases hI0 : I = ⊥; · simpa [pow_succ, hI0] using h
  simp only [← Submodule.span_singleton_le_iff_mem, Ideal.submodule_span_eq, ← Ideal.dvd_iff_le, ←
    Ideal.span_singleton_mul_span_singleton] at h ⊢
  by_cases ha : I ∣ span {a}
  · exact Or.inl ha
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : IsPrime I
a b : R
n✝ : ℕ
hI0 : ¬I = ⊥
h : I ^ Nat.succ n✝ ∣ span {a} * span {b}
ha : ¬I ∣ span {a}
⊢ I ∣ span {a} ∨ I ^ Nat.succ n✝ ∣ span {b}

Next tactic:
rw [mul_comm] at h
theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
    (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by
  cases n; · simp
  by_cases hI0 : I = ⊥; · simpa [pow_succ, hI0] using h
  simp only [← Submodule.span_singleton_le_iff_mem, Ideal.submodule_span_eq, ← Ideal.dvd_iff_le, ←
    Ideal.span_singleton_mul_span_singleton] at h ⊢
  by_cases ha : I ∣ span {a}
  · exact Or.inl ha
  rw [mul_comm] at h
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : IsPrime I
a b : R
n✝ : ℕ
hI0 : ¬I = ⊥
h : I ^ Nat.succ n✝ ∣ span {b} * span {a}
ha : ¬I ∣ span {a}
⊢ I ∣ span {a} ∨ I ^ Nat.succ n✝ ∣ span {b}

Next tactic:
exact Or.inr (Prime.pow_dvd_of_dvd_mul_right ((Ideal.prime_iff_isPrime hI0).mpr hI) _ ha h)
theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
    (coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K := by
  

Proof state:
R : Type u_1
A : Type u_2
K✝ : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K✝
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J K : Ideal R
coprime : ∀ (P : Ideal R), J ≤ P → K ≤ P → ¬IsPrime P
hJ : I ≤ J
hK : I ≤ K
⊢ I ≤ J * K

Next tactic:
simp only [← Ideal.dvd_iff_le] at coprime hJ hK ⊢
theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
    (coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K := by
  simp only [← Ideal.dvd_iff_le] at coprime hJ hK ⊢
  

Proof state:
R : Type u_1
A : Type u_2
K✝ : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K✝
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬IsPrime P
hJ : J ∣ I
hK : K ∣ I
⊢ J * K ∣ I

Next tactic:
by_cases hJ0 : J = 0
theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
    (coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K := by
  simp only [← Ideal.dvd_iff_le] at coprime hJ hK ⊢
  by_cases hJ0 : J = 0
  · 

Proof state:
case pos
R : Type u_1
A : Type u_2
K✝ : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K✝
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬IsPrime P
hJ : J ∣ I
hK : K ∣ I
hJ0 : J = 0
⊢ J * K ∣ I

Next tactic:
simpa only [hJ0, zero_mul] using hJ
theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
    (coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K := by
  simp only [← Ideal.dvd_iff_le] at coprime hJ hK ⊢
  by_cases hJ0 : J = 0
  · simpa only [hJ0, zero_mul] using hJ
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K✝ : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K✝
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬IsPrime P
hJ : J ∣ I
hK : K ∣ I
hJ0 : ¬J = 0
⊢ J * K ∣ I

Next tactic:
obtain ⟨I', rfl⟩ := hK
theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
    (coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K := by
  simp only [← Ideal.dvd_iff_le] at coprime hJ hK ⊢
  by_cases hJ0 : J = 0
  · simpa only [hJ0, zero_mul] using hJ
  obtain ⟨I', rfl⟩ := hK
  

Proof state:
case neg.intro
R : Type u_1
A : Type u_2
K✝ : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K✝
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬IsPrime P
hJ0 : ¬J = 0
I' : Ideal R
hJ : J ∣ K * I'
⊢ J * K ∣ K * I'

Next tactic:
rw [mul_comm]