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not_maximal_of_lt_prime in dim_le_one_of_dimLEOne

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GTBarkley 2023-06-14 22:36:08 +00:00
parent bdcfc9d821
commit 08711d7689
1 changed files with 3 additions and 5 deletions
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CommAlg/krull.lean
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@ -176,8 +176,8 @@ lemma dim_eq_zero_iff [Nontrivial R] : krullDim R = 0 ↔ ∀ I : PrimeSpectrum
rw [←WithBot.coe_le_coe,←hn] at this rw [←WithBot.coe_le_coe,←hn] at this
change (0 : WithBot ℕ∞) ≤ _ at this change (0 : WithBot ℕ∞) ≤ _ at this
constructor <;> intro h' constructor <;> intro h'
rw [h'] . rw [h']
exact le_antisymm h' this . exact le_antisymm h' this
@[simp] @[simp]
lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by
@ -259,9 +259,7 @@ lemma dim_le_one_of_dimLEOne : Ring.DimensionLEOne R → krullDim R ≤ 1 := by
change q0.asIdeal < q1.asIdeal at hc1 change q0.asIdeal < q1.asIdeal at hc1
have q1nbot := Trans.trans (bot_le : ⊥ ≤ q0.asIdeal) hc1 have q1nbot := Trans.trans (bot_le : ⊥ ≤ q0.asIdeal) hc1
specialize H q1.asIdeal (bot_lt_iff_ne_bot.mp q1nbot) q1.IsPrime specialize H q1.asIdeal (bot_lt_iff_ne_bot.mp q1nbot) q1.IsPrime
apply IsPrime.ne_top p.IsPrime exact (not_maximal_of_lt_prime p.IsPrime hc2) H
apply (IsCoatom.lt_iff H.out).mp
exact hc2
lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1 := by lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1 := by
rw [dim_le_one_iff] rw [dim_le_one_iff]