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proved dim_eq_zero_iff

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GTBarkley 2023-06-14 21:35:36 +00:00
parent 3c4cfeab65
commit 4690889149
1 changed files with 10 additions and 5 deletions
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CommAlg/krull.lean
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@ -169,11 +169,16 @@ lemma dim_le_zero_iff : krullDim R ≤ 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal
exact hc2
lemma dim_eq_zero_iff [Nontrivial R] : krullDim R = 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal I.asIdeal := by
constructor <;> intro h
. intro I
sorry
. sorry
rw [←dim_le_zero_iff]
obtain ⟨n, hn⟩ := krullDim_nonneg_of_nontrivial R
have : n ≥ 0 := zero_le n
change _ ≤ _ at this
rw [←WithBot.coe_le_coe,←hn] at this
change (0 : WithBot ℕ∞) ≤ _ at this
constructor <;> intro h'
rw [h']
exact le_antisymm h' this
@[simp]
lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by
constructor