not_lt_lt — Mathlib

English

If p0 < p1 < p2 are primes in a ring of dimension ≤ 1, then p0 = 0.

Русский

Если p0 < p1 < p2 — простые идеалы, тогда p0 = ⊥.

LaTeX

$$CompleteLattice.instOmegaCompletePartialOrder.lt p0 p1 ∧ CompleteLattice.instOmegaCompletePartialOrder.lt p1 p2 → p0 = ⊥$$

Lean4

theorem not_lt_lt [Ring.DimensionLEOne R] (p₀ p₁ p₂ : Ideal R) [hp₁ : p₁.IsPrime] [hp₂ : p₂.IsPrime] :
    ¬(p₀ < p₁ ∧ p₁ < p₂)
  | ⟨h01, h12⟩ => h12.ne ((hp₁.isMaximal (bot_le.trans_lt h01).ne').eq_of_le hp₂.ne_top h12.le)

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