exists_disjoint_powers_of_span_eq_top

English

If I and S are disjoint, there exists a prime p containing I with p disjoint from S.

Русский

Если I и S не пересекаются, существует primeside p содержательный I и disjoint с S.

LaTeX

$$$\exists p:\ Ideal\ α, p.IsPrime \land I \le p \land Disjoint (p) S$$$

Lean4

theorem exists_disjoint_powers_of_span_eq_top (s : Set α) (hs : span s = ⊤) (I : Ideal α) (hI : I ≠ ⊤) :
    ∃ r ∈ s, Disjoint (I : Set α) (Submonoid.powers r) :=
  by
  have ⟨M, hM, le⟩ := exists_le_maximal I hI
  have := hM.1.1
  rw [Ne, eq_top_iff, ← hs, span_le, Set.not_subset] at this
  have ⟨a, has, haM⟩ := this
  exact ⟨a, has, Set.disjoint_left.mpr fun x hx ⟨n, hn⟩ ↦ haM (hM.isPrime.mem_of_pow_mem _ (le <| hn ▸ hx))⟩

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