maximalIdeal_eq_span_two_three — Mathlib

English

The maximal ideal equals the ideal generated by 2 and 3: maximalIdeal ℕ = span {2,3}.

Русский

Максимальный идеал ℕ равен идеалу, порождаемому числами 2 и 3: maximalIdeal ℕ = span {2,3}.

LaTeX

$$$\maximalIdeal \mathbb{N} = \mathrm{span}\{2,3\}$$$

Lean4

theorem maximalIdeal_eq_span_two_three : maximalIdeal ℕ = span { 2, 3 } :=
  by
  refine le_antisymm (fun n h ↦ ?_) (span_le.mpr <| Set.pair_subset (by simp) (by simp))
  obtain lt | lt := (mem_maximalIdeal_iff.mp h).lt_or_gt
  · obtain rfl := lt_one_iff.mp lt; exact zero_mem _
  exact mem_span_pair.mpr <| exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le 2 3 n (by simp) (show 2 ≤ n by cutsat)

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