one_mem_span_iff — Mathlib

English

For any set s of natural numbers, 1 is in the span of s if and only if 1 is in s.

Русский

Для любого множества s натуральных чисел верно: 1 ∈ span(s) тогда и только тогда, когда 1 ∈ s.

LaTeX

$$$1 \in \mathrm{span}(s) \iff 1 \in s$$$

Lean4

theorem one_mem_span_iff {s : Set ℕ} : 1 ∈ span s ↔ 1 ∈ s :=
  by
  rw [← SetLike.mem_coe, ← not_iff_not]
  simp_rw [← Set.mem_compl_iff, ← Set.singleton_subset_iff, Set.subset_compl_comm, ← coe_maximalIdeal,
    SetLike.coe_subset_coe, span_le]

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