zero_smul_filter — Mathlib

English

For filters l1, l2 with l1 and l2 nontrivial, the product membership of the complement of the diagonal corresponds to disjointness: (diagonal α)^c ∈ l1 ×ˢ l2 iff l1 ⊥ l2.

Русский

Для не тривиальных фильтров l1, l2 принадлежность комплемента диагонали их произведению эквивалентна несовпадению: (диагональ α)^c ∈ l1 ×ˢ l2 ↔ Disjoint l1 l2.

LaTeX

$$$(\\ diagonal(\\alpha))^{c} \\in l_1 ×ˢ l_2 \\iff \\mathrm{Disjoint}(l_1,l_2)$$$

Lean4

theorem zero_smul_filter (hg : g.NeBot) : (0 : α) • g = 0 :=
  zero_smul_filter_nonpos.antisymm <|
    le_map_iff.2 fun s hs => by
      simp_rw [zero_smul, (hg.nonempty_of_mem hs).image_const]
      exact zero_mem_zero

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