exists_forall_notMem_of_forall_ne_top

English

If a finite family of submodules are all proper, there exists a vector outside all of them.

Русский

Если конечная семейство подмодулей всеProper, существует вектор вне каждого из них.

LaTeX

$$$$ \\exists x, \\forall i, x \\notin p(i). $$$$

Lean4

theorem exists_forall_notMem_of_forall_ne_top (p : ι → Submodule K M) (h : ∀ i, p i ≠ ⊤) : ∃ x, ∀ i, x ∉ p i :=
  by
  let _i : Fintype ι := Fintype.ofFinite ι
  suffices ⋃ i, (p i : Set M) ⊂ univ by simpa [ssubset_univ_iff, iUnion_eq_univ_iff] using this
  simpa using iUnion_ssubset_of_forall_ne_top_of_card_lt Finset.univ p h (by simp)

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