instInfSet — Mathlib

English

The proof constructs InfSet for Subspace K V by taking the infimum over images of a subset under the inclusion map.

Русский

Доказательство строит InfSet для Subspace K V через наименьшее множество, содержащее преобразования включения.

LaTeX

$$$\forall S\subseteq\mathrm{Subspace}(K,V),\; \inf S \in \mathrm{InfSet}(\mathrm{Subspace}(K,V))$$$

Lean4

/-- Infimums of arbitrary collections of subspaces exist. -/
instance instInfSet : InfSet (Subspace K V) :=
  ⟨fun A =>
    ⟨sInf (SetLike.coe '' A), fun v w hv hw hvw h1 h2 t =>
      by
      rintro ⟨s, hs, rfl⟩
      exact s.mem_add v w hv hw _ (h1 s ⟨s, hs, rfl⟩) (h2 s ⟨s, hs, rfl⟩)⟩⟩

Похожие утверждения