instHasSupportComap — Mathlib

English

If L has a well-defined support, then for any embedding f: γ ↪ β, the comapped filter L.comap f has a well-defined support.

Русский

Если у L есть определяемая опора, то для любого вложения f: γ ↪ β фильтр L.comap f также имеет определяемую опору.

LaTeX

$$$\\forall L\\,[\\text{HasSupport }L]\\;\\forall f:\\gamma\\hookrightarrow\\beta:\\;\\text{HasSupport}(L.comap f)$$$

Lean4

/-- If `L` has well-defined support, then so does its comap along an embedding. -/
instance (L : SummationFilter β) [HasSupport L] (f : γ ↪ β) : HasSupport (L.comap f) :=
  by
  constructor
  simp only [support_comap, comap_filter, eventually_map, Finset.coe_preimage]
  filter_upwards [L.eventually_le_support] with a using Set.preimage_mono

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