forall_exists_of_atTop — Mathlib

English

If for every x there exists y ∈ s with x ≤ y, then atTop equals the generate of Ici'' s.

Русский

Если для каждого x существует y ∈ s с x ≤ y, то atTop равен generate (Ici'' s).

LaTeX

$$$\\forall x,\\exists y\\in s:\\ x\\le y \\Rightarrow (atTop) = generate(\\mathrm{Ici}'' s).$$$

Lean4

theorem forall_exists_of_atTop [Preorder α] {p : α → Prop} (h : ∃ᶠ x in atTop, p x) (a : α) : ∃ b ≥ a, p b :=
  by
  rw [Filter.Frequently] at h
  contrapose! h
  exact (eventually_ge_atTop a).mono h

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