atTop_Ioi_eq — Mathlib

English

Under a Galois-connection-like condition above b, a monotone f maps atTop to atTop.

Русский

При условии, аналогичном гало-связи над b, монотонная f отображает atTop в atTop.

LaTeX

$$$\forall f:\alpha\to\beta,\ (hf:\mathrm{Monotone}(f))\to\, \forall b:\beta,\ \mathrm{map}\ f\ atTop = atTop$$$

Lean4

/-- The `atTop` filter for an open interval `Ioi a` comes from the `atTop` filter in the ambient
order. -/
theorem atTop_Ioi_eq [Preorder α] [IsDirected α (· ≤ ·)] (a : α) : atTop = comap ((↑) : Ioi a → α) atTop :=
  by
  rcases isEmpty_or_nonempty (Ioi a) with h | ⟨⟨b, hb⟩⟩
  · subsingleton
  · rw [← map_val_atTop_of_Ici_subset (Ici_subset_Ioi.2 hb), comap_map Subtype.coe_injective]

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