equicontinuousAt_iInf_dom — Mathlib

English

If hk is EquicontinuousAt with tX := t k, then EquicontinuousAt with the infimum t is obtained.

Русский

Если hk эконтинуированностьAt при tX = t_k, то при t = inf_k t_k получаем.

LaTeX

$$$\\operatorname{EquicontinuousAt} F x_0 \\Rightarrow \\operatorname{EquicontinuousAt} F x_0$ (under infimum of t_k).$$

Lean4

theorem equicontinuousAt_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α} {x₀ : X'} {k : κ}
    (hk : EquicontinuousAt (tX := t k) F x₀) : EquicontinuousAt (tX := ⨅ k, t k) F x₀ :=
  by
  rw [← equicontinuousWithinAt_univ (tX := _)] at hk ⊢
  exact equicontinuousWithinAt_iInf_dom hk

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