equicontinuousOn_iInf_rng — Mathlib

English

EquicontinuousOn F S is equivalent to the iInf of componentwise equicontinuity.

Русский

ЭконтинуированностьOn через iInf эквивалентна компонентной эконтинуированности.

LaTeX

$$$\\operatorname{EquicontinuousOn} F S \\iff \\operatorname{equicontinuousWithinAt\_iInf\_rng}\\text{(F,S)}.$$$

Lean4

theorem equicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} {S : Set X} :
    EquicontinuousOn (uα := ⨅ k, u k) F S ↔ ∀ k, EquicontinuousOn (uα := u k) F S := by
  simp_rw [EquicontinuousOn, equicontinuousWithinAt_iInf_rng, @forall_swap _ κ]

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