equicontinuous_iInf_rng — Mathlib

English

EquicontinuousOn F S is equivalent to ∀ k, EquicontinuousOn F S.

Русский

ЭконтинуированностьOn через iInf равносильна каждому компоненту.

LaTeX

$$$\\operatorname{EquicontinuousOn} F S \\iff \\forall k, \\operatorname{EquicontinuousOn} F S.$$$

Lean4

theorem equicontinuous_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} :
    Equicontinuous (uα := ⨅ k, u k) F ↔ ∀ k, Equicontinuous (uα := u k) F :=
  by
  simp_rw [equicontinuous_iff_continuous (uα := _), UniformFun.topologicalSpace]
  rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, continuous_iInf_rng]

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