equicontinuousAt_iInf_rng — Mathlib

English

EquicontinuousAt F at x0 is equivalent to the infimum range decomposition: EquicontinuousAt F x0 iff ∀ k, EquicontinuousAt F x0.

Русский

ЭконтинуированностьAt при x0 равна бесконечному объединению по iInf.

LaTeX

$$$\\operatorname{EquicontinuousAt} F x_0 \\iff \\forall k, \\operatorname{EquicontinuousAt} F x_0.$$$

Lean4

theorem equicontinuousAt_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} {x₀ : X} :
    EquicontinuousAt (uα := ⨅ k, u k) F x₀ ↔ ∀ k, EquicontinuousAt (uα := u k) F x₀ := by
  simp only [← equicontinuousWithinAt_univ (uα := _), equicontinuousWithinAt_iInf_rng]

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