equicontinuousWithinAt_iInf_rng — Mathlib

English

Equicontinuous F iff Equicontinuous ((↑) : range F → X → α).

Русский

Эконтинуированность F эквивалентна эконтинуированности через диапазон.

LaTeX

$$$\\operatorname{Equicontinuous} F \\iff \\operatorname{Equicontinuous} \\bigl((\\uparrow) : \\operatorname{range} F \\to X \\to \\alpha\\bigr).$$$

Lean4

theorem equicontinuousWithinAt_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} {S : Set X} {x₀ : X} :
    EquicontinuousWithinAt (uα := ⨅ k, u k) F S x₀ ↔ ∀ k, EquicontinuousWithinAt (uα := u k) F S x₀ :=
  by
  simp only [equicontinuousWithinAt_iff_continuousWithinAt (uα := _), topologicalSpace]
  unfold ContinuousWithinAt
  rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, nhds_iInf, tendsto_iInf]

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