uniformEquicontinuous_iff_range — Mathlib

English

A family F is UniformEquicontinuous iff the range family is UniformEquicontinuous.

Русский

Единаковое эконтинуированное семейство эквивалентно диапазону.

LaTeX

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

Lean4

/-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff `range 𝓕` is uniformly equicontinuous,
i.e the family `(↑) : range F → β → α` is uniformly equicontinuous. -/
theorem uniformEquicontinuous_iff_range {F : ι → β → α} :
    UniformEquicontinuous F ↔ UniformEquicontinuous ((↑) : range F → β → α) :=
  ⟨fun h => by rw [← comp_rangeSplitting F]; exact h.comp _, fun h => h.comp (rangeFactorization F)⟩

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