English
If a family F is uniformly equicontinuous with respect to each uniform structure u_k on β', then it is uniformly equicontinuous with respect to the infimum of these uniform structures ⨅k, u_k.
Русский
Если семейство F равноудельно равномерно носитEquicontinuous относительно каждой uniforme структуры u_k, то оно равноудельно равномерно относительно infimum этих структур ⨅k, u_k.
LaTeX
$$$\\text{If } hk: \\operatorname{UniformEquicontinuous}(F) \\,\\text{ w.r.t. } u_k \\text{ for all } k, \\text{ then } \\operatorname{UniformEquicontinuous}(F) \\text{ w.r.t. } \\inf_k u_k.$$$
Lean4
theorem uniformEquicontinuous_iInf_dom {u : κ → UniformSpace β'} {F : ι → β' → α} {k : κ}
(hk : UniformEquicontinuous (uβ := u k) F) : UniformEquicontinuous (uβ := ⨅ k, u k) F :=
by
simp_rw [uniformEquicontinuous_iff_uniformContinuous (uβ := _)] at hk ⊢
exact uniformContinuous_iInf_dom hk
