English
Composing on the right by a function preserves uniform Cauchy sequences: if F is a UniformCauchySeqOnFilter F p p' and g: γ → α, then UniformCauchySeqOnFilter (F n ∘ g) p (p'.comap g).
Русский
Сохранение равномерной坤 последовательности через композицию справа: если F — равномерно Cauchy по p и p', и g: γ → α, то F n ∘ g образуют равномерно Cauchy последовательность по p и p'∘g.
LaTeX
$$UniformCauchySeqOnFilter F p p' → ∀ g: γ → α, UniformCauchySeqOnFilter (λ n, F n ∘ g) p (p'.comap g)$$
Lean4
/-- Composing on the right by a function preserves uniform Cauchy sequences -/
theorem comp {γ : Type*} (hf : UniformCauchySeqOnFilter F p p') (g : γ → α) :
UniformCauchySeqOnFilter (fun n => F n ∘ g) p (p'.comap g) := fun u hu =>
by
obtain ⟨pa, hpa, pb, hpb, hpapb⟩ := eventually_prod_iff.mp (hf u hu)
rw [eventually_prod_iff]
refine ⟨pa, hpa, pb ∘ g, ?_, fun hx _ hy => hpapb hx hy⟩
exact eventually_comap.mpr (hpb.mono fun x hx y hy => by simp only [hx, hy, Function.comp_apply])
