English
Continuity of a bundle map in coordinates follows from continuity in total space and base.
Русский
Непрерывность отображения в координатах следует из непрерывности в полной конфигурации и базисе.
LaTeX
$$$\\text{ContinuousWithinAt}(\\text{fun } m \\mapsto \\text{TotalSpace.mk'}(\\cdots)(b m)(\\psi m)) s x \\rightarrow \\cdots$$$
Lean4
/-- Consider a `C^n` map `v : M → E₁` to a vector bundle, over a basemap `b : M → B`, and
linear maps `ϕ m : E₁ (b m) → E₂ (b m)` depending smoothly on `m`.
One can apply `ϕ m` to `v m`, and the resulting map is `C^n`. -/
theorem clm_bundle_apply
(hϕ : Continuous (fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂) (E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x)) (b m) (ϕ m)))
(hv : Continuous (fun m ↦ TotalSpace.mk' F₁ (b m) (v m))) :
Continuous (fun m ↦ TotalSpace.mk' F₂ (b m) (ϕ m (v m))) :=
by
simp only [← continuousOn_univ] at hϕ hv ⊢
exact hϕ.clm_bundle_apply hv
