English
If hϕ and hv, hw are continuous-on maps, then their bundle composition is continuous-on.
Русский
Если hϕ и hv, hw далее продолжаются как непрерывно на области, то составная карта непрерывна.
LaTeX
$$$\\text{ContinuousOn}(\\lambda m, TotalSpace.mk'(...) (b m)(\\psi m)) s \\Rightarrow \\cdots$$$
Lean4
/-- Consider `C^n` maps `v : M → E₁` and `v : M → E₂` to vector bundles, over a basemap
`b : M → B`, and bilinear maps `ψ m : E₁ (b m) → E₂ (b m) → E₃ (b m)` depending smoothly on `m`.
One can apply `ψ m` to `v m` and `w m`, and the resulting map is `C^n`. -/
theorem clm_bundle_apply₂
(hψ :
ContinuousOn
(fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂ →L[𝕜] F₃) (E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x →L[𝕜] E₃ x)) (b m) (ψ m)) s)
(hv : ContinuousOn (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) s)
(hw : ContinuousOn (fun m ↦ TotalSpace.mk' F₂ (b m) (w m)) s) :
ContinuousOn (fun m ↦ TotalSpace.mk' F₃ (b m) (ψ m (v m) (w m))) s := fun x hx ↦
(hψ x hx).clm_bundle_apply₂ (hv x hx) (hw x hx)
