English
A bilinear map applied to bundles yields a bundle map continuous within a specified set.
Русский
Согласованное билинейное отображение даёт непрерывное отображение в рамках множества.
LaTeX
$$$\\text{ContinuousWithinAt }\\text{TotalSpace.mk'}(F_1 \\toL F_2)(b m)(\\psi m) s x \\rightarrow \\text{ ... }$$$
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 : ContinuousOn (fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂) (E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x)) (b m) (ϕ m)) s)
(hv : ContinuousOn (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) s) :
ContinuousOn (fun m ↦ TotalSpace.mk' F₂ (b m) (ϕ m (v m))) s := fun x hx ↦ (hϕ x hx).clm_bundle_apply (hv x hx)
