English
If hϕ and hv are continuous, then the resulting bundle map is continuous.
Русский
Если hϕ и hv непрерывны, то полученное отображение расслоения непрерывно.
LaTeX
$$$\\text{Continuous}(\\lambda m, TotalSpace.mk'(\\cdots)(b m)(\\psi m))$$$
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 :
Continuous
(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)))
(hv : Continuous (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)))
(hw : Continuous (fun m ↦ TotalSpace.mk' F₂ (b m) (w m))) :
Continuous (fun m ↦ TotalSpace.mk' F₃ (b m) (ψ m (v m) (w m))) :=
by
simp only [← continuousOn_univ] at hψ hv hw ⊢
exact hψ.clm_bundle_apply₂ hv hw
