English
On a set s, the contMDiff property of a fiberwise continuous linear map bundle morphism reduces to contMDiff of base and coordinate maps on s.
Русский
На множестве s свойство contMDiff для отображения векторного расслоения сводится к contMDiff по базе и по координатам на s.
LaTeX
$$$\\text{ContMDiffOn}$ for $(F_1,E_1)\\to (F_2,E_2)$ is equivalent to base and coordinate contMDiff on s.$$
Lean4
/-- Consider a `C^n` map `v : M → E₁` to a vector bundle, over a base map `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`.
We give here a version of this statement on a set. -/
theorem clm_bundle_apply
(hϕ :
ContMDiffOn IM (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂)) n
(fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂) (E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x)) (b m) (ϕ m)) s)
(hv : ContMDiffOn IM (IB.prod 𝓘(𝕜, F₁)) n (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) s) :
ContMDiffOn IM (IB.prod 𝓘(𝕜, F₂)) n (fun m ↦ TotalSpace.mk' F₂ (b m) (ϕ m (v m))) s := fun x hx ↦
(hϕ x hx).clm_bundle_apply (hv x hx)
