English
Within a set, a similar decomposition holds: differentiability of a bundle morphism within a set s is equivalent to differentiability of its base part and its fiber-coordinate change within appropriate local coordinates.
Русский
Внутри множества сохраняется разложение: дифференцируемость морфизма расслоения внутри s равносильна дифференцируемости базовой части и координатного изменения по волокну в локальных координатах.
LaTeX
$$$\\text{ContMDiffWithinAt}_{IM} (f) \\iff \\text{ContMDiffWithinAt}_{IM} (f.\\text{proj}) \\land \\text{ContMDiffWithinAt}(inCoordinates(f))$$$
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 within a set at a point. -/
theorem clm_bundle_apply
(hϕ :
ContMDiffWithinAt 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 x)
(hv : ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₁)) n (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) s x) :
ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₂)) n (fun m ↦ TotalSpace.mk' F₂ (b m) (ϕ m (v m))) s x :=
by
simp only [contMDiffWithinAt_hom_bundle] at hϕ
exact hϕ.2.clm_apply_of_inCoordinates hv hϕ.1
