English
The differentiability of a bundle morphism at a point x0 decomposes into differentiability of its base projection and the induced fiber-coordinates part at x0.
Русский
Дифференцируемость морфизма расслоения в точке распадается на дифференцируемость базового отображения и волокнистой координатной части в точке.
LaTeX
$$$\\text{contMDiffAt}(f) \\iff \\text{contMDiffAt}(f.\\text{proj}) \\land \\text{contMDiffAt}(inCoordinates)\\!$$$
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 at a point. -/
theorem clm_bundle_apply
(hϕ :
ContMDiffAt 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)) x)
(hv : ContMDiffAt IM (IB.prod 𝓘(𝕜, F₁)) n (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) x) :
ContMDiffAt IM (IB.prod 𝓘(𝕜, F₂)) n (fun m ↦ TotalSpace.mk' F₂ (b m) (ϕ m (v m))) x :=
ContMDiffWithinAt.clm_bundle_apply hϕ hv
