English
The σ-twisted trivialization preserves source and target under ContinuousAt maps.
Русский
σ-огибающая тривиализация сохраняет источник и цель при отображениях непрерывности.
LaTeX
$$$\\text{If } f: M \\to (F_1 \\toSL[\\sigma] F_2) \\text{ is ContinuousAt at } x_0, \\ldots$$$
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 : ContinuousAt (fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂) (E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x)) (b m) (ϕ m)) x)
(hv : ContinuousAt (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) x) :
ContinuousAt (fun m ↦ TotalSpace.mk' F₂ (b m) (ϕ m (v m))) x :=
by
simp only [← continuousWithinAt_univ] at hϕ hv ⊢
exact hϕ.clm_bundle_apply hv
