English
Let ψ m be a bilinear map depending smoothly on m and v m, w m differentiable in a neighborhood; then the map m ↦ ψ(m)(v(m), w(m)) is C^n-differentiable at x.
Русский
Пусть билинейное отображение ψ m зависит гладко от m, а v(m), w(m) — дифференцируемы на окрестности x. Тогда m ↦ ψ(m)(v(m), w(m)) дифференцируемо до порядка C^n в x.
LaTeX
$$$$\text{ContMDiffWithinAt IM (IB. prod 𝓘(\mathbb{k}, F_1 \to_L F_2 \to_L F_3)) n (m \mapsto \mathrm{TotalSpace.mk}' (F_1 \to_L F_2 \to_L F_3) (b(m)) (\psi(m)))} s x \rightarrow \; \text{...}$$$$
Lean4
/-- Consider differentiable maps `v : M → E₁` and `v : M → E₂` to vector bundles, over a base map
`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 differentiable.
We give here a version of this statement at a point. -/
theorem clm_bundle_apply₂
(hψ :
MDifferentiableAt IM (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂ →L[𝕜] F₃))
(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)) x)
(hv : MDifferentiableAt IM (IB.prod 𝓘(𝕜, F₁)) (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) x)
(hw : MDifferentiableAt IM (IB.prod 𝓘(𝕜, F₂)) (fun m ↦ TotalSpace.mk' F₂ (b m) (w m)) x) :
MDifferentiableAt IM (IB.prod 𝓘(𝕜, F₃)) (fun m ↦ TotalSpace.mk' F₃ (b m) (ψ m (v m) (w m))) x :=
MDifferentiableWithinAt.clm_bundle_apply₂ hψ hv hw
