English
Let ψ m, v m, w m be C^n on M; then the map m ↦ ψ(m)(v(m), w(m)) is C^n on M.
Русский
Пусть ψ m, v(m), w(m) дифференцируемы до порядка C^n на M; тогда m ↦ ψ(m)(v(m), w(m)) — C^n на M.
LaTeX
$$$$\text{ContMDiff } IM (IB. prod 𝓘(\mathbb{k}, F_1 \to_L F_2 \to_L F_3)) n (m \mapsto \mathrm{TotalSpace.mk}' F_3 (b(m)) (\psi(m)(v(m), w(m)))) x.$$$$
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. -/
theorem clm_bundle_apply₂
(hψ :
MDifferentiable 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)))
(hv : MDifferentiable IM (IB.prod 𝓘(𝕜, F₁)) (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)))
(hw : MDifferentiable IM (IB.prod 𝓘(𝕜, F₂)) (fun m ↦ TotalSpace.mk' F₂ (b m) (w m))) :
MDifferentiable IM (IB.prod 𝓘(𝕜, F₃)) (fun m ↦ TotalSpace.mk' F₃ (b m) (ψ m (v m) (w m))) := fun x ↦
(hψ x).clm_bundle_apply₂ (hv x) (hw x)
