English
On a set, if ψ m, v m, w m are C^n-differentiable along the set, then the composed map m ↦ ψ(m)(v(m), w(m)) is C^n-differentiable on that set.
Русский
На множестве если ψ m, v m, w m дифференцируемы на уровне C^n, то композиция ψ(m)(v(m), w(m)) дифференцируема на этом множестве в смысле C^n.
LaTeX
$$$$\text{ContMDiffOn } IM (IB.\prod 𝓘(\mathbb{k}, F_1 \to_L F_2 \to_L F_3)) n (m \mapsto \mathrm{TotalSpace.mk}' F_3 (b(m)) (\psi m (v m) (w m))) s.$$$$
Lean4
/-- Consider `C^n` 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 `C^n`.
We give here a version of this statement on a set. -/
theorem clm_bundle_apply₂
(hψ :
ContMDiffOn IM (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂ →L[𝕜] F₃)) n
(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)) s)
(hv : ContMDiffOn IM (IB.prod 𝓘(𝕜, F₁)) n (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) s)
(hw : ContMDiffOn IM (IB.prod 𝓘(𝕜, F₂)) n (fun m ↦ TotalSpace.mk' F₂ (b m) (w m)) s) :
ContMDiffOn IM (IB.prod 𝓘(𝕜, F₃)) n (fun m ↦ TotalSpace.mk' F₃ (b m) (ψ m (v m) (w m))) s := fun x hx ↦
(hψ x hx).clm_bundle_apply₂ (hv x hx) (hw x hx)
