clm_comp — Mathlib

English

Precomposing f with Prod.snd is C^n at (x, y); that is, ContDiffAt 𝕜 n f y implies ContDiffAt 𝕜 n (λx, f x.2) (x, y).

Русский

Предпосылка: f: F → G C^n; затем x ↦ f(x.2) бесконечно гладкая в точке (x,y).

LaTeX

$$$\mathrm{ContDiffAt}_{\mathbb{k}}\ n\ f\ y \Rightarrow \mathrm{ContDiffAt}_{\mathbb{k}}\ n\ (\lambda x: (f x).2)\ (x, y).$$$

Lean4

@[fun_prop]
theorem clm_comp {g : X → F →L[𝕜] G} {f : X → E →L[𝕜] F} (hg : ContDiff 𝕜 n g) (hf : ContDiff 𝕜 n f) :
    ContDiff 𝕜 n fun x => (g x).comp (f x) :=
  isBoundedBilinearMap_comp.contDiff.comp₂ (g := fun p => p.1.comp p.2) hg hf

Похожие утверждения