clm_precomp — Mathlib

English

If f: M → F2 →L F3 is ContMDiffAt, then the map x ↦ f(x).postcomp F1 is ContMDiffAt in the appropriate space.

Русский

Если f: M → F2 →L F3 гладко в точке, то $x \\mapsto f(x).postcomp F_1$ гладко.

LaTeX

$$$\\text{If } f: M \\to F_2 \\to_L F_3 \\text{ is ContMDiffAt } I\\, 𝓘(𝕜, F_2 \\to_L F_3)\\, n f x,\\ \\text{then } x \\mapsto (f(x)).postcomp F_1\\text{ is ContMDiffAt } I\\, 𝓘(𝕜, (F_1 \\to_L F_2) \\to_L F_3)\\, n.$$$

Lean4

nonrec theorem clm_precomp {f : M → F₁ →L[𝕜] F₂} {x : M} (hf : ContMDiffAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n f x) :
    ContMDiffAt I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) n
      (fun y ↦ (f y).precomp F₃ : M → (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) x :=
  hf.clm_precomp

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