clm_precomp — Mathlib

English

Let f: M → F2 →L F3 be ContMDiffAt. Then the map x ↦ f(x).postcomp F1 is ContMDiffAt in the appropriate space.

Русский

Пусть f: M → F2 →L F3 гладко; тогда x ↦ f(x).postcomp F1 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

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

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