fst' — Mathlib

English

If g is C^n at y and f is ContDiffWithinAt at x with f(x)=y, then g ∘ f is ContDiffWithinAt at x.

Русский

Если g — C^n в y и f — ContDiffWithinAt в x с f(x)=y, то g∘f — ContDiffWithinAt в x.

LaTeX

$$$\forall x,\; \ContDiffAt 𝕜 n g y \rightarrow \ContDiffWithinAt 𝕜 n f s x \rightarrow f x = y \rightarrow \ContDiffWithinAt 𝕜 n (g \circ f) s x$$$

Lean4

/-- Precomposing `f` with `Prod.fst` is `C^n` -/
theorem fst' {f : E → G} (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x : E × F => f x.1 :=
  hf.comp contDiff_fst

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