differentiableWithinAt_snd — Mathlib

English

If h is a Fréchet derivative with respect to a scalar field 𝕜′, then restricting scalars to 𝕜 gives a derivative with respect to 𝕜, i.e., HasFDerivAt f f' x implies HasFDerivAt f (f'.restrictScalars 𝕜) x.

Русский

Если есть производная по полю 𝕜′, то ограничение по скалярам до 𝕜 дает производную по 𝕜: HasFDerivAt f f' x ⇒ HasFDerivAt f (f'.restrictScalars 𝕜) x.

LaTeX

$$$\text{HasFDerivAt } f f' x \Rightarrow \text{HasFDerivAt } f (f'.restrictScalars 𝕜) x$$$

Lean4

@[fun_prop]
theorem differentiableWithinAt_snd {s : Set (E × F)} : DifferentiableWithinAt 𝕜 Prod.snd s p :=
  differentiableAt_snd.differentiableWithinAt

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