fderivWithin_congr_set — Mathlib

English

If c is differentiable at x, then for any u, the map y ↦ (c(y))u is differentiable with derivative given by flipAlternating of the derivative of c.

Русский

Если c дифференцируема в x, то для любого u функция y ↦ (c(y))u дифференцируема, производная задаётся через flipAlternating производной c.

LaTeX

$$$\\text{If } c \\text{ is differentiable at } x, \\forall u,\\text{ differentiable } (y \\mapsto (c(y))u) \\text{ with derivative } (fderiv\\, c(x))\\,(u).$$$

Lean4

theorem fderivWithin_congr_set (h : s =ᶠ[𝓝 x] t) : fderivWithin 𝕜 f s x = fderivWithin 𝕜 f t x :=
  fderivWithin_congr_set' x <| h.filter_mono inf_le_left

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