comp_hasMFDerivWithinAt — Mathlib

English

If f1 coincides with f on s, then mfderivWithin I I' f1 s x = mfderivWithin I I' f s x for all x in s.

Русский

Если f1 совпадает с f на s, то mfderivWithin I I' f1 s x = mfderivWithin I I' f s x для всех x в s.

LaTeX

$$$\\forall x\\in s,\\; f_1(x)=f(x) \\Rightarrow mfderivWithin I I' f_1 s x = mfderivWithin I I' f s x$$$

Lean4

theorem comp_hasMFDerivWithinAt (hg : HasMFDerivAt I' I'' g (f x) g') (hf : HasMFDerivWithinAt I I' f s x f') :
    HasMFDerivWithinAt I I'' (g ∘ f) s x (g'.comp f') :=
  by
  rw [← hasMFDerivWithinAt_univ] at *
  exact HasMFDerivWithinAt.comp x (hg.mono (subset_univ _)) hf subset_preimage_univ

Похожие утверждения