mapsTo_tangent_cone — Mathlib

English

If f is an open partial homeomorphism with invertible derivative, then the inverse has derivative the inverse derivative at the corresponding point.

Русский

Если f — открытое частичное гомеоморфизм с обратимой производной, то обратная функция имеет производную, равную обратной производной.

LaTeX

$$$$ \\HasFDerivAt_{\\mathrm{symm}} $$$$

Lean4

/-- The image of a tangent cone under the differential of a map is included in the tangent cone to
the image. -/
theorem mapsTo_tangent_cone (h : HasFDerivWithinAt f f' s x) :
    MapsTo f' (tangentConeAt 𝕜 s x) (tangentConeAt 𝕜 (f '' s) (f x)) :=
  by
  rintro v ⟨c, d, dtop, clim, cdlim⟩
  refine ⟨c, fun n => f (x + d n) - f x, mem_of_superset dtop ?_, clim, h.lim atTop dtop clim cdlim⟩
  simp +contextual [-mem_image, mem_image_of_mem]

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