image_denseRange — Mathlib

English

If a set s has UniqueMDiffOn and f is differentiable on s with MF-derivative, then image has UniqueMDiffOn.

Русский

Если множество s имеет UniqueMDiffOn и f дифференцируема на s с MF-деривативом, тогда образ имеет UniqueMDiffOn.

LaTeX

$$Same content as 119521$$

Lean4

/-- If `s` has the unique differential property, `f` is differentiable on `s` and its derivative
at every point of `s` has dense range, then `f '' s` has the unique differential property. -/
theorem image_denseRange (hs : UniqueMDiffOn I s) {f : M → M'} (hf : MDifferentiableOn I I' f s)
    (hd : ∀ x ∈ s, DenseRange (mfderivWithin I I' f s x)) : UniqueMDiffOn I' (f '' s) :=
  hs.image_denseRange' (fun x hx ↦ (hf x hx).hasMFDerivWithinAt) hd

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