English
Under standard hypotheses including invertible mfderiv and y0 = f(x0), the mpullbackWithin_vectorField_inter is ContMDiffWithinAt.
Русский
При стандартных предпосылках, включая обратимость mfderiv и равенство y0 = f(x0), mpullbackWithin_vectorField_inter является ContMDiffWithinAt.
LaTeX
$$$\text{Under invertible derivative and equality } y_0=f(x_0), \ mpullbackWithin\_vectorField\_inter \text{ is ContMDiffWithinAt.}$$$
Lean4
/-- The pullback of a `C^m` vector field by a `C^n` function with invertible derivative and
with `m + 1 ≤ n` is `C^m`.
Version within a set at a point. -/
protected theorem _root_.ContMDiffWithinAt.mpullbackWithin_vectorField
(hV : ContMDiffWithinAt I' I'.tangent m (fun (y : M') ↦ (V y : TangentBundle I' M')) t (f x₀))
(hf : ContMDiffWithinAt I I' n f s x₀) (hf' : (mfderivWithin I I' f s x₀).IsInvertible) (hx₀ : x₀ ∈ s)
(hs : UniqueMDiffOn I s) (hmn : m + 1 ≤ n) (hst : MapsTo f s t) :
ContMDiffWithinAt I I.tangent m (fun (y : M) ↦ (mpullbackWithin I I' f V s y : TangentBundle I M)) s x₀ :=
ContMDiffWithinAt.mpullbackWithin_vectorField_of_mem hV hf hf' hx₀ hs hmn hst.preimage_mem_nhdsWithin
