English
If y0 = f(x0), the equality propagates through the pullback, yielding a differentiable pullback at x0 under the stated hypotheses.
Русский
Если y0 = f(x0), равенство сохраняется при пуллбаке, что даёт дифференцируемый пуллбэк в точке x0 под данными предпосылками.
LaTeX
$$$\text{If } y_0 = f(x_0), \ mpullback\_vectorField\_preimage \text{ is differentiable at } x_0.$$$
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_of_eq
(hV : ContMDiffWithinAt I' I'.tangent m (fun (y : M') ↦ (V y : TangentBundle I' M')) t y₀)
(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) (h : f x₀ = y₀) :
ContMDiffWithinAt I I.tangent m (fun (y : M) ↦ (mpullbackWithin I I' f V s y : TangentBundle I M)) s x₀ :=
by
subst h
exact ContMDiffWithinAt.mpullbackWithin_vectorField hV hf hf' hx₀ hs hmn hst
