English
If hV, hf, hf' and hy0 such that f x0 = y0, then mpullbackWithin_vectorField_inter_of_eq holds for the pulled back field.
Русский
Если hV, hf, hf' и hy0 такие, что f(x0) = y0, то mpullbackWithin_vectorField_inter_of_eq выполняется для pulled back поля.
LaTeX
$$$\text{If } f(x_0)=y_0, \text{ then } mpullbackWithin\_vectorField\_inter\_of\_eq\' \\text{gives differentiability on the pullback domain.}$$$
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_mem_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 : f ⁻¹' t ∈ 𝓝[s] x₀) (hy₀ : 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 hy₀
exact ContMDiffWithinAt.mpullbackWithin_vectorField_of_mem hV hf hf' hx₀ hs hmn hst
