English
Same as previous: the derivative within extChartAt_symm is invertible at y.
Русский
Точно так же, як и ранее, производная принадлежащая к extChartAt_symm обратима.
LaTeX
$$$(mfderivWithin 𝓘(𝕜,E) I (extChartAt I x).symm (range I) y).IsInvertible$$$
Lean4
theorem mdifferentiableWithinAt_extChartAt_symm (h : z ∈ (extChartAt I x).target) :
MDifferentiableWithinAt 𝓘(𝕜, E) I (extChartAt I x).symm (range I) z :=
by
have Z := I.mdifferentiableWithinAt_symm (extChartAt_target_subset_range x h)
apply MDifferentiableAt.comp_mdifferentiableWithinAt (I' := I) _ _ Z
apply mdifferentiableAt_atlas_symm (ChartedSpace.chart_mem_atlas x)
simp only [extChartAt, OpenPartialHomeomorph.extend, PartialEquiv.trans_target, ModelWithCorners.target_eq,
ModelWithCorners.toPartialEquiv_coe_symm, mem_inter_iff, mem_range, mem_preimage] at h
exact h.2
