English
If a section s is C^n in a neighborhood of x0 (ContMDiffWithinAt), then the section x ↦ -s(x) is also C^n in that neighborhood.
Русский
Если секция s гладкая до порядка n в окрестности точки x0, то секция x ↦ -s(x) тоже гладкая до порядка n в этой окрестности.
LaTeX
$$ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ TotalSpace.mk' F x (-s x)) u x0$$
Lean4
theorem neg_section (hs : ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ TotalSpace.mk' F x (s x)) u x₀) :
ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ TotalSpace.mk' F x (-s x)) u x₀ :=
by
rw [contMDiffWithinAt_section] at hs ⊢
set e := trivializationAt F V x₀
refine hs.neg.congr_of_eventuallyEq ?_ ?_
· apply eventually_of_mem (U := e.baseSet)
· exact mem_nhdsWithin_of_mem_nhds <| (e.open_baseSet.mem_nhds <| mem_baseSet_trivializationAt F V x₀)
· intro x hx
apply (e.linear 𝕜 hx).map_neg
· apply (e.linear 𝕜 (FiberBundle.mem_baseSet_trivializationAt' x₀)).map_neg
