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Boundary smoothness is independent of charts and extensions
Statement
The local-extension definition of a smooth map between manifolds with boundary is independent of the chosen boundary charts and of all chosen extensions.
Facts & Assumptions
Given: A continuous map between smooth manifolds with boundary, two compatible source charts, two compatible target charts, and any local Euclidean extensions of the resulting coordinate representatives.
Boundary-chart transition maps are smooth in the local-extension sense (Smooth charts, atlases, and structures with boundary).
Smooth half-space maps are closed under composition and satisfy the chain rule (Chain rule for smooth half-space maps).
Agreeing smooth Euclidean extensions have identical derivatives on their common half-space domain (Half-space extensions agreeing on a relatively open set have the same derivatives there).
Proof
On every common domain, the two coordinate representatives differ by composition on the left and right with the source and target transition maps, which are smooth by [L1].
By [L2], one representative is extension-smooth exactly when the other is, because the transition maps are diffeomorphisms with smooth inverses. By [L3], all derivatives obtained from different extensions agree on the half-space. Thus neither the charts nor the extensions affect the definition.