Riemannian manifold as a metric space

Isn't this the Myers-Steenrod theorem? "If $(M,g)$ and $(N,h)$ are connected Riemannian manifolds and $f:(M,d_g)\to(N,d_h)$ is an isometry, then $f:(M,g)\to(N,h)$ is a smooth isometry"

It was proven by Dick Palais.

MR0088000 (19,451a) Reviewed Palais, Richard S. On the differentiability of isometries. Proc. Amer. Math. Soc. 8 (1957), 805–807. 53.2X

MathSciNet

@article {MR88000, AUTHOR = {Palais, Richard S.}, TITLE = {On the differentiability of isometries}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {8}, YEAR = {1957}, PAGES = {805--807}, ISSN = {0002-9939}, MRCLASS = {53.2X}, MRNUMBER = {88000}, MRREVIEWER = {K. Krickeberg}, DOI = {10.2307/2033302}, URL = {https://doi-org.ucc.idm.oclc.org/10.2307/2033302}, }

According to Palais, if I read his paper correctly, Myers and Steenrod proved the differentiability of isometries but Palais obtained an explicit description of smooth functions on the manifold from the metric geometry.