If a mapping is a diffeomorphism, is it necessarily regular? That is, is its tangent map 1-1?

Question

I am working on a proof regarding a surface in $R^3$ that is mapped to another surface in $R^3$ by a diffeomorphism. The hint for this problem seems to assume that I know that a diffeomorphism mapping is necessarily regular. This seems reasonable but I have not found it as a theorem in the places I have looked. I see that if the rank of the Jacobian is equal to the dimension of the space, then the mapping is regular. But does being a diffeomorphism necessarily imply that the rank of the Jacobian is n?

Answer 1

Yes. If $f\circ f^{-1} = Id$ and both $f$ and $f^{-1}$ are differentiable, then, by the chain rule, $$Df|_{f^{-1}(x)}D(f^{-1}(x)) = Id$$ and this implies that both $Df$ and $D(f^{-1})$ have at least the same rank as $Id$.