This question was asked in my assignment on Manifolds and I was not able to prove it.
Question: Prove that if $\phi: M \to N$ is $C^{\infty}$, 1-1 , onto and everywhere non-singular then $\phi$ is a diffeomorphism.
Attempt: Definition of diffeomorphism is as follows: If $\phi$ is a $C^{\infty}$ map from M to N then if $\phi$ is 1-1 , onto and $\phi^{-1}$ is $C^{\infty}$ then $\phi$ is called a diffeomorphism.
$\phi$ is everywhere non-singular implies that kernel of $d \phi$ is {0}. But I am not able think on how to relate non singularity with the $C^{\infty}$ condition and hence I am not able to make any significant progress.
Kindly give some hints!