Is this proof of RSA's correctness sufficient?

Question

In a lecture at my university, the following proof of correctness of RSA is given (the lecture is not mainly on cryptography or even computer science):

$m^{ed} \equiv m^{ee^{-1}} \equiv m^{1} \equiv m \ \textrm{mod} \ N$

The given reasoning is: $d$ is chosen as the multiplicative inverse of $e$ in the modulo ring $\mathbb{Z}_{\phi(N)}$, therefore this holds.

Surely, this cannot be a proof, considering that $e$ is only an inverse of $d$ in the modulo ring $\mathbb{Z}_{\phi(N)}$ and not in the integers or even $\mathbb{Z}_N$, right?

Am I mistaken or is this proof insufficient? The given proofs on wikipedia and other lectures are significantly longer and involve either Fermat's, Euler's or the Chinese Remainder Theorem.

Answer 1

You're right it is not sufficient, see this question and answer here for a considerably more complicated argument.

In practice $m$ for typical parameters with very large primes $m$ lies in the multiplicative group with overwhelming probability so the issue does not come up. But this is irrelevant to the correctness of the argument.