elementary number theory - Proof of Fermat's Little Theorem using Primitive Roots

I just learned about primitive roots today, and then I thought of this proof of Fermat's Little Theorem. Seeing that most proofs of this theorem aren't simple, I think I'm either completely wrong in my application of primitive roots (must have missed something fundamental, having just learned about them), or primitive roots are extremely powerful. Which one is it?

Proof: We want to prove that if $\gcd(a,p)=1$, with $p$ a prime, then $a^{p-1}\equiv 1\pmod p$. Since $p$ is prime there exists a primitive root $\operatorname{mod}\, p$, say $j$. It is well known that we can write every least residue $\operatorname{mod}\, p$ as a power of $j$, so e.g. we can write $a\equiv j^k$. Thus, it suffices to prove that $a^{p-1}\equiv j^{k(p-1)}\equiv 1\pmod p$. But this is obvious because $j^{k(p-1)}\equiv (j^{p-1})^k\equiv (1)^k\equiv 1\pmod p$ since $\operatorname{ord}_p(j)=p-1$ by definition.

QED

Thanks for your help guys!