P-adic Valuation of Difference of Powers with Coprime Exponent/Proof 2

Proof
We have $x^n-y^n=(x-y)(x^{n-1}+\cdots+y^{n-1})$.

We have to prove that $p\nmid x^{n-1}+\cdots+y^{n-1}$.

Let $P(u)=u^n-y^n$.

If $p\mid x^{n-1}+\cdots+y^{n-1}$, $x$ would be a double root of $P$ modulo $p$.

By Double Root of Polynomial is Root of Derivative (or a version of this in modular arithmetic), $p\mid P'(x)=nx^{n-1}$, which is impossible.

Therefore, $p\nmid x^{n-1}+\cdots+y^{n-1}$.