P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 4

Theorem
Let $p, q$ be prime numbers such that $p \neq q$.

Let $x_1 \in \Z_{\gt 0}: p \nmid x_1$

Let $a \in \Z$ be any integer.

Let $f \paren{X} \in \Z [X]$ be the polynomial $X^q - a$

Let $f' \paren{X} \in \Z [X]$ be the formal derivative of $f \paren{X}$.

Then:
 * $\map {f’} {x_1} \not \equiv 0 \pmod p$

Proof
By assumption then:
 * $p \nmid x_1$

By the definition of a prime number then:
 * $p \nmid q$

By Euclid's Lemma for Prime Divisors then:
 * $p \nmid qx_1^{q-1}$

Hence:
 * $qx_1^{q-1} \not \equiv 0 \mod p$

The formal derivative $f' \paren{X} \in \Z [X]$ of $f \paren{X}$ is by definition:
 * $qX^{q-1}$

Then:
 * $\map {f’} {x_1} = qx_1^{q-1} \not \equiv 0 \pmod p$