Euclid's Lemma for Prime Divisors/General Result/Proof 3

Lemma
Let $p$ be a prime number.

Let $\displaystyle n = \prod_{i \mathop = 1}^r a_i$.

Then if $p$ divides $n$, it follows that $p$ divides $a_i$ for some $i$ such that $1 \le i \le r$.

That is:
 * $p \mathop \backslash a_1 a_2 \ldots a_n \implies p \mathop \backslash a_1 \lor p \mathop \backslash a_2 \lor \cdots \lor p \mathop \backslash a_n$

Proof
Suppose:
 * $\forall i \in \left\{{1, 2, \ldots, r}\right\}: p \nmid a_i$

By Prime not Divisor implies Coprime:
 * $\forall i \in \left\{{1, 2, \ldots, r}\right\}: p \perp a_i$

By Integer Coprime to all Factors is Coprime to Whole:
 * $p \perp n$

By definition of coprime:
 * $p \nmid n$