Wilson's Theorem/Necessary Condition/Proof 3

Proof
Let $p$ be prime.

Consider $\struct {\Z_p, +, \times}$, the ring of integers modulo $m$.

From Ring of Integers Modulo Prime is Field, $\struct {\Z_p, +, \times}$ is a field.

Hence, apart from $\eqclass 0 p$, all elements of $\struct {\Z_p, +, \times}$ are units

As $\struct {\Z_p, +, \times}$ is a field, it is also by definition an integral domain, we can apply:

From Product of Units of Integral Domain with Finite Number of Units, the product of all elements of $\struct {\Z_p, +, \times}$ is $-1$.

But the product of all elements of $\struct {\Z_p, +, \times}$ is $\paren {p - 1}!$

The result follows.