Invertible Integers under Multiplication

Theorem
The only invertible elements of $\Z$ for multiplication (that is, units of $\Z$) are $1$ and $-1$.

Proof
Let $x > 0$ and $x y > 0$.

first that $y \le 0$.

Then from Multiplicative Ordering on Integers and Ring Product with Zero:
 * $x y \le x \, 0 = 0$

From this contradiction we deduce that $y > 0$.

Let $x > 0$ and $x y = 1$.

Then:
 * $y > 0$

and by Natural Numbers are Non-Negative Integers:
 * $y \in \N$

Hence by Invertible Elements under Natural Number Multiplication:
 * $x = 1$

Thus $1$ is the only element of $\N$ that is invertible for multiplication.

Therefore by Natural Numbers are Non-Negative Integers and Product with Ring Negative, the result follows.