Natural Numbers have No Proper Zero Divisors

Theorem
Let $\N$ be the natural numbers.

Then for all $m, n \in \N$:


 * $m \times n = 0 \iff m = 0 \lor n = 0$

That is, $\N$ has no proper zero divisors.

Necessary Condition
Suppose that $n = 0$ or $m = 0$.

Then from Zero is Zero Element for Natural Number Multiplication:


 * $m \times n = 0$

Sufficient Condition
Let $m \times n = 0$.

Suppose WLOG that $n \ne 0$.

But as:
 * $m \times \left({n - 1}\right) \circ m = m \times n = 0$

it follows that:
 * $0 \le m \le 0$

and so as $\le$ is antisymmetric, it follows that $m = 0$.