Zero Product of Numbers implies Factors are Zero

Theorem
On all the number systems:
 * natural numbers $\N$
 * integers $\Z$
 * rational numbers $\Q$
 * real numbers $\R$
 * complex numbers $\C$

the following holds.

Let $a \times b = 0$.

Then either $a = 0$ or $b = 0$.

Proof
From Natural Numbers have No Proper Zero Divisors
 * $\forall a, b \in \N: a \times b = 0 \implies a = 0 \text { or } b = 0$

We have:
 * Integers form Integral Domain
 * Rational Numbers form Integral Domain
 * Real Numbers form Integral Domain
 * Complex Numbers form Integral Domain

Hence by definition of integral domain:


 * $a \times b = 0 \implies a = 0 \text { or } b = 0$

where $a, b \in \Z, \Q, \R, \C$.