Integer Multiplication Distributes over Addition/Corollary

Corollary to Integer Multiplication Distributes over Addition

The operation of multiplication on the set of integers $\Z$ is distributive over subtraction:

$\forall x, y, z \in \Z: x \times \left({y - z}\right) = \left({x \times y}\right) - \left({x \times z}\right)$
$\forall x, y, z \in \Z: \left({y - z}\right) \times x = \left({y \times x}\right) - \left({z \times x}\right)$

Proof

 $\displaystyle x \times \left({y - z}\right)$ $=$ $\displaystyle x \times \left({y + \left({- z}\right)}\right)$ Definition of Integer Subtraction $\displaystyle$ $=$ $\displaystyle x \times y + x \times \left({- z}\right)$ $\displaystyle$ $=$ $\displaystyle x \times y + \left({- \left({x \times z}\right)}\right)$ Product with Ring Negative $\displaystyle$ $=$ $\displaystyle x \times y - x \times z$ Definition of Integer Subtraction

$\Box$

 $\displaystyle \left({y - z}\right) \times x$ $=$ $\displaystyle x \times \left({y - z}\right)$ Integer Multiplication is Commutative $\displaystyle$ $=$ $\displaystyle x \times y - x \times z$ from above $\displaystyle$ $=$ $\displaystyle y \times z - z \times x$ Integer Multiplication is Commutative

$\blacksquare$