Integer Multiplication Distributes over Addition/Corollary

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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

$\blacksquare$