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:

$\forall x, y, z \in \Z: x \times \paren {y - z} = \paren {x \times y} - \paren {x \times z}$
$\forall x, y, z \in \Z: \paren {y - z} \times x = \paren {y \times x} - \paren {z \times x}$


\(\ds x \times \paren {y - z}\) \(=\) \(\ds x \times \paren {y + \paren {- z} }\) Definition of Integer Subtraction
\(\ds \) \(=\) \(\ds x \times y + x \times \paren {- z}\)
\(\ds \) \(=\) \(\ds x \times y + \paren {- \paren {x \times z} }\) Product with Ring Negative
\(\ds \) \(=\) \(\ds x \times y - x \times z\) Definition of Integer Subtraction


\(\ds \paren {y - z} \times x\) \(=\) \(\ds x \times \paren {y - z}\) Integer Multiplication is Commutative
\(\ds \) \(=\) \(\ds x \times y - x \times z\) from above
\(\ds \) \(=\) \(\ds y \times z - z \times x\) Integer Multiplication is Commutative