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:
- $\forall x, y, z \in \Z: x \times \paren {y - z} = \paren {x \times y} - \paren {x \times z}$
- $\forall x, y, z \in \Z: \paren {y - z} \times x = \paren {y \times x} - \paren {z \times x}$
Proof
\(\ds x \times \paren {y - z}\) | \(=\) | \(\ds x \times \paren {y + \paren {- z} }\) | 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 |
$\Box$
\(\ds \paren {y - z} \times x\) | \(=\) | \(\ds x \times \paren {y - z}\) | 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 |
$\blacksquare$