Integer Multiplication Distributes over Addition

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Theorem

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

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


Corollary

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

Let us define $\Z$ as in the formal definition of integers.

That is, $\Z$ is an inverse completion of $\N$.


From Natural Numbers form Commutative Semiring, we have that:

All elements of $\N$ are cancellable for addition
Addition and multiplication are commutative and associative on the natural numbers $\N$
Natural number multiplication is distributive over natural number addition.

The result follows from the Extension Theorem for Distributive Operations.

$\blacksquare$


Sources

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