Integer Multiplication Distributes over Addition

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


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

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.