Left Distributive Law for Natural Numbers

Theorem
The operation of multiplication is left distributive over addition on the set of natural numbers $\N$:
 * $\forall x, y, n \in \N: n \times \left({x + y}\right) = \left({n \times x}\right) + \left({n \times y}\right)$

Proof
Using the axiom schema:

Let us cast the proposition in the form:
 * $\forall a, b, n \in \N: a \times \left({b + n}\right) = \left({a \times b}\right) + \left({a \times n}\right)$

For all $n \in \N$, let $P \left({n}\right)$ be the proposition:
 * $\forall a, b \in \N: a \times \left({b + n}\right) = \left({a \times b}\right) + \left({a \times n}\right)$

$P \left({0}\right)$ is the case:

and so $P \left({0}\right)$ holds.

Basis for the Induction
$P \left({1}\right)$ is the case:

and so $P \left({1}\right)$ holds.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 0$, then it logically follows that $P \left({k + 1}\right)$ is true.

So this is our induction hypothesis:
 * $\forall a, b \in \N: a \times \left({b + k}\right) = \left({a \times b}\right) + \left({a \times k}\right)$

Then we need to show:
 * $\forall a, b \in \N: a \times \left({b + \left({k + 1}\right)}\right) = \left({a \times b}\right) + \left({a \times \left({k + 1}\right)}\right)$

Induction Step
This is our induction step:

The result follows by the Principle of Mathematical Induction.

Also see

 * Right Distributive Law for Natural Numbers