Left Distributive Law for Natural Numbers

Theorem
The operation of multiplication is left distributive over addition on the set of natural numbers $\N$:
 * $\forall x, y, n \in \N_{> 0}: n \times \paren {x + y} = \paren {n \times x} + \paren {n \times y}$

Proof
Using the axiomatization:

Let us cast the proposition in the form:
 * $\forall a, b, n \in \N_{> 0}: a \times \paren {b + n} = \paren {a \times b} + \paren {a \times n}$

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
 * $\forall a, b \in \N_{> 0}: a \times \paren {b + n} = \paren {a \times b} + \paren {a \times n}$

Basis for the Induction
$\map P 1$ is the case:

and so $\map P 1$ holds.

This is our basis for the induction.

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

So this is our induction hypothesis:
 * $\forall a, b \in \N_{> 0}: a \times \paren {b + k} = \paren {a \times b} + \paren {a \times k}$

Then we need to show:
 * $\forall a, b \in \N_{> 0}: a \times \paren {b + \paren {k + 1} } = \paren {a \times b} + \paren {a \times \paren {k + 1} }$

Induction Step
This is our induction step:

The result follows by the Principle of Mathematical Induction.

Also see

 * Right Distributive Law for Natural Numbers