Exponential of Product/Proof 3

Proof
This proof assumes the Exponent of Sum property.

First, for $n \in \Z_{\ge 0}$:

That is:


 * $\forall n \in \Z_{\ge 0}: \exp \left({n y}\right) = \left({\exp y}\right)^n$

Now let $n \in \Z_{<0}$.

It follows that $-n \in \Z_{>0}$, so:

Thus:
 * $\forall m \in \Z: \exp \left({m y}\right) = \left({\exp y}\right)^m$

Call this result $\left({2}\right)$.

Next, for $n \in \Z_{>0}$:

So fix $r \in \Q$.

Let $r = \dfrac m n$, where $m \in \Z$ is an integer and $n \in \Z_{>0}$ is a strictly positive integer.

From above:

Thus, from the definition of $\left({\exp y}\right)^x$ as the unique continuous extension of $r \mapsto \left({\exp y}\right)^r$ from $\Q$ to $\R$:
 * $\exp \left({x y}\right) = \left({\exp y}\right)^x$