Exponential of Product/Proof 3

Proof
This proof assumes the Exponential of Sum property.

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

That is:


 * $\forall n \in \Z_{\ge 0}: \map \exp {n y} = \paren {\exp y}^n$

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

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

Thus:
 * $(2): \quad \forall m \in \Z: \map \exp {m y} = \paren {\exp y}^m$

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 the above:

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