Exponent Combination Laws/Power of Power/Proof 2

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Theorem

Let $a \in \R_{>0}$ be a (strictly) positive real number.


Let $x, y \in \R$ be real numbers.

Let $a^x$ be defined as $a$ to the power of $x$.


Then:

$\left({a^x}\right)^y = a^{xy}$

Proof

We will show that:

$\forall \epsilon \in \R_{>0}: \left\vert{a^{xy} - \left({a^x}\right)^y}\right\vert < \epsilon$

Without loss of generality, suppose that $x < y$.

Consider $I := \left[{x \,.\,.\, y}\right]$.

Let $I_\Q = I \cap \Q$.

Let $M = \max \left\{ {\left\vert{x}\right\vert, \left\vert{y}\right\vert} \right\}$

Fix $\epsilon \in \R_{>0}$.

From Polynomial is Continuous:

$\exists \delta' \in \R_{>0} : \left\vert{a^x - a^{x'} }\right\vert < \delta' \implies \left\vert{\left({a^x}\right)^{y'} - \left({ a^{x'} }\right)^{y'} }\right\vert < \dfrac \epsilon 4$

From Power Function on Strictly Positive Base is Continuous:

\(\displaystyle \exists \delta_1 \in \R_{>0} : \ \ \) \(\displaystyle \left\vert{x x' - y y'}\right\vert \ \ \) \(\, \displaystyle < \, \) \(\displaystyle \delta_1\) \(\implies\) \(\, \displaystyle \left\vert{ a^{x x'} - a^{x y'} }\right\vert \, \) \(\, \displaystyle <\, \) \(\displaystyle \dfrac \epsilon 4\) $\quad$ $\quad$
\(\displaystyle \exists \delta_2 \in \R_{>0} : \ \ \) \(\displaystyle \left\vert{x y' - x' y'}\right\vert \ \ \) \(\, \displaystyle < \, \) \(\displaystyle \delta_2\) \(\implies\) \(\, \displaystyle \left\vert{a^{x y'} - a^{x' y'} }\right\vert \, \) \(\, \displaystyle <\, \) \(\displaystyle \dfrac \epsilon 4\) $\quad$ $\quad$
\(\displaystyle \exists \delta_3 \in \R_{>0} : \ \ \) \(\displaystyle \left\vert{x' - x}\right\vert \ \ \) \(\, \displaystyle < \, \) \(\displaystyle \delta_3\) \(\implies\) \(\, \displaystyle \left\vert{a^{x'} - a^x}\right\vert \, \) \(\, \displaystyle <\, \) \(\displaystyle \delta'\) $\quad$ $\quad$
\(\displaystyle \) \(\implies\) \(\, \displaystyle \left\vert{\left({a^{x} }\right)^{y'} - \left({ a^{x'} }\right)^{y'} }\right\vert \, \) \(\, \displaystyle <\, \) \(\displaystyle \dfrac \epsilon 4\) $\quad$ $\quad$
\(\displaystyle \exists \delta_4 \in \R_{>0} : \ \ \) \(\displaystyle \left\vert{y' - y}\right\vert \ \ \) \(\, \displaystyle < \, \) \(\displaystyle \delta_4\) \(\implies\) \(\, \displaystyle \left\vert{\left({ a^{x} }\right)^{y'} - \left({ a^{x} }\right)^{y} }\right\vert \, \) \(\, \displaystyle <\, \) \(\displaystyle \dfrac \epsilon 4\) $\quad$ $\quad$


Further:

\(\displaystyle \left\vert{y - y'}\right\vert\) \(<\) \(\displaystyle \frac {\delta_1} {\left\vert{x}\right\vert}\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \left\vert{x y - x y'}\right\vert\) \(=\) \(\displaystyle \left\vert{x}\right\vert \left\vert{y - y'}\right\vert\) $\quad$ Absolute Value Function is Completely Multiplicative $\quad$
\(\displaystyle \) \(<\) \(\displaystyle \left\vert{x}\right\vert \frac{\delta_1}{ \left\vert{x}\right\vert }\) $\quad$ multiplying both sides by $\left\vert{x}\right\vert \ge 0$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \delta_1\) $\quad$ $\quad$


And:

\(\displaystyle \left\vert{x - x'}\right\vert\) \(<\) \(\displaystyle \frac {\delta_2} M\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \left\vert{x y' - x'y'}\right\vert\) \(=\) \(\displaystyle \left\vert{y'}\right\vert \left\vert{x - x'}\right\vert\) $\quad$ Absolute Value Function is Completely Multiplicative $\quad$
\(\displaystyle \) \(\leq\) \(\displaystyle M \left\vert{x - x'}\right\vert\) $\quad$ Real Number Ordering is Compatible with Multiplication $\quad$
\(\displaystyle \) \(<\) \(\displaystyle M \frac {\delta_1} M\) $\quad$ multiplying both sides by $M \ge 0$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \delta_2\) $\quad$ $\quad$


Let $\delta = \max \left\{ {\dfrac {\delta_1} {\left\vert{x}\right\vert}, \dfrac {\delta_2} M, \delta_3, \delta_4} \right\}$.

From Closure of Rational Interval is Closed Real Interval:

$\exists r, s \in I_\Q: \left\vert{x - r}\right\vert < \delta \land \left\vert{y - s}\right\vert < \delta$


Thus:

\(\displaystyle \left\vert{a^{x y} - \left({a^x}\right)^y}\right\vert\) \(\le\) \(\displaystyle \left\vert{a^{x y} - a^{x s} }\right\vert + \left\vert{a^{x s} - a^{r s} }\right\vert + \left\vert{a^{r s} - \left({a^r}\right)^s}\right\vert + \left\vert{\left({a^r}\right)^s - \left({a^x}\right)^s}\right\vert + \left\vert{\left({a^x}\right)^s - \left({a^x}\right)^y}\right\vert\) $\quad$ Triangle Inequality for Real Numbers $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{a^{x y} - a^{x s} }\right\vert + \left\vert{a^{x s} - a^{r s} }\right\vert + \left\vert{ \left({a^r}\right)^s - \left({a^x}\right)^s}\right\vert + \left\vert{\left({a^x}\right)^s - \left({a^x}\right)^y}\right\vert\) $\quad$ Product of Indices of Real Number: Rational Numbers $\quad$
\(\displaystyle \) \(<\) \(\displaystyle \frac \epsilon 4 + \frac \epsilon 4 + \frac \epsilon 4 + \frac \epsilon 4\) $\quad$ Definition of $r$ and $s$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \epsilon\) $\quad$ $\quad$


Hence the result, by Real Plus Epsilon.

$\blacksquare$