Ordinal Power of Power

Theorem
Let $x$, $y$, and $z$ be ordinals.

Then:


 * $\left({x^y}\right)^z = x^{y \mathop \times z}$

Proof
The proof shall proceed by Transfinite Induction on $z$.

Basis for the Induction
This proves the basis for the induction.

Induction Step
Suppose that $\left({x^y}\right)^z = x^{y \mathop \times z}$.

Then:

This proves the induction step.

Limit Case
Suppose $\left({x^y}\right)^w = x^{y \mathop \times w}$ for all $w \in z$ where $z$ is a limit ordinal.

Then:

To prove the statement, it is necessary and sufficient to prove that:


 * $\displaystyle \bigcup_{w \mathop \in z} x^{y \mathop \times w} = x^{y \mathop \times z}$

The proof of this shall proceed by cases:

Case 1
If $y = 0$, it follows that:

Case 2
If $y \ne 0$:

Therefore, by Supremum Inequality for Ordinals, it follows that:


 * $\bigcup_{w \mathop \in z} x^{y \mathop \times w} \le x^{y \mathop \times z}$

Conversely:

By Supremum Inequality for Ordinals, it follows that:


 * $\displaystyle \bigcup_{v \mathop \in y \mathop \times z} x^v \le \bigcup_{w \mathop \in z} x^{y \mathop \times w}$

Therefore, by the definition of ordinal exponentiation:

This proves the limit case.