# Ordinal Exponentiation of Terms

## Theorem

Let $x, y, z$ be ordinals.

Let $n$ be a finite ordinal.

Let $x$ be a limit ordinal.

Let $y, z, n$ all be greater than $0$.

Then:

$\left({x^y \times n}\right)^z = x^{y \mathop \times z} \times n$ if $z$ is not a limit ordinal
$\left({x^y \times n }\right)^z = x^{y \mathop \times z}$ if $z$ is a limit ordinal

## Proof

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

### Basis for the Induction

The hypothesis requires that $z \ne 0$, so the induction starts at $z = 1$.

 $\displaystyle \left({ x^y \times n }\right) ^1$ $=$ $\displaystyle x^y \times n$ Definition of Ordinal Exponentiation $\displaystyle$ $=$ $\displaystyle x^{y \mathop \times 1} \times n$ Ordinal Multiplication by One

This proves the basis for the induction.

$\Box$

### Induction Step

 $\displaystyle \left({ x^y \times n }\right) ^z$ $=$ $\displaystyle x^{y \mathop \times z} \times n$ Inductive Hypothesis $\displaystyle \implies \ \$ $\displaystyle \left({ x^y \times n }\right) ^{z^+}$ $=$ $\displaystyle \left({ x^y \times n }\right) ^z \times x^y \times n$ Definition of Ordinal Exponentiation $\displaystyle$ $=$ $\displaystyle x^{y \mathop \times z} \times n \times x^y \times n$ Inductive Hypothesis $\displaystyle$ $=$ $\displaystyle x^{y \mathop \times z} \times x^y \times n$ Finite Ordinal Times Ordinal $\displaystyle$ $=$ $\displaystyle x^{y \mathop \times z + y} \times n$ Ordinal Sum of Powers $\displaystyle$ $=$ $\displaystyle x^{y \mathop \times z^+} \times n$ Definition of Ordinal Multiplication

This proves the induction step.

$\Box$

### Limit Case

Suppose that this statement holds for all $w \in z$ where $z$ is a limit ordinal.

Then:

 $\displaystyle x^{y \mathop \times z}$ $=$ $\displaystyle \left({x^y}\right)^z$ Ordinal Power of Power $\displaystyle$ $\le$ $\displaystyle \left({x^y \times n}\right)^z$ Subset is Right Compatible with Ordinal Exponentiation $\displaystyle$ $=$ $\displaystyle \bigcup_{w \mathop \in z} \left({x^y \times n}\right)^w$ Definition of Ordinal Exponentiation $\displaystyle$ $\le$ $\displaystyle \bigcup_{w \mathop \in z} x^{y \mathop \times w} \times n$ Proof by Cases where $w \in K_I$ or $w \in K_{II}$ $\displaystyle$ $\le$ $\displaystyle \bigcup_{w \mathop \in z} x^{y \mathop \times w + 1}$ Membership is Left Compatible with Ordinal Exponentiation $\displaystyle$ $\le$ $\displaystyle \bigcup_{w \mathop \in z} x^{y \mathop \times w^+}$ Membership is Left Compatible with Ordinal Multiplication and Membership is Left Compatible with Ordinal Exponentiation $\displaystyle$ $=$ $\displaystyle \bigcup_{w \mathop \in z} \left({ x^y }\right)^{w^+}$ Ordinal Power of Power $\displaystyle$ $=$ $\displaystyle \left({ x^y }\right)^z$ Definition of Ordinal Exponentiation

This proves the limit case.

$\blacksquare$