Ordinal Exponentiation of Terms

Theorem

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

Let $n$ be a natural number.

Let $x$ be a limit ordinal.

Let $y$, $z$ and $n$ 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 $z = 1$.

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

This proves the basis for the induction.

$\Box$

Induction Step

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \left({ x^y \times n }\right) ^z$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle x^{y \mathop \times z} \times n$$ $$\displaystyle$$ $$\displaystyle$$ Inductive Hypothesis $$\displaystyle$$ $$\displaystyle \implies$$ $$\displaystyle$$ $$\displaystyle \left({ x^y \times n }\right) ^{z^+}$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle \left({ x^y \times n }\right) ^z \times x^y \times n$$ $$\displaystyle$$ $$\displaystyle$$ definition of ordinal exponentiation $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle x^{y \mathop \times z} \times n \times x^y \times n$$ $$\displaystyle$$ $$\displaystyle$$ Inductive Hypothesis $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle x^{y \mathop \times z} \times x^y \times n$$ $$\displaystyle$$ $$\displaystyle$$ Natural Number Times Ordinal $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle x^{y \mathop \times z + y} \times n$$ $$\displaystyle$$ $$\displaystyle$$ Ordinal Sum of Powers $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle x^{y \mathop \times z^+} \times n$$ $$\displaystyle$$ $$\displaystyle$$ 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:

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

This proves the limit case.

$\blacksquare$