Ordinal Exponentiation of Terms

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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.

$\blacksquare$


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