# Ordinal Sum of Powers

## Theorem

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

Then:

- $x^y \times x^z = x^{y + z}$

## Proof

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

### Basis for the Induction

- $x^0 = 1$ for all $x$

\(\displaystyle x^y \times x^z\) | \(=\) | \(\displaystyle x^y\) | by Ordinal Multiplication by Zero | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x^{y + z}\) | by Ordinal Addition by Zero |

This proves the basis for the induction.

$\Box$

### Induction Step

Suppose that $x^y \times x^z = x^{y + z}$.

Then:

\(\displaystyle x^y \times x^{z^+}\) | \(=\) | \(\displaystyle x^y \times \left({ x^z \times x }\right)\) | definition of ordinal exponentiation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({ x^y \times x^z }\right) \times x\) | by Ordinal Multiplication is Associative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x^{y + z} \times x\) | by Inductive Hypothesis | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x^{\left({y + z}\right)^+}\) | definition of ordinal exponentiation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x^{\left({y + z^+}\right)}\) | definition of ordinal addition |

This proves the induction step.

$\Box$

### Limit Case

Suppose that $\forall w \in z: x^y \times x^w = x^{y + w}$ for limit ordinal $z$.

\(\displaystyle \forall w \in z: \ \ \) | \(\displaystyle \left({x^y \times x^w}\right)\) | \(\le\) | \(\displaystyle \left({x^y \times x^z}\right)\) | by Membership is Left Compatible with Ordinal Multiplication | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \bigcup_{w \in z} \left({ x^y \times x^w }\right)\) | \(\le\) | \(\displaystyle \left({ x^y \times x^z }\right)\) | by Indexed Union Subset |

Conversely:

\(\displaystyle w \in \left({ x^y \times x^z }\right)\) | \(\implies\) | \(\displaystyle \exists u \in x^z: w \in \left({ x^y \times u }\right)\) | by Ordinal is Less than Ordinal times Limit | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle \exists v \in z: \exists u \in x^v: w \in \left({ x^y \times u }\right)\) | by Ordinal is Less than Ordinal to Limit Power |

But this means that $u$ is bounded above by $x^v$ for some $v \in z$.

Thus there exists a $v \in z$ such that:

- $w \le \left({ x^y \times x^v }\right)$

By Supremum Inequality for Ordinals, it follows that:

- $\left({ x^y \times x^z }\right) \le \bigcup_{w \in z} \left({ x^y \times x^w }\right)$

\(\displaystyle \left({ x^y \times x^z }\right)\) | \(=\) | \(\displaystyle \bigcup_{w \in z} \left({ x^y \times x^w }\right)\) | Definition of Set Equality | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \bigcup_{w \in z} x^{y+w}\) | Inductive Hypothesis for the Limit Case |

$\Box$

\(\displaystyle \forall w \in z: \ \ \) | \(\displaystyle y + w\) | \(\le\) | \(\displaystyle y + z\) | by Membership is Left Compatible with Ordinal Addition | |||||||||

\(\displaystyle \forall w \in z: \ \ \) | \(\displaystyle x^{y + w}\) | \(\le\) | \(\displaystyle x^{y + z}\) | by Membership is Left Compatible with Ordinal Exponentiation | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \bigcup_{w \in z} x^{y + w}\) | \(\le\) | \(\displaystyle x^{y + z}\) | by Indexed Union Subset |

Conversely:

\(\displaystyle w \in x^{y + z}\) | \(\implies\) | \(\displaystyle \exists u \in \left({y + z}\right): w \in x^u\) | definition of ordinal exponentiation | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle \exists v \in z: \exists u \in \left({y + v}\right): w \in x^u\) | definition of ordinal addition |

Thus, $u$ is bounded above by $\left({ y + v }\right)$ for some $v \in z$.

Therefore:

- $x^u \le x^{y + v}$

By Supremum Inequality for Ordinals, it follows that:

- $x^{y + z} \le \bigcup_{w \in z} x^{y + w}$

Thus, by definition of set equality:

- $x^{y + z} = \bigcup_{w \in z} x^{y + w}$

$\Box$

Combining the results of the first and second lemmas for the limit case:

- $x^{y+z} = x^y \times x^z$

This proves the limit case.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 8.41$