Ordinal Exponentiation via Cantor Normal Form/Limit Exponents

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

Let $x$ and $y$ be limit ordinals.

Let $\left\langle{a_i}\right\rangle$ be a sequence of ordinals that is strictly decreasing on $1 \le i \le n$.

Let $\left\langle{b_i}\right\rangle$ be a sequence of natural numbers.

Then:


 * $\displaystyle \left({\sum_{i \mathop = 1}^n x^{a_i} \times b_i}\right)^y = x^{a_1 \mathop \times y}$

Proof
By Upper Bound of Ordinal Sum:
 * $\displaystyle \sum_{i \mathop = 1}^n \left({ x^{a_i} \times b_i }\right) \le x^{a_1} \times \left({b_1 + 1}\right)$

Furthermore:
 * $\displaystyle x^{a_1} \le \sum_{i \mathop = 1}^n \left({ x^{a_i} \times b_i }\right)$

It follows that:

It follows that:

Also see

 * Definition:Cantor Normal Form