Ordinal Exponentiation via Cantor Normal Form/Limit Exponents

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

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

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

Let $\sequence {b_i}$ be a sequence of natural numbers.

Then:


 * $\displaystyle \paren {\sum_{i \mathop = 1}^n x^{a_i} \times b_i}^y = x^{a_1 \mathop \times y}$

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

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

It follows that:

It follows that:

Also see

 * Definition:Cantor Normal Form