Finite Ordinal Times Ordinal

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Theorem

Let $m$ and $n$ be finite ordinals.

Let $m \ne 0$, where $0$ is the zero ordinal.

Let $x$ be a limit ordinal.

Then:

$m \times \left({ x + n }\right) = x + \left({ m \times n }\right)$

Proof

By Ordinal Multiplication is Left Distributive, it follows that:

$m \times \left({ x + n }\right) = \left({ m \times x }\right) + \left({ m \times n }\right)$

It remains to prove that $x = \left({ m \times x }\right)$.

Since $x$ is a limit ordinal, it follows that:

 $\, \displaystyle \exists y \in \operatorname{On}: \,$ $\displaystyle x$ $=$ $\displaystyle \left({\omega \times y}\right)$ Factorization of Limit Ordinals $\displaystyle m \times x$ $=$ $\displaystyle m \times \left({\omega \times y}\right)$ Substitutivity of Class Equality $\displaystyle m \times x$ $=$ $\displaystyle \left({m \times \omega}\right) \times y$ Ordinal Multiplication is Associative $\displaystyle$ $=$ $\displaystyle \omega \times y$ by lemma

$\blacksquare$