Here is an excerpt from my textbook Introduction to Set Theory by Hrbacek and Jech. Although I'm able to prove these three equalities, my proof is up to a page long containing even a lemma (If you are interested, you can find it here).
I retype here.
First we observe that the ordinal functions $\alpha+\beta,\alpha\cdot\beta$, and $\alpha^\beta$ are continuous in the second variable: If $\gamma$ is a limit ordinal and $\beta=\sup\limits_{\nu<\gamma}\beta_\nu$, then
$$\alpha+\beta=\sup\limits_{\nu<\gamma}(\alpha+\beta_\nu), \alpha\cdot\beta=\sup\limits_{\nu<\gamma}(\alpha\cdot\beta_\nu), \alpha^\beta=\sup\limits_{\nu<\gamma}(\alpha^{\beta_\nu})$$
This follows directly from Definitions 5.1, 5.6, and 5.9.
I add Definitions 5.1, 5.6, and 5.9 here for reference.
I can not understand why the authors say This follows directly.... From my proof, I don't see it directly at all.
Could you please elaborate/explain what authors want to communicate by the sentence This follows directly from Definitions 5.1, 5.6, and 5.9?
Thank you so much for your help!