I'm trying to prove $\alpha^{\beta+\gamma}=\alpha^\beta\alpha^\gamma$ where I adopt the recursive definition for ordinal exponent (Kunen p26 Def 9.5).
The natural approach is to use transfinite induction WRT $\gamma$. The base step and inductive step are more or less straight forward, but I'm a bit unsure regarding the limit case. If $\gamma$ is a limit ordinal, then so is $\beta + \gamma$. Here's my proof when $\gamma$ is a limit ordinal: $$ \alpha^{\beta + \gamma} = \sup\{\alpha^{\xi}:\xi < \beta + \gamma\} \stackrel!= \sup\{\alpha^{\beta + \xi}: \xi < \gamma\} = \bigcup\{\alpha^\beta\alpha^\xi:\xi < \gamma\} = \alpha^\beta\bigcup\{\alpha^\xi:\xi < \gamma\} = \alpha^\beta\sup\{\alpha^\xi:\xi < \gamma\} = \alpha^\beta\alpha^\gamma $$ I'm wondering is it reasonable to establish equality marked with (!) here?
Many thanks