I'm looking for the proof of the following fact:
Assume axiom of determinacy. Then $\omega^V_1$ is a strongly inaccessible cardinal in $L$.
I have seen this result mentioned in few places (e.g. here, page 12), however I can't seem to find any actual proof.
By myself I can prove that $\omega^V_1$ is regular in $L$, since it is regular in $V$ as well: suppose $\alpha_1,\alpha_2,...$ converges to $\omega_1$. For any $\beta<\omega_1$, let $S_\beta$ be the set of countable well-orderings of order type $\beta$ under some fixed encoding. Since AD implies countable choice for subsets of $\Bbb R$, we can choose a specific ordering of $\Bbb N$ of order type $\alpha_i$ for each $i$ simultaneously, and then glue them together to a well-order of $\Bbb N^2$ with order type $\omega_1$, which is impossible.
Hence we only need to show that $\omega^V_1$ is strong limit in $L$. It is mentioned in the paper I linked that it is related to perfect set property, but I see no way to prove it's even weakly inaccessible.
Thanks in advance.
Using the perfect set property gives a quick proof that $\omega_1^V$ is a limit cardinal in $L$, this in combination of the facts that $L$ satisfies $\sf GCH$ and that if $\kappa$ is regular in $V$ then it is regular in $L$ give us that $\omega_1^V$ is inaccessible in $L$.
You can find the proof that the perfect set property implies that $\omega_1$ is a strong limit in Kanamori's "The Higher Infinite", 11.3 to 11.5. The result is due to Specker from 1957.
Now using the fact that determinacy implies the perfect set property the proof is complete.