Checking a Proof of a Theorem
Your proof looks OK to me. But don't rush to publish $-$ it seems to me that Theorem 1.2 (available at this link) depends on the Taniyama–Shimura–Weil conjecture (or the modularity theorem, as it should now be called), just like Wiles' proof of Fermat's Last Theorem did.
So your proof is $-$ roughly speaking $-$ using a consequence of Fermat's Last Theorem to prove Fermat's Last Theorem.