I find that I am using the term "by the definition of" a lot in my proofs, especially after && in align* environments. Is there a symbol that I can use instead for conciseness?
Example:
Having established that $x$ is odd, I would say $x=2k+1$ for some $k\in\mathbb{Z}$, $\textit{by the definition of odd}$.
Usually, if something follows by definition in a (well-written) mathematical text you don't need to call that out. Your example (and I appreciate it's probably contrived) is actually a good example of that: having established that $x$ is odd you can just write that $x=2k+1$ without saying "by definition of odd" (by the way, either oddness, or oddity if you want to be mildly amusing, but "odd" alone is wrong).
If you want to remind the reader that something was defined further up, before something interesting/complicated was discussed, then slipping "(by definition)" in is sufficient to indicate that if they don't remember it they should go back and check.