The best ways to cite a theorem whose assumptions/conditions are written earlier in the source?
There are a number of ways to handle situations like this, depending on how much prominence you want to give to the result you are citing.
If the cited theorem is quite important to your work, and you think the reader will benefit from seeing it written out, then you can include a complete statement:

Theorem 1.2 [Smith 1953].  Suppose $X$ is compact, Artinian, and hyperfinite, and $\sqrt{Q(X)-57} &gt; \pi/4$.  Then $U(X) \sim \gamma$.

[Oh right, we don't have MathJax here. :( ]
Then elsewhere in your paper you can say &quot;By Theorem 1.2, ...&quot;
If the hypotheses of the theorem are particularly complicated and you don't feel it's helpful to include them in your paper, you could write:

Theorem 1.2 [Smith 1953].  If $X$ satisfies the assumptions of Section 5 of [Smith 1953], then  $U(X) \sim \gamma$.

Then you can write elsewhere &quot;We can easily check that $Y$ satisfies the conditions in [Smith 1953], so by Theorem 1.2., $U(Y) \sim \gamma$&quot;.
If you don't care to say much about Smith's result (perhaps it's well known, or you're just using it as a black box), you can just cite it, perhaps with a brief comment about its assumptions:

Since $X$ is hyperfinite, by Theorem 5.6.7 of [Smith 1967], ...
By Theorem 5.6.7 of [Smith 1967] (which requires $X$ to be hyperfinite)...