In a mathematics article, how much of the argument can be given outside of "proof" environments?
It is hard to judge in the abstract, of course. But there are a number of ways to organize such a paper. It is probably best to be clear when you are giving formal arguments (i.e. proofs) and when you are just giving informal discussion. 

One way is to list the main theorem early and state that you intend to prove it. You can precede this with some of your supporting lemmas (with proof, perhaps) and follow it with others. You can then show how the lemmas fit together to prove the theorem if needed. 

But, I think, that a sequence of lemmas with no reveal about why you are stating and proving them followed by the theorem statement isn't very compelling reading. 

Organize it so that your reader wants to read it and so that the purpose of each supporting piece is clear. 

Sometimes the lemmas have independent interest. But sometimes the technique of the proof is much more interesting than the statement of any theorem. Take that into consideration. 

You don't want your readers constantly asking themselves "What is the point of this?".