mathematics - Referee says the proof is wrong, but it is not, what to do?

About 3 months ago, my co-author and I submitted a mathematical paper to a journal. Yesterday, we got the editor's response and the paper has been rejected. The reason is that the referee thinks that our results are wrong or trivial.

However, we carefully read the referee's report and we conclude that the "errors" pointed out by the referee are not errors at all and that our proofs are correct. In all modesty, we think that the referee understood very poorly even the statements of our theorems.

Personally, this is the first time that such a situation occurs to me. I already have four papers published in such journal and the reports I got from the referees have been always really accurate.

What do you think is the best thing to do? Forget about and try another journal? Write an e-mail to the editor explaining why we disagree with the decision? Other ideas?

EDIT: Thank you very much for all your answers. At the end, we submitted our paper to another journal with only some slight edits we have already planned to do independently by the referee's report.

Answer

I must disagree with some of the answers.

Some assume that it is a problem with the person asking the question and giving suggestive answers with a very limited number of fixes ("I would strongly advise performing a major revision on your presentation of the proofs"; "two potential problems. One is that you may be in error, [...]. The other is presentation[...]").

If we are attentive enough, we all know well what the source of our problems are: The magic word is feedback. If people complain consistently that our papers are hard to read, the suggestion to work it over has a point. If again and again referees find errors in the manuscript, we will take much more care to avoid errors (proof-read, double-check, triple-check).

But if we get an outlier answer and cannot fathom what exactly should be the problem, it is very, very likely that we are not the source of the problem. Working the paper over in this case is a waste of time.

The reason(s) that a referee rejects a paper are virtually unlimited. He is wrong, she does not care, he is busy, she is right, but explains it badly. Whatever.

But what you can do is to read very carefully the answer and ask yourself under what circumstances a person would give the answer. Is it aggressive, dismissive, bored, sad ? It is rather short or more detailed ? Does it look like the referee invested time or it is hastily written with boilerplate passages ? On this you could often discern what the real reason for the rejection is.

Everyone will sooner or later have a paper rejected for inexplicable reasons, it is normal. Nothing to be afraid of.

That said, the best strategy already mentioned by other answers is to cut the losses and simply go to another journal. I would advise to challenge the decision if and only if the editor made it clear that the decision is not final and then if and only if it is objectively important to have it published in this specific journal. In that case do everything in your ability to check and recheck your paper and hone it to perfection because you will only get exactly one chance to convince the editor.