I don't publish in pure mathematics journals (I'm a computer science researcher) but at least in my field, there is no affiliation requirement in order to submit to peer-reviewed journals.
First a quick comment: you haven't told us the subject of your proof. I don't mean to sound negative or discouraging, but if it relates to a famous long-standing conjecture (Riemann hypothesis, twin prime conjecture, etc), there is very little chance your paper will be taken seriously, as these problems have been so well studied that the odds of an amateur resolving them in 6 pages is very very low. If you do find yourself in this situation, then surely along the way to proving the famous conjecture you have also developed new theory and partial results that are interesting in their own right -- I would focus on getting some of these published first.
The first step would be for you to identify a relevant journal, and read their submission instructions. All journals have very specific instructions about how your submission should be formatted and how it should be submitted/uploaded to them. Many have LaTeX templates you'll be able to use.
Next, the editor will determine whether to send the article out for peer review, or reject it immediately. To minimize the chances of outright rejection, be sure you structure and format your paper professionally. Reading accepted papers from the journal is surely the best way to learn the standards of form, but a few quick tips:
- Format your document using LaTeX. If you don't know LaTeX, learning it and converting your paper to it is well worth the few weeks' investment. I do not recommend submitting a paper written in MS Equation Editor, even if the journal allows it, as it will make an unprofessional first impression.
- Be sure your paper has an appropriate abstract, outlining in ~100 words what you prove and how it fits into the existing body of work.
- Throughout the paper, and in the introduction in particular, be sure you've appropriately acknowledged (and cited) all relevant existing work. Ignorance of existing published partial or related results is one of the brightest red flags of crankery, so you want to leave the editor with no doubts about whether you have done your homework.
If your submission looks and smells like a professional paper, it will be submitted for peer review. The usefulness of those reviews is always uncertain, but with any luck you will get good feedback on how to improve your paper, even in the unfortunate even that the reviewers recommend rejection.
Best Answer
Here is my comment with more details:
Quoting from https://arxiv.org/help/endorsement
I.e. pick a paper relevant to the work (possibly one you cite) and find it on the arXiv (assuming it is there). Then find an author who is an endorser using the link mentioned (may take a few tries depending on how lucky you are). Then contact this person about your work and ask to be endorsed.
Here is some general advice when asking to be endorsed:
Make sure you explain precisely in the email what results you have proven, and preferably also the methods used. This will allow the person to get a first impression on the suitability.
As part of this, make sure the paper is presented nicely (use $\LaTeX$!) You will need to show the paper to the person (it is also mentioned on the arXiv page I linked that people should not endorse someone without checking the paper). But they need not do a full review of the paper, they just need to check that it seems like you know what you are doing (this is specifically mentioned on that page). So the time requirement is not particularly high.
That said, it may still easily take you several tries to find someone who wants to take the time to look at the paper (and each one may end up taking a long time to reply, simply because this sort of request is not of the highest priority).
To mitigate the above, make sure you pick the person you can find who is the most relevant, so they can see that they have not been picked at random. It may also help if they seem unlikely to receive a lot of such requests (not sure how to gauge this though). For example, I would probably not mind looking over a paper in my field for this purpose, but I am not an endorser, so I have no idea how many such requests people typically get. It will certainly also depend on the field (I am in a field where very few amateurs publish, whereas someone in number theory will be much more likely to be swamped by requests).