Professor Isn't Teaching the Course we Signed up for Okay, so it is kind of a weird situation and we don't know what to do. The situation is this:

I am currently doing a math major and some of courses have two different (but equivalent) options -- application based, or a proof based option.

I am working on a pure math degree and so I opted in for the proof based version of the course, since I think it will better prepare me for later math courses.

However, the professor assigned to teach the course this term has never taught the proof based section before and straight up said in the first lecture that he was going to teach us the application based version of the course despite our choice.

A few of us went to complain to the chair of the dept. but to no avail, he simply addressed the class and said "If you don't like my lectures, don't come." Is there anything that can be done about this situation?
In the absence of really bad things, department chairs are going to defer to the course teacher, just as you saw. The department chair may reprimand the teacher, but this will be in private and you will never know it happened. Bringing the issue to the Dean will not result in any changes as they will also simply defer to the department chair, unless of course it is something really bad. The Dean may reprimand the Chair, but you will never know.

In terms of is there anything you can do, there is nothing you can do to change the course. The Chair, however, should hopefully work with you (and possibly the department administrator or undergraduate/graduate chair) to withdraw without it showing up on your transcript, getting a tuition refund (if appropriate), helping you add a different class, and helping you get into the version you want next semester.