Do I have to prove all the theorems I mention in my paper?
Generally speaking, every statement in a published mathematics paper needs to have a proof somewhere that the reader can find it. (If this paper is for a class assignment rather than for publication, ask your instructor. The rules are also a little different for "expository" papers which are mainly intended to discuss previous work, rather than to present new results.)


If you use a result that is well known, you can mention it by name. It is not necessary to include a formal statement or proof,  nor a reference. This would apply to results that are found in most standard textbooks in your area, and that most people reading the paper would be familiar with.
If you use a result previously proved by someone else, give a reference to the paper containing their proof. It is often also helpful to give a formal statement of their theorem. You do not need to include the proof in your paper unless you think it will be specifically helpful to the reader (for instance, if you are planning to extend the same ideas yourself).
For an original theorem, not previously proved by anyone else,  give a complete and careful proof.