A mathematical theorem is a rule or fact in math that has been shown to be true using logic and other math facts.

The process of showing a result to be true is called a proof.

Mathematicians tend to only call very important results theorems. They call other results a lemma, proposition, or corollary.

The fundamental assumptions in mathematics, that is the things we don't prove and just take as true, are called axioms.

Example

One famous theorem is the Pythagorean Theorem. It says:

This rule works for any right triangle, where $a$ and $b$ are the shorter sides, and $c$ is the longest side (called the hypotenuse). This theorem was proved thousands of years ago using logical steps.