Basics – How did mathematicians prove things before ZFC or Peano?

I do not get it. How was a + b = b + a or a (b + c) = ab + ac proved by ancient mathematicians prior to set theory or Peano axioms? If you can not logically prove the laws of arithmetic without Peano or Theorem Theory, how did the ancient mathematicians trust in laws of mathematics?

I have seen a video (on Youtube) that contains a proofb = bUse intuitive graphical methods, etc., eg. For example, picture a square with lengths whose units are multiplied by the two numbers, and then rotate themb = bon. What I consider as proof, because otherwise they have proved it. But commentators say that this is no proof that she does not use set theory. If mathematics is based on evidence and can prove everything about your claims, what was then in front of set theory? A bunch of no proven guesses? How do mathematicians trust in ab = bc as mathematical proof? Surely they had a chance to prove it (ab = ba) 2000 years ago without set theory or Peano axioms, since there was neither theorem nor Peano at that time? And if so, how did you prove it?