In logic class we kinda did prove most of the integer operations, but it was more like (extremely shortened and not properly written)
If 1+1=2 and 1+1+1=3 then prove that 1+2=3
2 was just a shortened representation of 1+1 so technically you were proving that 1+1 plus 1 equals 1+1+1.
Really fun stuff. It took a long while to reach division
Yeah, that's what meant with "2 is just the shortened representation of 1+1".
Same with 1+1+1=3, really. We need to decide the value of 1,2,3,4... Before we can do anything. In hindsight if you think about it, for someone that doesn't know the value of the symbols we use to represent numbers, any combination that mixes numbers requires the axiom of 1+1+1+1+... = X
I'd be surprised if someone proved that something equals 5 without any kind of axiom that already makes 5 equal to another thing.