I am trying to prove by induction the associativity of natural numbers. It is easy to see that if $n,m\in \mathbb{N}$, then $(mn)1=m(n1)$. If $p\in \mathbb{N}$ is such that $(mn)p= m(np)$, then $(mn)(p+1)=(mn)p+mn=m(np)+mn=m[np+n]=m(n(p+1)).$ But I am using the distributivity property here. Is that possible? Those properties should be independent.
Am I missing something?
when you write (mn)(p+1)=mn+mn+mn+..+mn (p+1)factors which is the definition of multiplication you get mn(p+1)=(mn)p+mn (without using the distribitivity) you can use the hypothesis of induction .thus mn(p+1)=m(np)+mn=m(np+n)(by the definition of multiplication and identification) so mn(p+1)=m(n(p+1)) and you can conclude.