Is my proof of $1+1=2$ correct?

Question

Here is the proof:

Note: I will denote the successor of a natural number $n$ by $(n++)$

If one assumes the Peano axioms then they may define addition as follows:

$0+m:=m$

$(n++)+m=(n+m)(++)$

$\forall n,m\in\mathbb{N}$

Using these definitions, here is a simple proof of $1+1=2$

$1+1$

$=(0++)+1$

$=(0+1)(++)$

$=(1++)$

$=2$

$\therefore{1+1=2}$

$Q.E.D$

Answer 1

Yes, the proof is correct. I think the only way to sort of "improve" it would be to include right away in the definitions that $1$ is the successor of $0$ and $2$ is the successor of $1$.