Is this a basis for the equational identities of the structure $(\mathbb{N};+,\cdot,0,1)$?

Question

Consider the equational identities of the algebraic structure $(\mathbb{N};+,\cdot,0,1)$. I believe that the following identities are a basis for it:

1. The commutative properties, of both addition and multiplication.
2. The associative properties, of both addition and multiplication.
3. The distributive property connecting addition and multiplication.
4. $x+0=x$
5. $x\cdot 1=x$
6. $x\cdot 0=0$

Is this true? Can all equational identities of that structure be generated from my set of identities? If so, what is the proof?