Proof of the canonical commutator relationship from $hat{p}=-ihbar nabla$

Given that $hat{r} psi = rpsi$ where $r$ is the position of a quantum particle, and where $hat{p}=-ihbar nabla$, the notes I have simply state that

$$[hat{r}_i, hat{p}_j] = ihbar delta_{ij}$$
where $delta$ is the Kronecker delta.
Is there a proof of this from the fundamental results i.e. the definitions of these operators?
I am also not sure what the deal with the subscripts $i, j$ is. Are they the $i,j$th particle in the system?
What I did seems stupid, but I want to know where I am going wrong:

so $$(hat{r}_ihat{p}_j – hat{p}_jhat{r}_i)psi = ihbar(rcdotnabla-nablacdot r)psi
=ihbar((xe_x+ye_y+ze_z)cdot(psi_x e_x+psi_y e_y+psi_z e_z)-((xpsi)_x+(ypsi)_y+(zpsi_z)))
=-3ihbar (psi)$$
Where I have said $r = xe_x+ye_y+ze_z$, and $nabla = frac{partial }{partial x}e_x + frac{partial }{partial y}e_y + frac{partial }{partial z}e_z$, where $e_i$ are standard cartesian unit vectors.

How does the definition of $nabla$ change with the particle$j$, if at all it does?

I would appreciate any advice you have for me.