Nielsen & Chuang Exercise 2.32: Show that the tensor product of two projectors is a projector

$newcommand{bra}[1]{left<#1right|}
newcommand{ket}[1]{left|#1right>}$Here is what I tried:

Given that we have two projectors:
$$
A = sum_i ket{i} bra{i}, hspace{2em}
B = sum_j ket{j} bra{j}
$$
The goal is to prove that:
$$
A otimes B = sum_k ket{k} bra{k}. tag1label1
$$
Plugging into eqref{1}, we get:
$$
A otimes B
= left( sum_i ket{i} bra{i} right) otimes left( sum_j ket{j} bra{j} right)
= sum_{i,j} ket{i} bra{i} otimes ket{j} bra{j}
tag{2}label{2}
$$
I’m not sure how to proceed from eqref{2}. It would be convenient if for every $ket{i}$ and $ket{j}$ there is a $ket{k}$ for which the following identity is true:
$$
ket{k} bra{k} = ket{i} bra{i} otimes ket{j} bra{j} tag{3}label{3}
$$
This would prove eqref{1} immediately. Is eqref{3} true though? If yes, why? If not, how else can we proceed to prove eqref{1}?