Query about another post $\operatorname{rank}(AB) \leq \min(\operatorname{rank}(A), \operatorname{rank}(B))$

For, $\operatorname{rank}(AB) \leq \min(\operatorname{rank}(A), \operatorname{rank}(B))$

On this post I don't understand why it is enough to prove $\dim \operatorname{range}(AB)\leq \dim R(A)$ and $\operatorname{range}(AB)\leq \dim R(B)$. Wouldn't the "min" part play a role as well?

• You don't need \operatorname{min}; you can just write \min. And in a "displayed", as opposed to "inline", context, that effects positioning of subscripts, thus: $$\min_{x\in\mathcal X} f(x)$$ – Michael Hardy Apr 12 '16 at 3:35

Proving both $a\le b$ and $a\le c$ is the same as proving $a\le\min\{b,c\}$.