linear algebra - If $A$ is $mtimes n$ matrix and $AA^T$ is non singular show that $text{rank}(A) = m$

$AA^T$ can be non singular only if the columns in $A$ are linear independent and they span the column space of $A$. Because the columns are linear independent then $A$ can be reduced to echelon form and $A$ will have $m$ pivots only if $m < n$. Because we have $m$ pivots, $\text{rank}(A) = m$.

Is this a valid prove for If $A$ is $m\times n$ matrix and $AA^T$ is non singular show that $\text{rank}(A) = m$?

Thanks ^_^