The Spearman coefficient is defined as following:$r_s = 1- \frac{6\Sigma d^2}{n(n^2 -1)}=1$ and the Person Coefficient is given by $r_p=\frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^n _{i=1}(x_i - \bar{x})^2} \sqrt{\sum ^n _{i=1}(y_i - \bar{y})^2}}$.
My question is : If we know that $r_s=1$, can we conclude that $r_p\neq 0$?