Is The Segre map an embedding between differentiable manifolds $\mathbb RP^n\times \mathbb RP^m$ and $\mathbb RP^{mn+m+n}$?

Question

Can I show that $\mathbb RP^n\times \mathbb RP^m$ is embedded in $\mathbb RP^{mn+m+n}$. Is The following map(Segre map) an embedding ? I do not know why it is an $C^{\infty}$ embedding .

$F: \mathbb{P}^m \times \mathbb{P}^n \to \mathbb{P}^{(m+1)(n+1)-1}$ \begin{equation} ([(x_0,\cdots,x_m)],[(y_0,\cdots,y_n)]) \mapsto [(x_0y_0,x_0y_1, \cdots,x_0y_m,x_1y_0,\cdots x_iy_j,\cdots, x_my_n)]\qquad \small\text{with lexicographical order} \end{equation}