Suppose $S_1$ and $S_2$ are regular submanifolds of $M_1$ and $M_2$, respectively. I would like to show that $S_1 \times S_2$ is a regular submanifold of $M_1 \times M_2$.
My attempt: Suppose $p_1 \in S_1$ and $p_2 \in S_2$. Then there exists coordinate map $(U_1, \phi_1)$ containing $p_1$ and another coordinate map $(U_2, \phi_2)$ containing $p_2$. Since a product of manifolds is a manifold, we have that $(p_1, p_2) \in (U_1, U_2)$ and that $(U_1 \times U_2, \phi_1 \times \phi_2)$ is a coordinate map on $M_1 \times M_2$. Since it was assumed that $S_1$ and $S_2$ are regular submanifolds, we have that on $U_1 \cap S$ a finitely many number of the coordinate functions vanish, and likewise for $U_2 \cap S_2$. Hence it follows that on $$(U_1 \times U_2) \cap (S_1 \times S_2) = (U_1 \cap S_1) \times (U_2 \times S_2)$$ a finitely many number of the coordinate functions also vanish. Since $(p_1, p_2)$ was arbitrary, we are done.
My proof feels a little incomplete. Am I missing anything or is this fine?