Transitivity of submanifolds

Question

If $P$ is a regular submanifold of a smooth manifold $N$, and in turn $N$ is a regular submanifold of a smooth manifold $M$, does it really follow that $P$ is a submanifold of $M$?

Answer 1

$P$ is a submanifold of $N$ if at each point $p$, given a coordinate system $\phi_P$ about $p$, there is a coordinate system of $N$ about $p$ such that $\phi_N = \phi_P \times u$ for some constant vector $u$ ,and $N$ is a submanifold of $M$ if there is a coordinate system $\phi_M$ of $M$ about $p$ such that $\phi_M = \phi_N \times v$ for some constant vector $v$.

But then $\phi_M = \phi_P \times (u \times v)$.