Invariant factors and minimal/characteristic polynomial

Question

$\DeclareMathOperator{\End}{End}\renewcommand\phi\varphi$ Let $V$ be a finite-dimensional $k$-vectorspace and $\phi \in \End_k(V)$. Then $V$ is also a $k[x]$-module (via $p\cdot v = (p(\phi))(v)$). Let $a_i\in k[x]$ with $1 \le i \le n$ with $a_i \mid a_{i+1}$ for $1 \le i < n$ be the invariant factors of $V$ ($V$ is finitely generated and $k[x]$ is a PID). I want to show that the minimal polynomial of $\phi$ is $a_n$, and the characteristic polynomial is $\prod_{i=1}^n a_i$.