Understanding the interplay between Fréchet derivative and ordinary derivative.

Question

Let $Y$ be a Banach space. Consider the real-valued fucnion $\phi \in C^\infty _c (\Omega)$, where $\Omega \subset \mathbb{R}^n$ is open. Let $y\in Y$. Then can we say $\phi y \in C^\infty _c (\Omega; Y)$? We note that $ C^\infty _c (\Omega; Y)$ is the space of all functions $u$ such that for each $x \in \Omega$, the function $u(x,.) \in Y$ and the mapping $ x \in \Omega \mapsto u(x,.) \in Y$ is infinitely differentiable in Fréchet derivative sense. The difficulty I am having is to justify that $\phi y$ is infinitely differentiable in Fréchet sense.