I have a problem in proving the existence of a convolution product of two distributions $T,S\in \mathcal{D}^{′}(\mathbb{R}^n)$ such that $S$ with compact support, that is to say: the linear form defined on $\mathcal{C}^{\infty}_{c}(\mathbb{R}^n)$ by $$ \phi\mapsto \langle T*S, \phi\rangle = \langle T_x \otimes S_y, (x,y)\mapsto \phi(x + y)\rangle $$
In a stage of the proof he tells us that : if $\phi \in \mathcal{C}^{\infty}_{c}(\mathbb{R}^n)$ the function : $$ y\mapsto \langle S, x\mapsto \phi(x + y)\rangle $$ is an element of $\mathcal{C}^{\infty}_{c}(\mathbb{R}^n)$. I did not understand why. An idea, please.