$$\left\|\sum A_iv_i\right\|=\sup_{\|x\|=1} \left\|\sum A_iv_ix\right\|\leq \sup_{\|x_i\|=|v_i|} \left\|\sum A_ix_i\right\|=\sup_{\|x_i\|=|v_i|} \left\|A \begin{pmatrix} x_1\\ \vdots\end{pmatrix}\right\|\\\leq \sup_{\|x_i\|=|v_i|} \|A\| \left\|\begin{pmatrix} x_1\\ \vdots\end{pmatrix}\right\|= \|A\| \|v\|.$$