Sum of many dilates

We show that for any coprime integers ${\lambda }_1,\dots ,{\lambda }_k$ and any finite $A\subset \mathbb{Z}$, one has $$\mathrm{\mid }{\lambda }_1\cdot A+\dots +{\lambda }_k\cdot A\mathrm{\mid }\ge (\mathrm{\mid }{\lambda }_1\mathrm{\mid }+\dots +\mathrm{\mid }{\lambda }_k\mathrm{\mid })\mathrm{\mid }A\mathrm{\mid }-C,$$ where $C$ only depends on ${\lambda }_1,\dots ,{\lambda }_k$.