Multilinear mapping/Distributivity law/Fact/Proof/Exercise

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V_{1},\ldots ,V_{n}}$ and ${\displaystyle {}W}$ be vector spaces over ${\displaystyle {}K}$. Let

${\displaystyle \Phi \colon V_{1}\times \cdots \times V_{n}\longrightarrow W}$

be a multilinear mapping, and let ${\displaystyle {}v_{1j},\ldots ,v_{m_{j}j}\in V_{j}}$ and ${\displaystyle {}a_{ij}\in K}$. Show that

${\displaystyle \Phi {\left(\sum _{i=1}^{m_{1}}a_{i1}v_{i1},\ldots ,\sum _{i=1}^{m_{n}}a_{in}v_{in}\right)}=\sum _{(i_{1},\ldots ,i_{n})\in \{1,\ldots ,m_{1}\}\times \cdots \times \{1,\ldots ,m_{n}\}}a_{i_{1}1}\cdots a_{i_{n}n}\Phi {\left(v_{i_{1}1},\ldots ,v_{i_{n}n}\right)}\,}$

holds.