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 ${\displaystyle {}v_{i1},\ldots ,v_{im_{i}}\in V_{i}}$ and ${\displaystyle {}a_{ij}\in K}$ for ${\displaystyle {}i=1,\ldots ,n}$ and ${\displaystyle {}j=1,\ldots ,m_{i}}$. Show that

${\displaystyle \Phi {\left(\sum _{j=1}^{m_{1}}a_{1j}v_{1j},\ldots ,\sum _{j=1}^{m_{n}}a_{nj}v_{nj}\right)}=\sum _{(j_{1},\ldots ,j_{n})\in \{1,\ldots ,m_{1}\}\times \cdots \times \{1,\ldots ,m_{n}\}}a_{1j_{1}}\cdots a_{nj_{n}}\Phi {\left(v_{1j_{1}},\ldots ,v_{nj_{n}}\right)}\,}$

holds.