Trigonalizable/Direct sum/Fact

Let $V_1,\dots ,V_n$ denote finite-dimensional vector spaces over the field $K$, let

${\phi }_i:V_i⟶V_i$

denote linear mappings, and let

$\phi ={\phi }_1\times \cdots \times {\phi }_n:V_1\times \cdots \times V_n⟶V_1\times \cdots \times V_n$

denote the product mapping.

Then $\phi$ is

trigonalizable

if and only if this holds for all ${\phi }_i$.

Write a proof