Quasirandom degree of group is bounded below by minimum of quasirandom degrees of generating subgroups

Statement in terms of quasirandom degrees
Suppose $$G$$ is a finite group that is generated by the union of subgroups $$G_1,G_2, \dots G_k$$ of $$G$$. Then, the quasirandom degree of $$G$$ is bounded by the minimum of the quasirandom degrees of $$G_1,G_2,\dots,G_k$$.

Statement in terms of $$D$$-quasirandom groups
Equivalently, if $$G_1,G_2,\dots,G_k$$ are all $$D$$-quasirandom groups for some positive integer $$D$$, then so is $$G$$.

Related facts

 * Quasirandom degree of quotient group is bounded below by quasirandom degree of whole group
 * Quasirandom degree of extension group is bounded below by minimum of quasirandom degrees of normal subgroup and quotient group

Proof
The key idea behind the proof is to note that if a representation restricts to the trivial representation on all the generating subgroups, it must be the trivial representation on the whole group.