  Let $V\subseteq A$ be a conformal inclusion of vertex operator algebras and
let $\mathcal{C}$ be a category of grading-restricted generalized $V$-modules
that admits the vertex algebraic braided tensor category structure of
Huang-Lepowsky-Zhang. We give conditions under which $\mathcal{C}$ inherits
semisimplicity from the category of grading-restricted generalized $A$-modules
in $\mathcal{C}$, and vice versa. The most important condition is that $A$ be a
rigid $V$-module in $\mathcal{C}$ with non-zero categorical dimension, that is,
we assume the index of $V$ as a subalgebra of $A$ is finite and non-zero. As a
consequence, we show that if $A$ is strongly rational, then $V$ is also
strongly rational under the following conditions: $A$ contains $V$ as a
$V$-module direct summand, $V$ is $C_2$-cofinite with a rigid tensor category
of modules, and $A$ has non-zero categorical dimension as a $V$-module. These
results are vertex operator algebra interpretations of theorems proved for
general commutative algebras in braided tensor categories. We also generalize
these results to the case that $A$ is a vertex operator superalgebra.
