  In this paper, we obtain a Le Potier-type isomorphism theorem twisted with
multiplier submodule sheaves, which relates a holomorphic vector bundle endowed
with a strongly Nakano semipositive singular Hermitian metric to the
tautological line bundle with the induced metric. As applications, we obtain a
Koll\'ar-type injectivity theorem, a Nadel-type vanishing theorem, and a
singular holomorphic Morse inequality for holomorphic vector bundles and so on.
