  We prove a conjecture of Colmez concerning the reduction modulo $p$ of
invariant lattices in irreducible admissible unitary $p$-adic Banach space
representations of $GL_2(Q_p)$ with $p\ge 5$. This enables us to restate nicely
the $p$-adic local Langlands correspondence for $GL_2(Q_p)$ and deduce a
conjecture of Breuil on irreducible admissible unitary completions of locally
algebraic representations.
