I am working with a product of matrices $ABC$ and the right kernel of this product. $A$ and $C$ are invertible matrices. Clearly then, if $\vec{v}$ is a an element of the null space of the product, then
$$ABC\vec{v}=0$$
gives
$$BC\vec{v}=0$$
because $A$ is invertible. Now, if we move into modular arithmetic, and suppose $A$, $B$ $C$ and $\vec{v}$ do not necessarily satisfy $ABC\vec{v}=0$ as above. If, however, we do have
$$ABC\vec{v}\equiv0\mod{n}$$
and thus
$$BC\vec{v}\equiv0\mod{n}\tag{Claim 1}$$
but
$$ABC\vec{v}\mod{n}=\\(A\mod n)(B\mod{n})(C\mod n)(\vec{v}\mod n)\equiv0\mod n\tag{Claim 2}$$
And thus from these claims (one of which is likely in error) it is clear that $A\mod{n}$ is invertible always.
I don't think in general an invertible matrix $A$ is always invertible $\mod{n}$ $\text{(Claim 3)}$, so what is my error here?