Cross Browser Mouse Wheel Event Handler

I'm trying to write a cross-browser (IE and firefox) event handler for
the mousewheel. Basically my web app is an image viewer, so if you
scroll "down" with the wheel the next image should be displayed and if
you scroll "up" the previous image should be displayed.

So far I have:

function handleMouseWheel(e)
{
if (!e) e = window.event;

if ( e.wheelDelta <= 0 || e.detail > 0) { changeImage(1); }
else { changeImage(-1); }
}

This code works in IE and in Firefox to the desired effect. The
problem I am having is with registering the event handler.

The following works for IE:
document.onmousewheel = handleMouseWheel;

And the following works for Firefox:
window.addEventListener("DOMMouseScroll", handleMouseWheel, false);

But when I add the firefox event handler to my code, IE chokes with an
error "Object doesn't support this property or method".

What do I need to do to "hide" the firefox event handler code from IE?