The stereo won't harm your inverter...I'm more worried about the inverter harming your stereo.
A power inverter doesn't put out the same pattern of power as what comes out of your wall...rather than being a sine wave or close-to-sine wave, it's actually a square wave. For most products, this doesn't really matter, but you need to remember that the power that goes in to your stereo is ultimately what makes the sound.
Inverter power is noisy and dirty, two things you do not want for a stereo. While it may work, and may produce sound, it is taking a horrible toll on the electronics in the stereo and might even produce a rather harsh sound.
I wouldn't recommend it, for any reason. If you need a 5.1 mobile solution, there are systems specifically designed to run off the 12V supply of your car.