Not enough information.
You may be able to, and then again, you may not. I notice you say that your friend has a DV6000 series.. just like you have a DV6000 series, but you didn't get any more specific than that regarding his model. If the specific model your friend has is close enough to yours, in that it uses the same processor family and chipset family (if they both have Intel processors, you should be fine), then it should just be a matter of downloading any missing drivers after the recovery discs have done their thing... and their may not even BE any driver issues... if the models are close enough.
However, if the chipsets are different, you may end up hitting a BSOD during the recovery process, or on the first subsequent boot, etc.
If we knew your friend's specific model number, we could look up your model specs at HP, and his model specs at HP, and see. Note, you can see that even with the information you provided, we still can't get to the specific model information.... you didn't say whether you had a DV6833us, or DV6833tx.
If there are no hardware conflicts, then yes. You can just use your Vista product key after all is said and done. However, since the HP recovery discs use an image of the installed OS and installed hardware drivers (as opposed to how Dell does it, with a Windows installation disc, and a disc of drivers), there could possibly be hardware conflicts that would cause issues. If you did use his AND it's the same version of Vista AND there were no conflicts to the point where you could boot to safe mode to uninstall any conflicting drivers, and then reboot normally to install any missing drivers... then as I said... you could just use your Vista product key after all is said and done.