people who are completely brain dead can be kept alive on life support for many years. The bodily functions do not require the brain when life support is turned on.
I'm really just being cheeky here... but i think they call it a 'vegetative state'
Being in a vegetative state does not equate to being brain dead. It equates to the higher functioning of the brain being lost or impeded in some way. The lower functions, the autonomic stuff, still functions. If the brain stops working completely, then body dies because there is nothing to tell the brain to beat, the lungs to contract, ect, etc.