Connect it up and switch on....
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The question isn't will it work off your batteries ... It is more a case of will it work when the engine is running or the shorepower charger is on when your battery voltage could rise to in excess of 15 volts!!!!!! You might be wise to get a dc / dc power supply from Maplins so that you do not make an expensive mistake.
It's probably more of a question of will it work when the voltage falls below 12 volts. Since you should never let the batteries fall below 12.2 volts then the TV switching itself off is a good indication that the batteries need charging.
Our 15" 12 volt Matsui from Currys takes much less AHrs than the computer.