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Check your 4047 ic,it might b fault or your MOSFETs.because that's exactly what happened when I started building inverters
To know how long the battery would last, multiply your battery volt and amps together and the divide by your total load valu(I.e load of 100w 12v * 7ah/100w.°. 84/100=0.84hours)
U can use a step down transformer just use the 12 side or 9v side and connect it to the MOSFET and that's all
5a 220v transformer is more than enough to power ur 15w bulb