@goldenways
Strictly, the OP is not using batteries to directly feed his equipment. He is using an inverter to convert the battery DC to mains-like AC. Almost certainly the inverter will contain a switched mode power supply, especially as it is a "pure sine-wave" inverter.
I have a set up like this in my motorhome. Interestingly the 2000-Watt sine-wave inverter I use will synchronize to mains frequency within a couple of cycles if mains is connected.
My lithium batteries are nominally 12-Volt but the actual voltage ranges anywhere from 14.7 if the sun is shining on my solar panels to about 11 just before the batteries shut themselves down. I hate to think how many switched mode power supplies are used inside my components to deliver the various voltages actually needed!