Device plugged into mains

a tribunal
you are confusing it with a battery charger; it's not a battery charger, but a direct AC power supply for the camera
the two are on sale and serve distinct purposes
and on the D5100, this AC power supply is shaped like the battery and fits in place of the battery, SO IT'S IMPOSSIBLE TO LEAVE THE BATTERY IN
you need to know the functions of the accessories before being assertive

no, I read that at Nikon eh 5b
AC adapter providing continuous power supply in place of batteries/rechargeable batteries.
and therefore without the batteries

Hello
Of course, if you have a charger (power adapter), it's for charging your batteries.
Otherwise, how do you expect to use your device once the batteries are discharged?
So when plugged in, your batteries will always be fully charged if you need to take your device with you.

Who loves instruction! Find the knowledge that comes from reflection!
P.S. Often create a restore point! It solves quite a few problems.

A battery charger delivers a voltage in frequency and peak above the battery voltage value, which allows for charging.
A direct power supply provides a continuous voltage equal to that of a battery; that's the difference, and it cannot be an effective battery charger.