Lightning Digital AV Adapter - power and picture

When I plug in the Lightning adapter to my Gen 1 iPad Pro, the picture does not show up on the projector unless I also plug the power adapter into the lightning port on the adapter and then plug that into the wall. However, that does not charge the iPad so the iPad dies after a short period of time. My questions are: shouldn't the adapter work without being also plugged into power. Second, shouldn't the iPad itself charge when plugged in via the Digital AV Adapter?