Question

how does a heat pump prove the law of conservation of energy?

Answer 1

the heat pump basically draws some heat from the surrounding and puts it into the room.

Answer 2

The heat pump conveys the heat from an area to another against the temperature gradient that means from a cooler region to hotter. The above picture is of Carnot engine and Th=W+Tc where Qh & Qc are the heat energies. the W is the work done by the body and that is equal to Th-Tc. from this we can say that the heat is conserved.