Heat pump physics question

Suppose we had a heat pump of COP x and we reversed it to an air conditioner.

We know that Qh = Qc + W

and

COPheat = Qh/W = (Qc + W)/W
COPcold = Qc/W

then
(W)(COPheat) = Qc + W

(W)(COPheat) - W = Qc

W(COPheat - 1) = Qc

COPheat - 1 = Qc/W = COPcold

Is my working correct and is this true for all heat pumps that can be reversed? Is COPheat always greater than COPcold by 1?

Yes.
COPheat= Qh/(Qh-Qc)
COPcold= Qc/(Qh -Qc)
which makes it easy to see that
COPheat -COPcold = 1