There are two clear arguments that prove that dropping the nuclear bombs on Japan saved lives:

1. Had they not been dropped, or even existed, the cities that received them would have suffered damage equal to or heavier with equal or heavier casualties to those caused by the nuclear weapons in conventional bombing attacks. Both cities were only spared up to the time of their bombing so the USAAF could better assess the damage of a nuclear strike.
If nuclear weapons were not available, there was no reason to spare either city. Given that the destruction and casualties in a single night's raid on Toyko caused about 40% more casualties and destroyed about twice the area of either nuclear strike using conventional weapons there is no doubt that both Hiroshima and Nagasaki would have suffered equally or worse.

2. They precluded the need for an invasion. This too saved lives.

But, in any case arguing that the primary rational for dropping or not dropping nuclear weapons is the casualty rate is simply insane. One uses nuclear weapons, or any other weapon of war, for one reason alone: To achieve victory. Only some Liberal / Leftist touchy feely moron with absolutely no conception of history or the strategy of conflict would think otherwise.