Out then safe then out...

[MatthewPV]

Situation (major league rules): 2 outs. ground ball to 2nd. Overthrown to first. Runner attempts to advance to 2nd. Slides and umpire calls the runner out. End of inning...but the umpire finally sees that the ball is on the ground. In the meantime the runner has taken 3-4 steps toward dugout. Umpire then declares runner safe. Fielder then tags stranded runner then umpire calls him out. My question is...are there any rules that the umpire messed up? It was suggested that the arguement could be made that the ball automatically becomes dead when the 3rd out is declared, and after the umpire sees his mistake he can award the runner second base. Any help would be helpful. Thanks!