Actually it's more important not to lie about the mistakes you have made. Years ago, in my first IT job, I screwed up. My boss asked me if I had done XYZ and I replied that I had. She commented that because I had told the truth there wasn't an issue but had I lied then I would have been fired.

For something like this, firing the GA isn't really going to achieve anything. It certainly won't stop the mistake happening again. Reviewing the processes or human factors that led to the wrong door being opened would be more productive, or a period of retraining or working under supervision for the GA. Working under a culture of fear, fearing that any simple screw-up will cost you your job isn't a particularly effective management strategy. We're all human. We're not infallible. We all make mistakes. How we handle those mistakes is what defines us.