Visible crew/equipment: When they are first reporting to the dockyard in the beginning, right after being waved through the gate, there are servicemen working on the ships. In one of the shots, a serviceman making sparks with a grinder for effect is wearing a white Adidas shoe on his left foot, with the logo fully visible on the heel. This is in the BluRay edition. (00:13:17)
Visible crew/equipment: Just as the SS-33 sets off we see the crew on board say "all ahead two thirds". As they say this the camera moves behind a ladder resulting in a camera shadow on the ladder. (00:20:50)
Answer: Technically he's not getting fired, but Dahlgren did not recommend Tyler for his own command, and he won't get a promotion without Dahlgren's recommendation. Therefore, Tyler's only choices are to keep being an executive officer (which is possible but not what he wants, not to mention Tyler feels betrayed by Dahlgren and isn't sure he wants to continue working with him), or leave the navy entirely.
Aerinah