Does the prayer of a follower become invalid if he bows and then prostrates, thinking that the Imam has prostrated for a recitation, then realizes his mistake and completes his prayer with the Imam?

If your statement "Then I stood up with him, and completed my prayer" means that you stood up from bowing, hunching until you reached the bowing limit, and then stood up, or that you went back to standing directly and then prostrated, then your prayer is valid.

Shaykh Muhammad Ulish al-Maliki mentioned in "Manh al-Jalil" that whoever omits rising from bowing should return hunching until he reaches the bowing limit, then rise with the intention of rising, or return to standing with the intention of rising from bowing and prostrate from his standing position.