Is this the real reason why the great Library of Alexandria was destroyed?

Actually, the library, although losing valuable volumes, wasn't destroyed then. Nor was it destroyed when attacked by zealot monks 4 centuries later. Several historians told varying accounts of a Arab army led by Amr ibn al 'Aas sacking the city in 642 after the Byzantine army was defeated at the Battle of Heliopolis, and that the commander asked the caliph Umar what to do with the library. He gave the famous answer: "They will either contradict the Koran, in which case they are heresy, or they will agree with it, so they are superfluous." The Arabs subsequently burned the books to heat bathwater for the soldiers. However, this is likely a myth as well, with many of texts being saved by muslims, transcribed and transported to libraries in Spain, where they remained and were rediscovered by Christendom in 1492, when Ferdinand and Isabella completed the reconquest.