RE: once a building is used for muslim worship, it becomes a mosque.
Is that an Islamic doctrine?
I tired to research this, and several articles say that a mosque must be a building that has been specially prepared for muslim prayer. So I may be wrong.
But we need to watch out! Both the Christian cathedral of John the Baptist in Damascus and the Hagia Sophia in Constantinople became mosques—the first after a long time period, and the second instantly—as soon as Mehmed the Conquerer prayed in it.