This claim is trivially true by everyday observation of English pronunciation, so no citation is needed.
The claim states an elementary linguistic fact about English orthography: the letter combination 'oo' is pronounced as a short /ʊ/ in 'poor' (or /ɔː/), a long /ɔː/ in 'door', and a long /uː/ in 'doom'. Because this is a direct, undeniable everyday observation of English words, it falls under COMMON_KNOWLEDGE per STEP 0.