Yes, Lucifer is Satan.
God cast him out of heaven like lightning for wanting to usurp God's authority!
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Luke 10:18-20
18 And He said to them, “I saw Satan fall like lightning from heaven. 19 Behold, I give you the authority to trample on serpents and scorpions, and over all the power of the enemy, and nothing shall by any means hurt you. 20 Nevertheless do not rejoice in this, that the spirits are subject to you, but [a]rather rejoice because your names are written in heaven.”
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