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Based on what you're put in the facts then no, you wouldn't own the car. The dealer can only transfer what title they had and since they never actually got the title then they can't transfer it.
This is a defense to a contract known as a "mistake of fact" and, assuming that they could prove it if you went to court, then the judge would rule in their favor.
They do have to give you back all of the money that you spent or put down on the car, they cannot charge mileage, etc.. If they want out of the contract then they have to put you back into the same position you would have been in had you never signed the papers.