What U. S document states the law for how Tyler became President?

John Tyler became president because he was the vice-president when the President died. The legal basis for his becoming president was the Us Constitution.

(The only question was whether he would actually become president or merely act as president amd carry out the necessary duties without actually being a real president. There was objection to his moving into the White House and at first no money was allocated for him to give state dinners or provide other necessary state functions.)