It's not a matter of assumption, and Paul wasn't refering to some thing that had never been referred to before. "The law" consistently refers to the pentateuch/torah in both the OT and the NT. Paul didn't slice it up and distinguish a particular part of the law but used the distinctives of circumcision and diet as shorthand for the whole law.
Paul's language has to be understood with the Old Testament context in mind and the law was never separated into parts, it was one body meant to be followed as a whole. There is no Scripture that disagrees with what I am presenting on this, just Scripture you fail to understand.
And yes, the 613 laws include the 10 commandments. But the New Testament writers re-establish 9 of the 10 commandments in their writings as part of Christ's law that supercedes the old covenant law.
Galatians, Colossians, and Hebrews all state the old covenant has been put away as does Romans when properly understood. Then there is statements like the one Paul made in Timothy about the law " understanding this, that the law is not laid down for the just but for the lawless and disobedient, for the ungodly and sinners, for the unholy and profane, for those who strike their fathers and mothers, for murderers, 10 the sexually immoral, men who practice homosexuality, enslavers,[
a] liars, perjurers, and whatever else is contrary to sound[
b] doctrine, 11 in accordance with the gospel of the glory of the blessed God with which I have been entrusted. "
So the question is are you one of the righteous that lives by faith, or are you lawless and disobedient needing a law to condemn you?