Polycarp testifies that Paul's teachings match the apostles.1
Ignatius testifies that Paul's teachings match the apostles.1
Clement of Rome testifies that Paul's teachings matched his own teaching. This is relevant Clement's teachings matched that of the apostles.
Contemporary Christians were saying that Paul “who once persecuted us is now preaching the faith which he once tried to destroy.” This is relevant because the faith that Paul was trying to destroy was apostolic Christianity (i.e. what the apostles were teaching).
Paul desired to have his teachings conform to the apostles' teachings.1 This is relevant because, if Paul wanted his teachings to conform to the apostles', it would have been very easy for him to do it.
But wait, couldn't it simply be that…
• …the Jerusalem churches' were not very well networked to Paul (i.e. their teachings were inaccessible to him), despite his desire?
No, …Gal 2 suggests Paul disputed with Peter over this (not submitting).
Paul believed that his teachings matched the Apostles, and that they were on the same team. This is relevant because Paul would not believe this if his teachings did not actually match the apostles.
The apostles at Jerusalem accepted Paul as a brother and co-worker.1
Paul was in the habit of submitting his teachings to the apostles This is relevant because the apostles would have corrected Paul's teachings if they were wrong.
Paul and his churches sent financial relief to the Jerusalem church. This is relevant because his churches were the Gentile churches, and Paul, and it was he who who collected the funds for.
Paul received his teachings from outside the Jerusalem church
David Garland: Paul reprises the opening verse (15:1) in 15:11. “Thus we preach” (κηρύσσομεν, kēryssomen) includes all of the apostles, and the present tense conveys that it continues to be their message. Christ’s resurrection is the common denominator on which all are in accord. It is nonnegotiable and cannot be jettisoned without gutting the Christian faith. [Baker Exegetical Commentary]
Peter disagreed with Peter over doctrine when Peter visited Antioch. This is relevant because it suggests Paul disagrees with Peter and James.
But so what? Couldn't it simply be that… …Peter granted Paul's admonishment
Even if Peter disagreed with Paul, it was over a minor issue that did not affect salvation?.; over the issue of table-fellowship with gentiles --Trivial disagreement: This was a disagreement over church politics [cf. [?]], it's not a topic the apostles would've been deemed an authority on (any more than their authority in farming techniques). Bart Ehrman: In Acts, Paul is portrayed as being in complete harmony with Peter and the other apostles; according to Paul, he had major disagreements with the Jerusalem apostles, especially Peter, in an ugly confrontation in the city of Antioch over significant implications of his gospel message 
"false brothers" (2 Cor 11:26; Gal 2:4; cf. 4:17; 5:12) "false apostles" (2 Cor 11:13) from Jerusalem