Yes you are, but understand he thinks he is right too, and for someone to say they were wrong, even if just to make someone feel better, is the hardest and scariest thing to ever do. And why not just say your sorry and hes right to calm him, the issue wont be swept away, once hes calmed down, we can get things on the right track and get things sorted out. Sometimes one person needs to be the bigger person and say they were wrong, even if its a lie, to calm the other one down, then they can sort things from there. I'm not saying for you to drop your argument, I'm saying for you to admit your wrong to calm him down, then pick up the argument again, but in a correct way.
And ignore the things that made you feel bad, it keeps your head leveled, and makes it so the corrupter can't hurt you.