Theorem addcompi

Description: Addition of positive integers is commutative. (Contributed by NM, 27-Aug-1995) (New usage is discouraged.)

	Ref	Expression
Assertion	addcompi	|- ( A +N B ) = ( B +N A )

Proof

Step	Hyp	Ref	Expression
1		pinn	|- ( A e. N. -> A e. _om )
2		pinn	|- ( B e. N. -> B e. _om )
3		nnacom	|- ( ( A e. _om /\ B e. _om ) -> ( A +o B ) = ( B +o A ) )
4	1 2 3	syl2an	|- ( ( A e. N. /\ B e. N. ) -> ( A +o B ) = ( B +o A ) )
5		addpiord	|- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( A +o B ) )
6		addpiord	|- ( ( B e. N. /\ A e. N. ) -> ( B +N A ) = ( B +o A ) )
7	6	ancoms	|- ( ( A e. N. /\ B e. N. ) -> ( B +N A ) = ( B +o A ) )
8	4 5 7	3eqtr4d	|- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( B +N A ) )
9		dmaddpi	|- dom +N = ( N. X. N. )
10	9	ndmovcom	|- ( -. ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( B +N A ) )
11	8 10	pm2.61i	|- ( A +N B ) = ( B +N A )