Theorem rngocom

Description: The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ringgcl.1	|- G = ( 1st ` R )
	ringgcl.2	|- X = ran G
Assertion	rngocom	|- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A G B ) = ( B G A ) )

Proof

Step	Hyp	Ref	Expression
1		ringgcl.1	|- G = ( 1st ` R )
2		ringgcl.2	|- X = ran G
3	1	rngoablo	|- ( R e. RingOps -> G e. AbelOp )
4	2	ablocom	|- ( ( G e. AbelOp /\ A e. X /\ B e. X ) -> ( A G B ) = ( B G A ) )
5	3 4	syl3an1	|- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A G B ) = ( B G A ) )