Theorem rngogcl

Description: Closure law for the addition (group) operation of a ring. (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	rngogcl	|- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A G B ) e. X )

Proof

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