Theorem rngonegcl

Description: A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010)

	Ref	Expression
Hypotheses	ringnegcl.1	|- G = ( 1st ` R )
	ringnegcl.2	|- X = ran G
	ringnegcl.3	|- N = ( inv ` G )
Assertion	rngonegcl	|- ( ( R e. RingOps /\ A e. X ) -> ( N ` A ) e. X )

Proof

Step	Hyp	Ref	Expression
1		ringnegcl.1	|- G = ( 1st ` R )
2		ringnegcl.2	|- X = ran G
3		ringnegcl.3	|- N = ( inv ` G )
4	1	rngogrpo	|- ( R e. RingOps -> G e. GrpOp )
5	2 3	grpoinvcl	|- ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) e. X )
6	4 5	sylan	|- ( ( R e. RingOps /\ A e. X ) -> ( N ` A ) e. X )