Theorem rngoaddneg1

Description: Adding the negative in a ring gives zero. (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 )
	ringaddneg.4	|- Z = ( GId ` G )
Assertion	rngoaddneg1	|- ( ( R e. RingOps /\ A e. X ) -> ( A G ( N ` A ) ) = Z )

Proof

Step	Hyp	Ref	Expression
1		ringnegcl.1	|- G = ( 1st ` R )
2		ringnegcl.2	|- X = ran G
3		ringnegcl.3	|- N = ( inv ` G )
4		ringaddneg.4	|- Z = ( GId ` G )
5	1	rngogrpo	|- ( R e. RingOps -> G e. GrpOp )
6	2 4 3	grporinv	|- ( ( G e. GrpOp /\ A e. X ) -> ( A G ( N ` A ) ) = Z )
7	5 6	sylan	|- ( ( R e. RingOps /\ A e. X ) -> ( A G ( N ` A ) ) = Z )