Theorem rngolcan

Description: Left cancellation law for the addition 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	rngolcan	|- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( C G A ) = ( C G B ) <-> A = B ) )

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	grpolcan	|- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( C G A ) = ( C G B ) <-> A = B ) )
5	3 4	sylan	|- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( C G A ) = ( C G B ) <-> A = B ) )