Theorem rngo0cl

Description: A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ring0cl.1	|- G = ( 1st ` R )
	ring0cl.2	|- X = ran G
	ring0cl.3	|- Z = ( GId ` G )
Assertion	rngo0cl	|- ( R e. RingOps -> Z e. X )

Proof

Step	Hyp	Ref	Expression
1		ring0cl.1	|- G = ( 1st ` R )
2		ring0cl.2	|- X = ran G
3		ring0cl.3	|- Z = ( GId ` G )
4	1	rngogrpo	|- ( R e. RingOps -> G e. GrpOp )
5	2 3	grpoidcl	|- ( G e. GrpOp -> Z e. X )
6	4 5	syl	|- ( R e. RingOps -> Z e. X )