Theorem ringideu

Description: The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)

	Ref	Expression
Hypotheses	ringcl.b	|- B = ( Base ` R )
	ringcl.t	|- .x. = ( .r ` R )
Assertion	ringideu	|- ( R e. Ring -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )

Proof

Step	Hyp	Ref	Expression
1		ringcl.b	|- B = ( Base ` R )
2		ringcl.t	|- .x. = ( .r ` R )
3		eqid	|- ( mulGrp ` R ) = ( mulGrp ` R )
4	3	ringmgp	|- ( R e. Ring -> ( mulGrp ` R ) e. Mnd )
5	3 1	mgpbas	|- B = ( Base ` ( mulGrp ` R ) )
6	3 2	mgpplusg	|- .x. = ( +g ` ( mulGrp ` R ) )
7	5 6	mndideu	|- ( ( mulGrp ` R ) e. Mnd -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )
8	4 7	syl	|- ( R e. Ring -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )