Theorem cmtidN

Description: Any element commutes with itself. ( cmidi analog.) (Contributed by NM, 6-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cmtid.b	|- B = ( Base ` K )
	cmtid.c	|- C = ( cm ` K )
Assertion	cmtidN	|- ( ( K e. OML /\ X e. B ) -> X C X )

Proof

Step	Hyp	Ref	Expression
1		cmtid.b	|- B = ( Base ` K )
2		cmtid.c	|- C = ( cm ` K )
3		omllat	|- ( K e. OML -> K e. Lat )
4		eqid	|- ( le ` K ) = ( le ` K )
5	1 4	latref	|- ( ( K e. Lat /\ X e. B ) -> X ( le ` K ) X )
6	3 5	sylan	|- ( ( K e. OML /\ X e. B ) -> X ( le ` K ) X )
7	1 4 2	lecmtN	|- ( ( K e. OML /\ X e. B /\ X e. B ) -> ( X ( le ` K ) X -> X C X ) )
8	7	3anidm23	|- ( ( K e. OML /\ X e. B ) -> ( X ( le ` K ) X -> X C X ) )
9	6 8	mpd	|- ( ( K e. OML /\ X e. B ) -> X C X )