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	$$⊢B={\mathrm{Base}}_K$$
	cmtid.c	$$⊢C=\mathrm{cm}\left(K\right)$$
Assertion	cmtidN	$$⊢\left(K\in \mathrm{OML}\wedge X\in B\right)\to XCX$$

Proof

Step	Hyp	Ref	Expression
1		cmtid.b	$$⊢B={\mathrm{Base}}_K$$
2		cmtid.c	$$⊢C=\mathrm{cm}\left(K\right)$$
3		omllat	$$⊢K\in \mathrm{OML}\to K\in \mathrm{Lat}$$
4		eqid	$$⊢{\le }_K={\le }_K$$
5	1 4	latref	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\right)\to X{\le }_{K}X$$
6	3 5	sylan	$$⊢\left(K\in \mathrm{OML}\wedge X\in B\right)\to X{\le }_{K}X$$
7	1 4 2	lecmtN	$$⊢\left(K\in \mathrm{OML}\wedge X\in B\wedge X\in B\right)\to \left(X{\le }_{K}X\to XCX\right)$$
8	7	3anidm23	$$⊢\left(K\in \mathrm{OML}\wedge X\in B\right)\to \left(X{\le }_{K}X\to XCX\right)$$
9	6 8	mpd	$$⊢\left(K\in \mathrm{OML}\wedge X\in B\right)\to XCX$$