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 ∈ OML ∧ 𝑋 ∈ 𝐵 ) → 𝑋 𝐶 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		cmtid.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cmtid.c	⊢ 𝐶 = ( cm ‘ 𝐾 )
3		omllat	⊢ ( 𝐾 ∈ OML → 𝐾 ∈ Lat )
4		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
5	1 4	latref	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
6	3 5	sylan	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
7	1 4 2	lecmtN	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑋 → 𝑋 𝐶 𝑋 ) )
8	7	3anidm23	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑋 → 𝑋 𝐶 𝑋 ) )
9	6 8	mpd	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ) → 𝑋 𝐶 𝑋 )