Theorem latmcom

Description: The join of a lattice commutes. (Contributed by NM, 6-Nov-2011)

	Ref	Expression
Hypotheses	latmcom.b	|- B = ( Base ` K )
	latmcom.m	|- ./\ = ( meet ` K )
Assertion	latmcom	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( Y ./\ X ) )

Proof

Step	Hyp	Ref	Expression
1		latmcom.b	|- B = ( Base ` K )
2		latmcom.m	|- ./\ = ( meet ` K )
3		opelxpi	|- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
4	3	3adant1	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
5		eqid	|- ( join ` K ) = ( join ` K )
6	1 5 2	islat	|- ( K e. Lat <-> ( K e. Poset /\ ( dom ( join ` K ) = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) )
7		simprr	|- ( ( K e. Poset /\ ( dom ( join ` K ) = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) -> dom ./\ = ( B X. B ) )
8	6 7	sylbi	|- ( K e. Lat -> dom ./\ = ( B X. B ) )
9	8	3ad2ant1	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> dom ./\ = ( B X. B ) )
10	4 9	eleqtrrd	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. X , Y >. e. dom ./\ )
11		opelxpi	|- ( ( Y e. B /\ X e. B ) -> <. Y , X >. e. ( B X. B ) )
12	11	ancoms	|- ( ( X e. B /\ Y e. B ) -> <. Y , X >. e. ( B X. B ) )
13	12	3adant1	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. Y , X >. e. ( B X. B ) )
14	13 9	eleqtrrd	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. Y , X >. e. dom ./\ )
15	10 14	jca	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( <. X , Y >. e. dom ./\ /\ <. Y , X >. e. dom ./\ ) )
16		latpos	|- ( K e. Lat -> K e. Poset )
17	1 2	meetcom	|- ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ ( <. X , Y >. e. dom ./\ /\ <. Y , X >. e. dom ./\ ) ) -> ( X ./\ Y ) = ( Y ./\ X ) )
18	16 17	syl3anl1	|- ( ( ( K e. Lat /\ X e. B /\ Y e. B ) /\ ( <. X , Y >. e. dom ./\ /\ <. Y , X >. e. dom ./\ ) ) -> ( X ./\ Y ) = ( Y ./\ X ) )
19	15 18	mpdan	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( Y ./\ X ) )