This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )