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Metamath Proof Explorer

Theorem latjcom

Description: The join of a lattice commutes. ( chjcom analog.) (Contributed by NM, 16-Sep-2011)

Ref Expression
Hypotheses latjcom.b 
|- B = ( Base ` K )
latjcom.j 
|- .\/ = ( join ` K )
Assertion latjcom 
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) = ( Y .\/ X ) )

Proof

Step Hyp Ref Expression
1 latjcom.b 
 |-  B = ( Base ` K )
2 latjcom.j 
 |-  .\/ = ( join ` 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 
 |-  ( meet ` K ) = ( meet ` K )
6 1 2 5 islat 
 |-  ( K e. Lat <-> ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ( meet ` K ) = ( B X. B ) ) ) )
7 simprl 
 |-  ( ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ( meet ` K ) = ( 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 joincom 
 |-  ( ( ( 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 ) )