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Theorem latmcom 15551
Description: The join of a lattice commutes. (Contributed by NM, 6-Nov-2011.)
Hypotheses
Ref Expression
latmcom.b  |-  B  =  ( Base `  K
)
latmcom.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latmcom  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )

Proof of Theorem latmcom
StepHypRef Expression
1 opelxpi 5023 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
213adant1 1009 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
3 latmcom.b . . . . . . 7  |-  B  =  ( Base `  K
)
4 eqid 2460 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
5 latmcom.m . . . . . . 7  |-  ./\  =  ( meet `  K )
63, 4, 5islat 15523 . . . . . 6  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  ( dom  ( join `  K )  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) ) )
7 simprr 756 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( dom  ( join `  K
)  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) )  ->  dom  ./\  =  ( B  X.  B
) )
86, 7sylbi 195 . . . . 5  |-  ( K  e.  Lat  ->  dom  ./\  =  ( B  X.  B ) )
983ad2ant1 1012 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  dom  ./\  =  ( B  X.  B ) )
102, 9eleqtrrd 2551 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e. 
dom  ./\  )
11 opelxpi 5023 . . . . . 6  |-  ( ( Y  e.  B  /\  X  e.  B )  -> 
<. Y ,  X >.  e.  ( B  X.  B
) )
1211ancoms 453 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. Y ,  X >.  e.  ( B  X.  B
) )
13123adant1 1009 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  -> 
<. Y ,  X >.  e.  ( B  X.  B
) )
1413, 9eleqtrrd 2551 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  -> 
<. Y ,  X >.  e. 
dom  ./\  )
1510, 14jca 532 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( <. X ,  Y >.  e.  dom  ./\  /\  <. Y ,  X >.  e.  dom  ./\  ) )
16 latpos 15526 . . 3  |-  ( K  e.  Lat  ->  K  e.  Poset )
173, 5meetcom 15508 . . 3  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  ( <. X ,  Y >.  e.  dom  ./\  /\  <. Y ,  X >.  e.  dom  ./\  ) )  ->  ( X  ./\  Y )  =  ( Y  ./\  X
) )
1816, 17syl3anl1 1271 . 2  |-  ( ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  /\  ( <. X ,  Y >.  e.  dom  ./\  /\  <. Y ,  X >.  e.  dom  ./\  ) )  ->  ( X  ./\  Y )  =  ( Y  ./\  X
) )
1915, 18mpdan 668 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   <.cop 4026    X. cxp 4990   dom cdm 4992   ` cfv 5579  (class class class)co 6275   Basecbs 14479   Posetcpo 15416   joincjn 15420   meetcmee 15421   Latclat 15521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-glb 15451  df-meet 15453  df-lat 15522
This theorem is referenced by:  latleeqm2  15556  latmlem2  15558  latmlej21  15568  latmlej22  15569  mod2ile  15582  olm12  33900  latm12  33902  latm32  33903  latmrot  33904  olm02  33909  omllaw2N  33916  cmtcomlemN  33920  cmtbr3N  33926  omlfh1N  33930  omlmod1i2N  33932  omlspjN  33933  cvlcvrp  34012  intnatN  34078  cvrexch  34091  cvrat4  34114  2atjm  34116  1cvrat  34147  2at0mat0  34196  dalem4  34336  dalem56  34399  atmod2i1  34532  atmod2i2  34533  llnmod2i2  34534  atmod3i1  34535  atmod3i2  34536  llnexchb2lem  34539  dalawlem3  34544  dalawlem4  34545  dalawlem6  34547  dalawlem9  34550  dalawlem11  34552  dalawlem12  34553  dalawlem15  34556  lhpmcvr  34694  4atexlemc  34740  cdleme20zN  34972  cdleme20d  34983  cdleme20l  34993  cdleme20m  34994  cdlemg12  35321  cdlemg17  35348  cdlemg19  35355  cdlemg44a  35402  dihmeetlem17N  35995  dihmeetlem20N  35998  dihmeetALTN  35999
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