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Theorem latlem12 14434
Description: An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (ssin 3506 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
latmle.b  |-  B  =  ( Base `  K
)
latmle.l  |-  .<_  =  ( le `  K )
latmle.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latlem12  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<_  Y  /\  X  .<_  Z )  <->  X  .<_  ( Y  ./\  Z )
) )

Proof of Theorem latlem12
StepHypRef Expression
1 3anrot 941 . 2  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  <->  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B )
)
2 latmle.b . . . . 5  |-  B  =  ( Base `  K
)
3 latmle.m . . . . 5  |-  ./\  =  ( meet `  K )
42, 3latmcl 14407 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  e.  B )
543adant3r3 1164 . . 3  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B
) )  ->  ( Y  ./\  Z )  e.  B )
6 latpos 14405 . . . 4  |-  ( K  e.  Lat  ->  K  e.  Poset )
7 latmle.l . . . . . 6  |-  .<_  =  ( le `  K )
82, 7, 3meetle 14384 . . . . 5  |-  ( ( K  e.  Poset  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B )  /\  ( Y  ./\  Z
)  e.  B )  ->  ( X  .<_  ( Y  ./\  Z )  <->  ( X  .<_  Y  /\  X  .<_  Z ) ) )
98bicomd 193 . . . 4  |-  ( ( K  e.  Poset  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B )  /\  ( Y  ./\  Z
)  e.  B )  ->  ( ( X 
.<_  Y  /\  X  .<_  Z )  <->  X  .<_  ( Y 
./\  Z ) ) )
106, 9syl3an1 1217 . . 3  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B
)  /\  ( Y  ./\ 
Z )  e.  B
)  ->  ( ( X  .<_  Y  /\  X  .<_  Z )  <->  X  .<_  ( Y  ./\  Z )
) )
115, 10mpd3an3 1280 . 2  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B
) )  ->  (
( X  .<_  Y  /\  X  .<_  Z )  <->  X  .<_  ( Y  ./\  Z )
) )
121, 11sylan2b 462 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<_  Y  /\  X  .<_  Z )  <->  X  .<_  ( Y  ./\  Z )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   Posetcpo 14324   meetcmee 14329   Latclat 14401
This theorem is referenced by:  latleeqm1  14435  latmlem1  14437  latmidm  14442  latledi  14445  mod1ile  14461  oldmm1  29332  olm01  29351  cmtbr4N  29370  atnle  29432  atlatmstc  29434  hlrelat2  29517  cvrval5  29529  cvrexchlem  29533  2atjm  29559  atbtwn  29560  ps-2b  29596  2atm  29641  2llnm4  29684  2llnmeqat  29685  dalemcea  29774  dalem21  29808  dalem54  29840  dalem55  29841  dalem57  29843  2atm2atN  29899  2llnma1b  29900  cdlemblem  29907  dalawlem2  29986  dalawlem3  29987  dalawlem6  29990  dalawlem11  29995  dalawlem12  29996  lhpocnle  30130  lhpmcvr4N  30140  lhpat3  30160  4atexlemcnd  30186  lautm  30208  trlval3  30301  cdlemc5  30309  cdleme3  30351  cdleme7ga  30362  cdleme7  30363  cdleme11k  30382  cdleme16e  30396  cdleme16f  30397  cdlemednpq  30413  cdleme22aa  30453  cdleme22b  30455  cdleme22cN  30456  cdleme23c  30465  cdlemeg46req  30643  cdlemf2  30676  cdlemg10c  30753  cdlemg12f  30762  cdlemg17dALTN  30778  cdlemg19a  30797  cdlemg27b  30810  cdlemi  30934  cdlemk15  30969  cdlemk50  31066  dia2dimlem1  31179  dihopelvalcpre  31363  dihord5b  31374  dihmeetlem1N  31405  dihglblem5apreN  31406  dihglblem2N  31409  dihmeetlem3N  31420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-glb 14359  df-meet 14361  df-lat 14402
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