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Theorem latlem12 14497
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 3555 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 14470 . . . 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 14468 . . . 4  |-  ( K  e.  Lat  ->  K  e.  Poset )
7 latmle.l . . . . . 6  |-  .<_  =  ( le `  K )
82, 7, 3meetle 14447 . . . . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13459   lecple 13526   Posetcpo 14387   meetcmee 14392   Latclat 14464
This theorem is referenced by:  latleeqm1  14498  latmlem1  14500  latmidm  14505  latledi  14508  mod1ile  14524  oldmm1  29916  olm01  29935  cmtbr4N  29954  atnle  30016  atlatmstc  30018  hlrelat2  30101  cvrval5  30113  cvrexchlem  30117  2atjm  30143  atbtwn  30144  ps-2b  30180  2atm  30225  2llnm4  30268  2llnmeqat  30269  dalemcea  30358  dalem21  30392  dalem54  30424  dalem55  30425  dalem57  30427  2atm2atN  30483  2llnma1b  30484  cdlemblem  30491  dalawlem2  30570  dalawlem3  30571  dalawlem6  30574  dalawlem11  30579  dalawlem12  30580  lhpocnle  30714  lhpmcvr4N  30724  lhpat3  30744  4atexlemcnd  30770  lautm  30792  trlval3  30885  cdlemc5  30893  cdleme3  30935  cdleme7ga  30946  cdleme7  30947  cdleme11k  30966  cdleme16e  30980  cdleme16f  30981  cdlemednpq  30997  cdleme22aa  31037  cdleme22b  31039  cdleme22cN  31040  cdleme23c  31049  cdlemeg46req  31227  cdlemf2  31260  cdlemg10c  31337  cdlemg12f  31346  cdlemg17dALTN  31362  cdlemg19a  31381  cdlemg27b  31394  cdlemi  31518  cdlemk15  31553  cdlemk50  31650  dia2dimlem1  31763  dihopelvalcpre  31947  dihord5b  31958  dihmeetlem1N  31989  dihglblem5apreN  31990  dihglblem2N  31993  dihmeetlem3N  32004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14393  df-glb 14422  df-meet 14424  df-lat 14465
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