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Theorem latlem12 14184
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 3391 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 939 . 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 14157 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  e.  B )
543adant3r3 1162 . . 3  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B
) )  ->  ( Y  ./\  Z )  e.  B )
6 latpos 14155 . . . 4  |-  ( K  e.  Lat  ->  K  e.  Poset )
7 latmle.l . . . . . 6  |-  .<_  =  ( le `  K )
82, 7, 3meetle 14134 . . . . 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 192 . . . 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 1215 . . 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 1278 . 2  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B
) )  ->  (
( X  .<_  Y  /\  X  .<_  Z )  <->  X  .<_  ( Y  ./\  Z )
) )
121, 11sylan2b 461 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   Posetcpo 14074   meetcmee 14079   Latclat 14151
This theorem is referenced by:  latleeqm1  14185  latmlem1  14187  latmidm  14192  latledi  14195  mod1ile  14211  oldmm1  29407  olm01  29426  cmtbr4N  29445  atnle  29507  atlatmstc  29509  hlrelat2  29592  cvrval5  29604  cvrexchlem  29608  2atjm  29634  atbtwn  29635  ps-2b  29671  2atm  29716  2llnm4  29759  2llnmeqat  29760  dalemcea  29849  dalem21  29883  dalem54  29915  dalem55  29916  dalem57  29918  2atm2atN  29974  2llnma1b  29975  cdlemblem  29982  dalawlem2  30061  dalawlem3  30062  dalawlem6  30065  dalawlem11  30070  dalawlem12  30071  lhpocnle  30205  lhpmcvr4N  30215  lhpat3  30235  4atexlemcnd  30261  lautm  30283  trlval3  30376  cdlemc5  30384  cdleme3  30426  cdleme7ga  30437  cdleme7  30438  cdleme11k  30457  cdleme16e  30471  cdleme16f  30472  cdlemednpq  30488  cdleme22aa  30528  cdleme22b  30530  cdleme22cN  30531  cdleme23c  30540  cdlemeg46req  30718  cdlemf2  30751  cdlemg10c  30828  cdlemg12f  30837  cdlemg17dALTN  30853  cdlemg19a  30872  cdlemg27b  30885  cdlemi  31009  cdlemk15  31044  cdlemk50  31141  dia2dimlem1  31254  dihopelvalcpre  31438  dihord5b  31449  dihmeetlem1N  31480  dihglblem5apreN  31481  dihglblem2N  31484  dihmeetlem3N  31495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-glb 14109  df-meet 14111  df-lat 14152
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