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Theorem latlem12 14200
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 3404 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 14173 . . . 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 14171 . . . 4  |-  ( K  e.  Lat  ->  K  e.  Poset )
7 latmle.l . . . . . 6  |-  .<_  =  ( le `  K )
82, 7, 3meetle 14150 . . . . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   Posetcpo 14090   meetcmee 14095   Latclat 14167
This theorem is referenced by:  latleeqm1  14201  latmlem1  14203  latmidm  14208  latledi  14211  mod1ile  14227  oldmm1  30029  olm01  30048  cmtbr4N  30067  atnle  30129  atlatmstc  30131  hlrelat2  30214  cvrval5  30226  cvrexchlem  30230  2atjm  30256  atbtwn  30257  ps-2b  30293  2atm  30338  2llnm4  30381  2llnmeqat  30382  dalemcea  30471  dalem21  30505  dalem54  30537  dalem55  30538  dalem57  30540  2atm2atN  30596  2llnma1b  30597  cdlemblem  30604  dalawlem2  30683  dalawlem3  30684  dalawlem6  30687  dalawlem11  30692  dalawlem12  30693  lhpocnle  30827  lhpmcvr4N  30837  lhpat3  30857  4atexlemcnd  30883  lautm  30905  trlval3  30998  cdlemc5  31006  cdleme3  31048  cdleme7ga  31059  cdleme7  31060  cdleme11k  31079  cdleme16e  31093  cdleme16f  31094  cdlemednpq  31110  cdleme22aa  31150  cdleme22b  31152  cdleme22cN  31153  cdleme23c  31162  cdlemeg46req  31340  cdlemf2  31373  cdlemg10c  31450  cdlemg12f  31459  cdlemg17dALTN  31475  cdlemg19a  31494  cdlemg27b  31507  cdlemi  31631  cdlemk15  31666  cdlemk50  31763  dia2dimlem1  31876  dihopelvalcpre  32060  dihord5b  32071  dihmeetlem1N  32102  dihglblem5apreN  32103  dihglblem2N  32106  dihmeetlem3N  32117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-glb 14125  df-meet 14127  df-lat 14168
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