MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  latlem12 Structured version   Unicode version

Theorem latlem12 15907
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. (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 latmle.b . 2  |-  B  =  ( Base `  K
)
2 latmle.l . 2  |-  .<_  =  ( le `  K )
3 latmle.m . 2  |-  ./\  =  ( meet `  K )
4 latpos 15879 . . 3  |-  ( K  e.  Lat  ->  K  e.  Poset )
54adantr 463 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Poset )
6 simpr2 1001 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
7 simpr3 1002 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
8 simpr1 1000 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
9 eqid 2454 . . . 4  |-  ( join `  K )  =  (
join `  K )
10 simpl 455 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
111, 9, 3, 10, 6, 7latcl2 15877 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( <. Y ,  Z >.  e. 
dom  ( join `  K
)  /\  <. Y ,  Z >.  e.  dom  ./\  )
)
1211simprd 461 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  <. Y ,  Z >.  e.  dom  ./\  )
131, 2, 3, 5, 6, 7, 8, 12meetle 15857 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<_  Y  /\  X  .<_  Z )  <->  X  .<_  ( Y  ./\  Z )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   <.cop 4022   class class class wbr 4439   dom cdm 4988   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   Posetcpo 15768   joincjn 15772   meetcmee 15773   Latclat 15874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-poset 15774  df-glb 15804  df-meet 15806  df-lat 15875
This theorem is referenced by:  latleeqm1  15908  latmlem1  15910  latmidm  15915  latledi  15918  mod1ile  15934  oldmm1  35339  olm01  35358  cmtbr4N  35377  atnle  35439  atlatmstc  35441  hlrelat2  35524  cvrval5  35536  cvrexchlem  35540  2atjm  35566  atbtwn  35567  ps-2b  35603  2atm  35648  2llnm4  35691  2llnmeqat  35692  dalemcea  35781  dalem21  35815  dalem54  35847  dalem55  35848  dalem57  35850  2atm2atN  35906  2llnma1b  35907  cdlemblem  35914  dalawlem2  35993  dalawlem3  35994  dalawlem6  35997  dalawlem11  36002  dalawlem12  36003  lhpocnle  36137  lhpmcvr4N  36147  lhpat3  36167  4atexlemcnd  36193  lautm  36215  trlval3  36309  cdlemc5  36317  cdleme3  36359  cdleme7ga  36370  cdleme7  36371  cdleme11k  36390  cdleme16e  36404  cdleme16f  36405  cdlemednpq  36421  cdleme22aa  36462  cdleme22b  36464  cdleme22cN  36465  cdleme23c  36474  cdlemeg46req  36652  cdlemf2  36685  cdlemg10c  36762  cdlemg12f  36771  cdlemg17dALTN  36787  cdlemg19a  36806  cdlemg27b  36819  cdlemi  36943  cdlemk15  36978  cdlemk50  37075  dia2dimlem1  37188  dihopelvalcpre  37372  dihord5b  37383  dihmeetlem1N  37414  dihglblem5apreN  37415  dihglblem2N  37418  dihmeetlem3N  37429
  Copyright terms: Public domain W3C validator