MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  latleeqm2 Structured version   Unicode version

Theorem latleeqm2 14510
Description: Less-than-or-equal-to in terms of meet. (sseqin2 3561 analog.) (Contributed by NM, 7-Nov-2011.)
Hypotheses
Ref Expression
latmle.b  |-  B  =  ( Base `  K
)
latmle.l  |-  .<_  =  ( le `  K )
latmle.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latleeqm2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( Y  ./\ 
X )  =  X ) )

Proof of Theorem latleeqm2
StepHypRef Expression
1 latmle.b . . 3  |-  B  =  ( Base `  K
)
2 latmle.l . . 3  |-  .<_  =  ( le `  K )
3 latmle.m . . 3  |-  ./\  =  ( meet `  K )
41, 2, 3latleeqm1 14509 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )
51, 3latmcom 14505 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
65eqeq1d 2445 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y )  =  X  <->  ( Y  ./\ 
X )  =  X ) )
74, 6bitrd 246 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( Y  ./\ 
X )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   meetcmee 14403   Latclat 14475
This theorem is referenced by:  cmtcomlemN  30047  omlmod1i2N  30059  2llnma3r  30586  dalawlem7  30675  dalawlem11  30679  dalawlem12  30680  lhp2at0  30830  lhp2atnle  30831  cdleme9  31051  cdleme11g  31063  cdleme35c  31249  cdlemh1  31613  dia2dimlem2  31864  dia2dimlem3  31865  dihmeetlem15N  32120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-glb 14433  df-meet 14435  df-lat 14476
  Copyright terms: Public domain W3C validator