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Theorem lhpmatb 30513
Description: An element covered by the lattice unit, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.)
Hypotheses
Ref Expression
lhpmat.l  |-  .<_  =  ( le `  K )
lhpmat.m  |-  ./\  =  ( meet `  K )
lhpmat.z  |-  .0.  =  ( 0. `  K )
lhpmat.a  |-  A  =  ( Atoms `  K )
lhpmat.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpmatb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A
)  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )

Proof of Theorem lhpmatb
StepHypRef Expression
1 lhpmat.l . . . 4  |-  .<_  =  ( le `  K )
2 lhpmat.m . . . 4  |-  ./\  =  ( meet `  K )
3 lhpmat.z . . . 4  |-  .0.  =  ( 0. `  K )
4 lhpmat.a . . . 4  |-  A  =  ( Atoms `  K )
5 lhpmat.h . . . 4  |-  H  =  ( LHyp `  K
)
61, 2, 3, 4, 5lhpmat 30512 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  .0.  )
76anassrs 630 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  -.  P  .<_  W )  -> 
( P  ./\  W
)  =  .0.  )
8 hlatl 29843 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
98ad3antrrr 711 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  K  e.  AtLat )
10 simplr 732 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  P  e.  A )
113, 4atn0 29791 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  .0.  )
1211necomd 2650 . . . . 5  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  .0.  =/=  P )
139, 10, 12syl2anc 643 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  .0.  =/=  P )
14 neeq1 2575 . . . . 5  |-  ( ( P  ./\  W )  =  .0.  ->  ( ( P  ./\  W )  =/= 
P  <->  .0.  =/=  P
) )
1514adantl 453 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  (
( P  ./\  W
)  =/=  P  <->  .0.  =/=  P ) )
1613, 15mpbird 224 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  ( P  ./\  W )  =/= 
P )
17 hllat 29846 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1817ad3antrrr 711 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  K  e.  Lat )
19 eqid 2404 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2019, 4atbase 29772 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2110, 20syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  P  e.  ( Base `  K
) )
2219, 5lhpbase 30480 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2322ad3antlr 712 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  W  e.  ( Base `  K
) )
2419, 1, 2latleeqm1 14463 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  ( P  .<_  W  <->  ( P  ./\ 
W )  =  P ) )
2518, 21, 23, 24syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  ( P  .<_  W  <->  ( P  ./\ 
W )  =  P ) )
2625necon3bbid 2601 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =/=  P
) )
2716, 26mpbird 224 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  -.  P  .<_  W )
287, 27impbida 806 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A
)  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   meetcmee 14357   0.cp0 14421   Latclat 14429   Atomscatm 29746   AtLatcal 29747   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdlemh  31299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-glb 14387  df-meet 14389  df-p0 14423  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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