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Theorem lhpmatb 30828
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 30827 . . 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 30158 . . . . . 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 30106 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  .0.  )
1211necomd 2687 . . . . 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 2609 . . . . 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 30161 . . . . . 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 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2019, 4atbase 30087 . . . . . 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 30795 . . . . . 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 14508 . . . . 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 2635 . . 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 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   meetcmee 14402   0.cp0 14466   Latclat 14474   Atomscatm 30061   AtLatcal 30062   HLchlt 30148   LHypclh 30781
This theorem is referenced by:  cdlemh  31614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-glb 14432  df-meet 14434  df-p0 14468  df-lat 14475  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785
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