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Theorem lhpmatb 35183
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 35182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  .0.  )
76anassrs 648 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  -.  P  .<_  W )  -> 
( P  ./\  W
)  =  .0.  )
8 hlatl 34513 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
98ad3antrrr 729 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  K  e.  AtLat )
10 simplr 754 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  P  e.  A )
113, 4atn0 34461 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  .0.  )
1211necomd 2738 . . . . 5  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  .0.  =/=  P )
139, 10, 12syl2anc 661 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  .0.  =/=  P )
14 neeq1 2748 . . . . 5  |-  ( ( P  ./\  W )  =  .0.  ->  ( ( P  ./\  W )  =/= 
P  <->  .0.  =/=  P
) )
1514adantl 466 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  (
( P  ./\  W
)  =/=  P  <->  .0.  =/=  P ) )
1613, 15mpbird 232 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  ( P  ./\  W )  =/= 
P )
17 hllat 34516 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1817ad3antrrr 729 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  K  e.  Lat )
19 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2019, 4atbase 34442 . . . . . 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 35150 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2322ad3antlr 730 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  W  e.  ( Base `  K
) )
2419, 1, 2latleeqm1 15583 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  ( P  .<_  W  <->  ( P  ./\ 
W )  =  P ) )
2518, 21, 23, 24syl3anc 1228 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  ( P  .<_  W  <->  ( P  ./\ 
W )  =  P ) )
2625necon3bbid 2714 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =/=  P
) )
2716, 26mpbird 232 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  -.  P  .<_  W )
287, 27impbida 830 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A
)  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   meetcmee 15449   0.cp0 15541   Latclat 15549   Atomscatm 34416   AtLatcal 34417   HLchlt 34503   LHypclh 35136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-covers 34419  df-ats 34420  df-atl 34451  df-cvlat 34475  df-hlat 34504  df-lhyp 35140
This theorem is referenced by:  cdlemh  35969
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