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Theorem lhpmatb 33673
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 33672 . . 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 33003 . . . . . 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 32951 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  .0.  )
1211necomd 2694 . . . . 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 2615 . . . . 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 33006 . . . . . 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 2442 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2019, 4atbase 32932 . . . . . 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 33640 . . . . . 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 15248 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  ( P  .<_  W  <->  ( P  ./\ 
W )  =  P ) )
2518, 21, 23, 24syl3anc 1218 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  /\  ( P  ./\  W )  =  .0.  )  ->  ( P  .<_  W  <->  ( P  ./\ 
W )  =  P ) )
2625necon3bbid 2641 . . 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 828 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 1369    e. wcel 1756    =/= wne 2605   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   Basecbs 14173   lecple 14244   meetcmee 15114   0.cp0 15206   Latclat 15214   Atomscatm 32906   AtLatcal 32907   HLchlt 32993   LHypclh 33626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-lat 15215  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-lhyp 33630
This theorem is referenced by:  cdlemh  34459
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