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Theorem lhpmat 36151
Description: An element covered by the lattice unit, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.)
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
lhpmat  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  .0.  )

Proof of Theorem lhpmat
StepHypRef Expression
1 simprr 755 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
2 hlatl 35482 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
32ad2antrr 723 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  AtLat )
4 simprl 754 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
5 eqid 2454 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
6 lhpmat.h . . . . 5  |-  H  =  ( LHyp `  K
)
75, 6lhpbase 36119 . . . 4  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
87ad2antlr 724 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K ) )
9 lhpmat.l . . . 4  |-  .<_  =  ( le `  K )
10 lhpmat.m . . . 4  |-  ./\  =  ( meet `  K )
11 lhpmat.z . . . 4  |-  .0.  =  ( 0. `  K )
12 lhpmat.a . . . 4  |-  A  =  ( Atoms `  K )
135, 9, 10, 11, 12atnle 35439 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  W  e.  ( Base `  K
) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
143, 4, 8, 13syl3anc 1226 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( -.  P  .<_  W  <-> 
( P  ./\  W
)  =  .0.  )
)
151, 14mpbid 210 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 367    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   meetcmee 15773   0.cp0 15866   Atomscatm 35385   AtLatcal 35386   HLchlt 35472   LHypclh 36105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-lhyp 36109
This theorem is referenced by:  lhpmatb  36152  lhp2at0  36153  lhpelim  36158  lhple  36163  idltrn  36271  ltrnmw  36272  trl0  36292  cdleme0e  36339  cdleme2  36350  cdleme7c  36367  cdleme22d  36466  cdlemefrs29pre00  36518  cdlemefrs29bpre0  36519  cdlemefrs29cpre1  36521  cdleme32fva  36560  cdleme35d  36575  cdleme42ke  36608  cdlemeg46frv  36648  cdleme50trn3  36676  cdlemg2fv2  36723  cdlemg8a  36750  cdlemg10bALTN  36759  cdlemh2  36939  cdlemk9  36962  cdlemk9bN  36963  dia2dimlem1  37188  dihvalcqat  37363  dihjatc1  37435
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