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Theorem lhpmat 30841
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 733 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
2 hlatl 30172 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
32ad2antrr 706 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  AtLat )
4 simprl 732 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
5 eqid 2296 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
6 lhpmat.h . . . . 5  |-  H  =  ( LHyp `  K
)
75, 6lhpbase 30809 . . . 4  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
87ad2antlr 707 . . 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 30129 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  W  e.  ( Base `  K
) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
143, 4, 8, 13syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( -.  P  .<_  W  <-> 
( P  ./\  W
)  =  .0.  )
)
151, 14mpbid 201 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   meetcmee 14095   0.cp0 14159   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  lhpmatb  30842  lhp2at0  30843  lhpelim  30848  lhple  30853  idltrn  30961  trl0  30981  cdleme0e  31028  cdleme2  31039  cdleme7c  31056  cdleme22d  31154  cdlemefrs29pre00  31206  cdlemefrs29bpre0  31207  cdlemefrs29cpre1  31209  cdleme32fva  31248  cdleme35d  31263  cdleme42ke  31296  cdlemeg46frv  31336  cdleme50trn3  31364  cdlemg2fv2  31411  cdlemg8a  31438  cdlemg10bALTN  31447  cdlemh2  31627  cdlemk9  31650  cdlemk9bN  31651  dia2dimlem1  31876  dihvalcqat  32051  dihjatc1  32123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-glb 14125  df-meet 14127  df-p0 14161  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799
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