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Theorem lhpmat 33666
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 774 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
2 hlatl 32997 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
32ad2antrr 740 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  AtLat )
4 simprl 772 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
5 eqid 2471 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
6 lhpmat.h . . . . 5  |-  H  =  ( LHyp `  K
)
75, 6lhpbase 33634 . . . 4  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
87ad2antlr 741 . . 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 32954 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  W  e.  ( Base `  K
) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
143, 4, 8, 13syl3anc 1292 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( -.  P  .<_  W  <-> 
( P  ./\  W
)  =  .0.  )
)
151, 14mpbid 215 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   lecple 15275   meetcmee 16268   0.cp0 16361   Atomscatm 32900   AtLatcal 32901   HLchlt 32987   LHypclh 33620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  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 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-lat 16370  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-lhyp 33624
This theorem is referenced by:  lhpmatb  33667  lhp2at0  33668  lhpelim  33673  lhple  33678  idltrn  33786  ltrnmw  33787  trl0  33807  cdleme0e  33854  cdleme2  33865  cdleme7c  33882  cdleme22d  33981  cdlemefrs29pre00  34033  cdlemefrs29bpre0  34034  cdlemefrs29cpre1  34036  cdleme32fva  34075  cdleme35d  34090  cdleme42ke  34123  cdlemeg46frv  34163  cdleme50trn3  34191  cdlemg2fv2  34238  cdlemg8a  34265  cdlemg10bALTN  34274  cdlemh2  34454  cdlemk9  34477  cdlemk9bN  34478  dia2dimlem1  34703  dihvalcqat  34878  dihjatc1  34950
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