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Theorem lhpmat 33983
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 756 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
2 hlatl 33314 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
32ad2antrr 725 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  AtLat )
4 simprl 755 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
5 eqid 2451 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
6 lhpmat.h . . . . 5  |-  H  =  ( LHyp `  K
)
75, 6lhpbase 33951 . . . 4  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
87ad2antlr 726 . . 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 33271 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  W  e.  ( Base `  K
) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
143, 4, 8, 13syl3anc 1219 . 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 369    = wceq 1370    e. wcel 1758   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   Basecbs 14285   lecple 14356   meetcmee 15226   0.cp0 15318   Atomscatm 33217   AtLatcal 33218   HLchlt 33304   LHypclh 33937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-lat 15327  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-lhyp 33941
This theorem is referenced by:  lhpmatb  33984  lhp2at0  33985  lhpelim  33990  lhple  33995  idltrn  34103  trl0  34123  cdleme0e  34170  cdleme2  34181  cdleme7c  34198  cdleme22d  34296  cdlemefrs29pre00  34348  cdlemefrs29bpre0  34349  cdlemefrs29cpre1  34351  cdleme32fva  34390  cdleme35d  34405  cdleme42ke  34438  cdlemeg46frv  34478  cdleme50trn3  34506  cdlemg2fv2  34553  cdlemg8a  34580  cdlemg10bALTN  34589  cdlemh2  34769  cdlemk9  34792  cdlemk9bN  34793  dia2dimlem1  35018  dihvalcqat  35193  dihjatc1  35265
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