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Theorem lhpocnel2 34815
Description: The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014.)
Hypotheses
Ref Expression
lhpocnel2.l  |-  .<_  =  ( le `  K )
lhpocnel2.a  |-  A  =  ( Atoms `  K )
lhpocnel2.h  |-  H  =  ( LHyp `  K
)
lhpocnel2.p  |-  P  =  ( ( oc `  K ) `  W
)
Assertion
Ref Expression
lhpocnel2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )

Proof of Theorem lhpocnel2
StepHypRef Expression
1 lhpocnel2.l . . 3  |-  .<_  =  ( le `  K )
2 eqid 2467 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
3 lhpocnel2.a . . 3  |-  A  =  ( Atoms `  K )
4 lhpocnel2.h . . 3  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 34814 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
6 lhpocnel2.p . . . 4  |-  P  =  ( ( oc `  K ) `  W
)
76eleq1i 2544 . . 3  |-  ( P  e.  A  <->  ( ( oc `  K ) `  W )  e.  A
)
86breq1i 4454 . . . 4  |-  ( P 
.<_  W  <->  ( ( oc
`  K ) `  W )  .<_  W )
98notbii 296 . . 3  |-  ( -.  P  .<_  W  <->  -.  (
( oc `  K
) `  W )  .<_  W )
107, 9anbi12i 697 . 2  |-  ( ( P  e.  A  /\  -.  P  .<_  W )  <-> 
( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
115, 10sylibr 212 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586   lecple 14555   occoc 14556   Atomscatm 34060   HLchlt 34147   LHypclh 34780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-lhyp 34784
This theorem is referenced by:  cdlemk56w  35769  diclspsn  35991  cdlemn3  35994  cdlemn4  35995  cdlemn4a  35996  cdlemn6  35999  cdlemn8  36001  cdlemn9  36002  cdlemn11a  36004  dihordlem7b  36012  dihopelvalcpre  36045  dihmeetlem1N  36087  dihglblem5apreN  36088  dihglbcpreN  36097  dihmeetlem4preN  36103  dihmeetlem13N  36116  dih1dimatlem0  36125  dih1dimatlem  36126  dihpN  36133  dihatexv  36135  dihjatcclem3  36217  dihjatcclem4  36218
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