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Theorem lhpocnel2 34002
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 2454 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
3 lhpocnel2.a . . 3  |-  A  =  ( Atoms `  K )
4 lhpocnel2.h . . 3  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 34001 . 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 2531 . . 3  |-  ( P  e.  A  <->  ( ( oc `  K ) `  W )  e.  A
)
86breq1i 4408 . . . 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 1370    e. wcel 1758   class class class wbr 4401   ` cfv 5527   lecple 14365   occoc 14366   Atomscatm 33247   HLchlt 33334   LHypclh 33967
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-lhyp 33971
This theorem is referenced by:  cdlemk56w  34956  diclspsn  35178  cdlemn3  35181  cdlemn4  35182  cdlemn4a  35183  cdlemn6  35186  cdlemn8  35188  cdlemn9  35189  cdlemn11a  35191  dihordlem7b  35199  dihopelvalcpre  35232  dihmeetlem1N  35274  dihglblem5apreN  35275  dihglbcpreN  35284  dihmeetlem4preN  35290  dihmeetlem13N  35303  dih1dimatlem0  35312  dih1dimatlem  35313  dihpN  35320  dihatexv  35322  dihjatcclem3  35404  dihjatcclem4  35405
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