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Theorem lhpocnle 30813
Description: The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
lhpocnle.l  |-  .<_  =  ( le `  K )
lhpocnle.o  |-  ._|_  =  ( oc `  K )
lhpocnle.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpocnle  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  -.  (  ._|_  `  W
)  .<_  W )

Proof of Theorem lhpocnle
StepHypRef Expression
1 hlatl 30158 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
21adantr 452 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  AtLat )
3 simpr 448 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  H )
4 eqid 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
5 lhpocnle.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
64, 5lhpbase 30795 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7 lhpocnle.o . . . . . . 7  |-  ._|_  =  ( oc `  K )
8 eqid 2436 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
94, 7, 8, 5lhpoc 30811 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  ( Base `  K ) )  -> 
( W  e.  H  <->  ( 
._|_  `  W )  e.  ( Atoms `  K )
) )
106, 9sylan2 461 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( W  e.  H  <->  ( 
._|_  `  W )  e.  ( Atoms `  K )
) )
113, 10mpbid 202 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  W )  e.  ( Atoms `  K
) )
12 eqid 2436 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
1312, 8atn0 30106 . . . 4  |-  ( ( K  e.  AtLat  /\  (  ._|_  `  W )  e.  ( Atoms `  K )
)  ->  (  ._|_  `  W )  =/=  ( 0. `  K ) )
142, 11, 13syl2anc 643 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  W )  =/=  ( 0. `  K ) )
1514neneqd 2617 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  -.  (  ._|_  `  W
)  =  ( 0.
`  K ) )
16 simpr 448 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  .<_  W )
17 hllat 30161 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1817ad2antrr 707 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  ->  K  e.  Lat )
19 hlop 30160 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
2019ad2antrr 707 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  ->  K  e.  OP )
216ad2antlr 708 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  ->  W  e.  ( Base `  K ) )
224, 7opoccl 29992 . . . . . . 7  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
(  ._|_  `  W )  e.  ( Base `  K
) )
2320, 21, 22syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  e.  ( Base `  K
) )
24 lhpocnle.l . . . . . . 7  |-  .<_  =  ( le `  K )
254, 24latref 14482 . . . . . 6  |-  ( ( K  e.  Lat  /\  (  ._|_  `  W )  e.  ( Base `  K
) )  ->  (  ._|_  `  W )  .<_  (  ._|_  `  W )
)
2618, 23, 25syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  .<_  (  ._|_  `  W ) )
27 eqid 2436 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
284, 24, 27latlem12 14507 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( (  ._|_  `  W
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
)  /\  (  ._|_  `  W )  e.  (
Base `  K )
) )  ->  (
( (  ._|_  `  W
)  .<_  W  /\  (  ._|_  `  W )  .<_  (  ._|_  `  W )
)  <->  (  ._|_  `  W
)  .<_  ( W (
meet `  K )
(  ._|_  `  W )
) ) )
2918, 23, 21, 23, 28syl13anc 1186 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
( ( (  ._|_  `  W )  .<_  W  /\  (  ._|_  `  W )  .<_  (  ._|_  `  W ) )  <->  (  ._|_  `  W
)  .<_  ( W (
meet `  K )
(  ._|_  `  W )
) ) )
3016, 26, 29mpbi2and 888 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  .<_  ( W ( meet `  K ) (  ._|_  `  W ) ) )
314, 7, 27, 12opnoncon 30006 . . . . 5  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( W ( meet `  K ) (  ._|_  `  W ) )  =  ( 0. `  K
) )
3220, 21, 31syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
( W ( meet `  K ) (  ._|_  `  W ) )  =  ( 0. `  K
) )
3330, 32breqtrd 4236 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  .<_  ( 0. `  K
) )
344, 24, 12ople0 29985 . . . 4  |-  ( ( K  e.  OP  /\  (  ._|_  `  W )  e.  ( Base `  K
) )  ->  (
(  ._|_  `  W )  .<_  ( 0. `  K
)  <->  (  ._|_  `  W
)  =  ( 0.
`  K ) ) )
3520, 23, 34syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
( (  ._|_  `  W
)  .<_  ( 0. `  K )  <->  (  ._|_  `  W )  =  ( 0. `  K ) ) )
3633, 35mpbid 202 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  =  ( 0. `  K ) )
3715, 36mtand 641 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  -.  (  ._|_  `  W
)  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   occoc 13537   meetcmee 14402   0.cp0 14466   Latclat 14474   OPcops 29970   Atomscatm 30061   AtLatcal 30062   HLchlt 30148   LHypclh 30781
This theorem is referenced by:  lhpocnel  30815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785
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