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Theorem lhpocnle 33756
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 33101 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
21adantr 465 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  AtLat )
3 simpr 461 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  H )
4 eqid 2443 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
5 lhpocnle.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
64, 5lhpbase 33738 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7 lhpocnle.o . . . . . . 7  |-  ._|_  =  ( oc `  K )
8 eqid 2443 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
94, 7, 8, 5lhpoc 33754 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  ( Base `  K ) )  -> 
( W  e.  H  <->  ( 
._|_  `  W )  e.  ( Atoms `  K )
) )
106, 9sylan2 474 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( W  e.  H  <->  ( 
._|_  `  W )  e.  ( Atoms `  K )
) )
113, 10mpbid 210 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  W )  e.  ( Atoms `  K
) )
12 eqid 2443 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
1312, 8atn0 33049 . . . 4  |-  ( ( K  e.  AtLat  /\  (  ._|_  `  W )  e.  ( Atoms `  K )
)  ->  (  ._|_  `  W )  =/=  ( 0. `  K ) )
142, 11, 13syl2anc 661 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  W )  =/=  ( 0. `  K ) )
1514neneqd 2627 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  -.  (  ._|_  `  W
)  =  ( 0.
`  K ) )
16 simpr 461 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  .<_  W )
17 hllat 33104 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1817ad2antrr 725 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  ->  K  e.  Lat )
19 hlop 33103 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
2019ad2antrr 725 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  ->  K  e.  OP )
216ad2antlr 726 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  ->  W  e.  ( Base `  K ) )
224, 7opoccl 32935 . . . . . . 7  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
(  ._|_  `  W )  e.  ( Base `  K
) )
2320, 21, 22syl2anc 661 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  e.  ( Base `  K
) )
24 lhpocnle.l . . . . . . 7  |-  .<_  =  ( le `  K )
254, 24latref 15244 . . . . . 6  |-  ( ( K  e.  Lat  /\  (  ._|_  `  W )  e.  ( Base `  K
) )  ->  (  ._|_  `  W )  .<_  (  ._|_  `  W )
)
2618, 23, 25syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  .<_  (  ._|_  `  W ) )
27 eqid 2443 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
284, 24, 27latlem12 15269 . . . . . 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 1220 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
( ( (  ._|_  `  W )  .<_  W  /\  (  ._|_  `  W )  .<_  (  ._|_  `  W ) )  <->  (  ._|_  `  W
)  .<_  ( W (
meet `  K )
(  ._|_  `  W )
) ) )
3016, 26, 29mpbi2and 912 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  .<_  ( W ( meet `  K ) (  ._|_  `  W ) ) )
314, 7, 27, 12opnoncon 32949 . . . . 5  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( W ( meet `  K ) (  ._|_  `  W ) )  =  ( 0. `  K
) )
3220, 21, 31syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
( W ( meet `  K ) (  ._|_  `  W ) )  =  ( 0. `  K
) )
3330, 32breqtrd 4337 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  .<_  ( 0. `  K
) )
344, 24, 12ople0 32928 . . . 4  |-  ( ( K  e.  OP  /\  (  ._|_  `  W )  e.  ( Base `  K
) )  ->  (
(  ._|_  `  W )  .<_  ( 0. `  K
)  <->  (  ._|_  `  W
)  =  ( 0.
`  K ) ) )
3520, 23, 34syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
( (  ._|_  `  W
)  .<_  ( 0. `  K )  <->  (  ._|_  `  W )  =  ( 0. `  K ) ) )
3633, 35mpbid 210 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  W
)  .<_  W )  -> 
(  ._|_  `  W )  =  ( 0. `  K ) )
3715, 36mtand 659 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  -.  (  ._|_  `  W
)  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   Basecbs 14195   lecple 14266   occoc 14267   meetcmee 15136   0.cp0 15228   Latclat 15236   OPcops 32913   Atomscatm 33004   AtLatcal 33005   HLchlt 33091   LHypclh 33724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-meet 15168  df-p0 15230  df-p1 15231  df-lat 15237  df-oposet 32917  df-ol 32919  df-oml 32920  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-lhyp 33728
This theorem is referenced by:  lhpocnel  33758
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