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Theorem lhpjat1 30879
Description: The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)
Hypotheses
Ref Expression
lhpjat.l  |-  .<_  =  ( le `  K )
lhpjat.j  |-  .\/  =  ( join `  K )
lhpjat.u  |-  .1.  =  ( 1. `  K )
lhpjat.a  |-  A  =  ( Atoms `  K )
lhpjat.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpjat1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  .1.  )

Proof of Theorem lhpjat1
StepHypRef Expression
1 simpll 732 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 eqid 2438 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 lhpjat.h . . . 4  |-  H  =  ( LHyp `  K
)
42, 3lhpbase 30857 . . 3  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
54ad2antlr 709 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K ) )
6 simprl 734 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
7 lhpjat.u . . . 4  |-  .1.  =  ( 1. `  K )
8 eqid 2438 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
97, 8, 3lhp1cvr 30858 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
)  .1.  )
109adantr 453 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W (  <o  `  K
)  .1.  )
11 simprr 735 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
12 lhpjat.l . . 3  |-  .<_  =  ( le `  K )
13 lhpjat.j . . 3  |-  .\/  =  ( join `  K )
14 lhpjat.a . . 3  |-  A  =  ( Atoms `  K )
152, 12, 13, 7, 8, 141cvrjat 30334 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  ( Base `  K )  /\  P  e.  A )  /\  ( W (  <o  `  K
)  .1.  /\  -.  P  .<_  W ) )  ->  ( W  .\/  P )  =  .1.  )
161, 5, 6, 10, 11, 15syl32anc 1193 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  .1.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   1.cp1 14469    <o ccvr 30122   Atomscatm 30123   HLchlt 30210   LHypclh 30843
This theorem is referenced by:  lhpjat2  30880  lhpj1  30881  trljat1  31025  trljat2  31026  cdlemc1  31050  cdlemc6  31055  cdleme20c  31170  cdleme20j  31177  trlcolem  31585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lhyp 30847
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