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Theorem lhpj1 30746
Description: The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)
Hypotheses
Ref Expression
lhpj1.b  |-  B  =  ( Base `  K
)
lhpj1.l  |-  .<_  =  ( le `  K )
lhpj1.j  |-  .\/  =  ( join `  K )
lhpj1.u  |-  .1.  =  ( 1. `  K )
lhpj1.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpj1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W  .\/  X
)  =  .1.  )

Proof of Theorem lhpj1
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simpll 731 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  K  e.  HL )
2 simpr 448 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  X  e.  B )
3 lhpj1.b . . . . . 6  |-  B  =  ( Base `  K
)
4 lhpj1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
53, 4lhpbase 30722 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
65ad2antlr 708 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  W  e.  B )
7 lhpj1.l . . . . 5  |-  .<_  =  ( le `  K )
8 eqid 2435 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
93, 7, 8hlrelat2 30127 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  W  e.  B )  ->  ( -.  X  .<_  W  <->  E. p  e.  ( Atoms `  K ) ( p  .<_  X  /\  -.  p  .<_  W ) ) )
101, 2, 6, 9syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  ( -.  X  .<_  W  <->  E. p  e.  ( Atoms `  K )
( p  .<_  X  /\  -.  p  .<_  W ) ) )
11 simp1l 981 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
12 simp2 958 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  e.  ( Atoms `  K )
)
13 simp3r 986 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  -.  p  .<_  W )
14 lhpj1.j . . . . . . . 8  |-  .\/  =  ( join `  K )
15 lhpj1.u . . . . . . . 8  |-  .1.  =  ( 1. `  K )
167, 14, 15, 8, 4lhpjat1 30744 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  ( W  .\/  p )  =  .1.  )
1711, 12, 13, 16syl12anc 1182 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( W  .\/  p )  =  .1.  )
18 simp3l 985 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  .<_  X )
19 simp1ll 1020 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  K  e.  HL )
20 hllat 30088 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
2119, 20syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  K  e.  Lat )
223, 8atbase 30014 . . . . . . . . 9  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  B )
23223ad2ant2 979 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  e.  B )
24 simp1r 982 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  X  e.  B )
2563ad2ant1 978 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  W  e.  B )
263, 7, 14latjlej2 14487 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( p  e.  B  /\  X  e.  B  /\  W  e.  B
) )  ->  (
p  .<_  X  ->  ( W  .\/  p )  .<_  ( W  .\/  X ) ) )
2721, 23, 24, 25, 26syl13anc 1186 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( p  .<_  X  ->  ( W  .\/  p )  .<_  ( W 
.\/  X ) ) )
2818, 27mpd 15 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( W  .\/  p )  .<_  ( W 
.\/  X ) )
2917, 28eqbrtrrd 4226 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  .1.  .<_  ( W 
.\/  X ) )
30 hlop 30087 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
3119, 30syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  K  e.  OP )
323, 14latjcl 14471 . . . . . . 7  |-  ( ( K  e.  Lat  /\  W  e.  B  /\  X  e.  B )  ->  ( W  .\/  X
)  e.  B )
3321, 25, 24, 32syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( W  .\/  X )  e.  B
)
343, 7, 15op1le 29917 . . . . . 6  |-  ( ( K  e.  OP  /\  ( W  .\/  X )  e.  B )  -> 
(  .1.  .<_  ( W 
.\/  X )  <->  ( W  .\/  X )  =  .1.  ) )
3531, 33, 34syl2anc 643 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  (  .1.  .<_  ( W  .\/  X )  <-> 
( W  .\/  X
)  =  .1.  )
)
3629, 35mpbid 202 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( W  .\/  X )  =  .1.  )
3736rexlimdv3a 2824 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  ( E. p  e.  ( Atoms `  K ) ( p 
.<_  X  /\  -.  p  .<_  W )  ->  ( W  .\/  X )  =  .1.  ) )
3810, 37sylbid 207 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  ( -.  X  .<_  W  ->  ( W  .\/  X )  =  .1.  ) )
3938impr 603 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W  .\/  X
)  =  .1.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   1.cp1 14459   Latclat 14466   OPcops 29897   Atomscatm 29988   HLchlt 30075   LHypclh 30708
This theorem is referenced by:  lhpmcvr  30747  cdleme30a  31102
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-lhyp 30712
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