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Theorem lhpj1 30211
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 730 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  K  e.  HL )
2 simpr 447 . . . 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 30187 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
65ad2antlr 707 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  W  e.  B )
7 lhpj1.l . . . . 5  |-  .<_  =  ( le `  K )
8 eqid 2283 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
93, 7, 8hlrelat2 29592 . . . 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 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  ( -.  X  .<_  W  <->  E. p  e.  ( Atoms `  K )
( p  .<_  X  /\  -.  p  .<_  W ) ) )
11 simp1l 979 . . . . . . 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 956 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  e.  ( Atoms `  K )
)
13 simp3r 984 . . . . . . 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 30209 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  ( W  .\/  p )  =  .1.  )
1711, 12, 13, 16syl12anc 1180 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( W  .\/  p )  =  .1.  )
18 simp3l 983 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  .<_  X )
19 simp1ll 1018 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  K  e.  HL )
20 hllat 29553 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
2119, 20syl 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  K  e.  Lat )
223, 8atbase 29479 . . . . . . . . 9  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  B )
23223ad2ant2 977 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  e.  B )
24 simp1r 980 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  X  e.  B )
2563ad2ant1 976 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  W  e.  B )
263, 7, 14latjlej2 14172 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( p  e.  B  /\  X  e.  B  /\  W  e.  B
) )  ->  (
p  .<_  X  ->  ( W  .\/  p )  .<_  ( W  .\/  X ) ) )
2721, 23, 24, 25, 26syl13anc 1184 . . . . . . 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 14 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( W  .\/  p )  .<_  ( W 
.\/  X ) )
2917, 28eqbrtrrd 4045 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  .1.  .<_  ( W 
.\/  X ) )
30 hlop 29552 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
3119, 30syl 15 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  K  e.  OP )
323, 14latjcl 14156 . . . . . . 7  |-  ( ( K  e.  Lat  /\  W  e.  B  /\  X  e.  B )  ->  ( W  .\/  X
)  e.  B )
3321, 25, 24, 32syl3anc 1182 . . . . . 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 29382 . . . . . 6  |-  ( ( K  e.  OP  /\  ( W  .\/  X )  e.  B )  -> 
(  .1.  .<_  ( W 
.\/  X )  <->  ( W  .\/  X )  =  .1.  ) )
3531, 33, 34syl2anc 642 . . . . 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 201 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( W  .\/  X )  =  .1.  )
3736rexlimdv3a 2669 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  ( E. p  e.  ( Atoms `  K ) ( p 
.<_  X  /\  -.  p  .<_  W )  ->  ( W  .\/  X )  =  .1.  ) )
3810, 37sylbid 206 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  ( -.  X  .<_  W  ->  ( W  .\/  X )  =  .1.  ) )
3938impr 602 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   1.cp1 14144   Latclat 14151   OPcops 29362   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  lhpmcvr  30212  cdleme30a  30567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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