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Theorem lhpj1 30833
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 30809 . . . . 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 2296 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
93, 7, 8hlrelat2 30214 . . . 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 30831 . . . . . . 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 30175 . . . . . . . . 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 30101 . . . . . . . . 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 14188 . . . . . . . 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 4061 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  .1.  .<_  ( W 
.\/  X ) )
30 hlop 30174 . . . . . . 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 14172 . . . . . . 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 30004 . . . . . 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 2682 . . 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 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   1.cp1 14160   Latclat 14167   OPcops 29984   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  lhpmcvr  30834  cdleme30a  31189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799
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