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Theorem lhpj1 35035
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 753 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  K  e.  HL )
2 simpr 461 . . . 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 35011 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
65ad2antlr 726 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  W  e.  B )
7 lhpj1.l . . . . 5  |-  .<_  =  ( le `  K )
8 eqid 2467 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
93, 7, 8hlrelat2 34416 . . . 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 1228 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  ( -.  X  .<_  W  <->  E. p  e.  ( Atoms `  K )
( p  .<_  X  /\  -.  p  .<_  W ) ) )
11 simp1l 1020 . . . . . . 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 997 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  e.  ( Atoms `  K )
)
13 simp3r 1025 . . . . . . 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 35033 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  ( W  .\/  p )  =  .1.  )
1711, 12, 13, 16syl12anc 1226 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( W  .\/  p )  =  .1.  )
18 simp3l 1024 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  .<_  X )
19 simp1ll 1059 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  K  e.  HL )
20 hllat 34377 . . . . . . . . 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 34303 . . . . . . . . 9  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  B )
23223ad2ant2 1018 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  e.  B )
24 simp1r 1021 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  X  e.  B )
2563ad2ant1 1017 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  W  e.  B )
263, 7, 14latjlej2 15556 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( p  e.  B  /\  X  e.  B  /\  W  e.  B
) )  ->  (
p  .<_  X  ->  ( W  .\/  p )  .<_  ( W  .\/  X ) ) )
2721, 23, 24, 25, 26syl13anc 1230 . . . . . . 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 4469 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  .1.  .<_  ( W 
.\/  X ) )
30 hlop 34376 . . . . . . 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 15541 . . . . . . 7  |-  ( ( K  e.  Lat  /\  W  e.  B  /\  X  e.  B )  ->  ( W  .\/  X
)  e.  B )
3321, 25, 24, 32syl3anc 1228 . . . . . 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 34206 . . . . . 6  |-  ( ( K  e.  OP  /\  ( W  .\/  X )  e.  B )  -> 
(  .1.  .<_  ( W 
.\/  X )  <->  ( W  .\/  X )  =  .1.  ) )
3531, 33, 34syl2anc 661 . . . . 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 210 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( W  .\/  X )  =  .1.  )
3736rexlimdv3a 2957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  ( E. p  e.  ( Atoms `  K ) ( p 
.<_  X  /\  -.  p  .<_  W )  ->  ( W  .\/  X )  =  .1.  ) )
3810, 37sylbid 215 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  ( -.  X  .<_  W  ->  ( W  .\/  X )  =  .1.  ) )
3938impr 619 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W  .\/  X
)  =  .1.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   1.cp1 15528   Latclat 15535   OPcops 34186   Atomscatm 34277   HLchlt 34364   LHypclh 34997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-p1 15530  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-lhyp 35001
This theorem is referenced by:  lhpmcvr  35036  cdleme30a  35391
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