Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpj1 Structured version   Unicode version

Theorem lhpj1 33506
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 33482 . . . . 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 2438 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
93, 7, 8hlrelat2 32887 . . . 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 1218 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B
)  ->  ( -.  X  .<_  W  <->  E. p  e.  ( Atoms `  K )
( p  .<_  X  /\  -.  p  .<_  W ) ) )
11 simp1l 1012 . . . . . . 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 989 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  e.  ( Atoms `  K )
)
13 simp3r 1017 . . . . . . 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 33504 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  ( W  .\/  p )  =  .1.  )
1711, 12, 13, 16syl12anc 1216 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  ( W  .\/  p )  =  .1.  )
18 simp3l 1016 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  .<_  X )
19 simp1ll 1051 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  K  e.  HL )
20 hllat 32848 . . . . . . . . 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 32774 . . . . . . . . 9  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  B )
23223ad2ant2 1010 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  p  e.  B )
24 simp1r 1013 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  X  e.  B )
2563ad2ant1 1009 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  W  e.  B )
263, 7, 14latjlej2 15228 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( p  e.  B  /\  X  e.  B  /\  W  e.  B
) )  ->  (
p  .<_  X  ->  ( W  .\/  p )  .<_  ( W  .\/  X ) ) )
2721, 23, 24, 25, 26syl13anc 1220 . . . . . . 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 4309 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  /\  p  e.  ( Atoms `  K )  /\  ( p  .<_  X  /\  -.  p  .<_  W ) )  ->  .1.  .<_  ( W 
.\/  X ) )
30 hlop 32847 . . . . . . 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 15213 . . . . . . 7  |-  ( ( K  e.  Lat  /\  W  e.  B  /\  X  e.  B )  ->  ( W  .\/  X
)  e.  B )
3321, 25, 24, 32syl3anc 1218 . . . . . 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 32677 . . . . . 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 2838 . . 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 965    = wceq 1369    e. wcel 1756   E.wrex 2711   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   1.cp1 15200   Latclat 15207   OPcops 32657   Atomscatm 32748   HLchlt 32835   LHypclh 33468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-lhyp 33472
This theorem is referenced by:  lhpmcvr  33507  cdleme30a  33862
  Copyright terms: Public domain W3C validator