Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hlhgt2 Unicode version

Theorem hlhgt2 29578
Description: A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011.)
Hypotheses
Ref Expression
hlhgt4.b  |-  B  =  ( Base `  K
)
hlhgt4.s  |-  .<  =  ( lt `  K )
hlhgt4.z  |-  .0.  =  ( 0. `  K )
hlhgt4.u  |-  .1.  =  ( 1. `  K )
Assertion
Ref Expression
hlhgt2  |-  ( K  e.  HL  ->  E. x  e.  B  (  .0.  .<  x  /\  x  .<  .1.  )
)
Distinct variable groups:    x, B    x, K
Allowed substitution hints:    .< ( x)    .1. ( x)    .0. (
x)

Proof of Theorem hlhgt2
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hlhgt4.b . . 3  |-  B  =  ( Base `  K
)
2 hlhgt4.s . . 3  |-  .<  =  ( lt `  K )
3 hlhgt4.z . . 3  |-  .0.  =  ( 0. `  K )
4 hlhgt4.u . . 3  |-  .1.  =  ( 1. `  K )
51, 2, 3, 4hlhgt4 29577 . 2  |-  ( K  e.  HL  ->  E. y  e.  B  E. x  e.  B  E. z  e.  B  ( (  .0.  .<  y  /\  y  .<  x )  /\  (
x  .<  z  /\  z  .<  .1.  ) ) )
6 hlpos 29555 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Poset )
76ad3antrrr 710 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  K  e.  Poset )
8 hlop 29552 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
98ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  K  e.  OP )
101, 3op0cl 29374 . . . . . . . 8  |-  ( K  e.  OP  ->  .0.  e.  B )
119, 10syl 15 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  .0.  e.  B )
12 simpllr 735 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  y  e.  B )
13 simplr 731 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  x  e.  B )
141, 2plttr 14104 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (  .0.  e.  B  /\  y  e.  B  /\  x  e.  B ) )  -> 
( (  .0.  .<  y  /\  y  .<  x
)  ->  .0.  .<  x
) )
157, 11, 12, 13, 14syl13anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  (
(  .0.  .<  y  /\  y  .<  x )  ->  .0.  .<  x ) )
16 simpr 447 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  z  e.  B )
171, 4op1cl 29375 . . . . . . . 8  |-  ( K  e.  OP  ->  .1.  e.  B )
189, 17syl 15 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  .1.  e.  B )
191, 2plttr 14104 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (
x  e.  B  /\  z  e.  B  /\  .1.  e.  B ) )  ->  ( ( x 
.<  z  /\  z  .<  .1.  )  ->  x  .<  .1.  ) )
207, 13, 16, 18, 19syl13anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  (
( x  .<  z  /\  z  .<  .1.  )  ->  x  .<  .1.  )
)
2115, 20anim12d 546 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  (
( (  .0.  .<  y  /\  y  .<  x
)  /\  ( x  .<  z  /\  z  .<  .1.  ) )  ->  (  .0.  .<  x  /\  x  .<  .1.  ) ) )
2221rexlimdva 2667 . . . 4  |-  ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B
)  ->  ( E. z  e.  B  (
(  .0.  .<  y  /\  y  .<  x )  /\  ( x  .<  z  /\  z  .<  .1.  )
)  ->  (  .0.  .<  x  /\  x  .<  .1.  )
) )
2322reximdva 2655 . . 3  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( E. x  e.  B  E. z  e.  B  ( (  .0. 
.<  y  /\  y  .<  x )  /\  (
x  .<  z  /\  z  .<  .1.  ) )  ->  E. x  e.  B  (  .0.  .<  x  /\  x  .<  .1.  ) )
)
2423rexlimdva 2667 . 2  |-  ( K  e.  HL  ->  ( E. y  e.  B  E. x  e.  B  E. z  e.  B  ( (  .0.  .<  y  /\  y  .<  x
)  /\  ( x  .<  z  /\  z  .<  .1.  ) )  ->  E. x  e.  B  (  .0.  .<  x  /\  x  .<  .1.  )
) )
255, 24mpd 14 1  |-  ( K  e.  HL  ->  E. x  e.  B  (  .0.  .<  x  /\  x  .<  .1.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   Posetcpo 14074   ltcplt 14075   0.cp0 14143   1.cp1 14144   OPcops 29362   HLchlt 29540
This theorem is referenced by:  hl0lt1N  29579  hl2at  29594
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-poset 14080  df-plt 14092  df-lat 14152  df-oposet 29366  df-ol 29368  df-oml 29369  df-atl 29488  df-cvlat 29512  df-hlat 29541
  Copyright terms: Public domain W3C validator