Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hlhgt4 Unicode version

Theorem hlhgt4 29577
Description: A Hilbert lattice has a height of at least 4. (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
hlhgt4  |-  ( K  e.  HL  ->  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) )
Distinct variable groups:    x, y,
z, B    x, K, y, z
Allowed substitution hints:    .< ( x, y,
z)    .1. ( x, y, z)    .0. ( x, y, z)

Proof of Theorem hlhgt4
StepHypRef Expression
1 hlhgt4.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 hlhgt4.s . . 3  |-  .<  =  ( lt `  K )
4 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
5 hlhgt4.z . . 3  |-  .0.  =  ( 0. `  K )
6 hlhgt4.u . . 3  |-  .1.  =  ( 1. `  K )
7 eqid 2283 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat2 29543 . 2  |-  ( K  e.  HL  <->  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
)  /\  ( A. x  e.  ( Atoms `  K ) A. y  e.  ( Atoms `  K )
( ( x  =/=  y  ->  E. z  e.  ( Atoms `  K )
( z  =/=  x  /\  z  =/=  y  /\  z ( le `  K ) ( x ( join `  K
) y ) ) )  /\  A. z  e.  B  ( ( -.  x ( le `  K ) z  /\  x ( le `  K ) ( z ( join `  K
) y ) )  ->  y ( le
`  K ) ( z ( join `  K
) x ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) ) )
9 simprr 733 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  ( A. x  e.  ( Atoms `  K ) A. y  e.  ( Atoms `  K ) ( ( x  =/=  y  ->  E. z  e.  ( Atoms `  K ) ( z  =/=  x  /\  z  =/=  y  /\  z
( le `  K
) ( x (
join `  K )
y ) ) )  /\  A. z  e.  B  ( ( -.  x ( le `  K ) z  /\  x ( le `  K ) ( z ( join `  K
) y ) )  ->  y ( le
`  K ) ( z ( join `  K
) x ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) ) )  ->  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) )
108, 9sylbi 187 1  |-  ( K  e.  HL  ->  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (  .0.  .<  x  /\  x  .<  y )  /\  (
y  .<  z  /\  z  .<  .1.  ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   ltcplt 14075   joincjn 14078   0.cp0 14143   1.cp1 14144   CLatccla 14213   OMLcoml 29365   Atomscatm 29453   AtLatcal 29454   HLchlt 29540
This theorem is referenced by:  hlhgt2  29578  athgt  29645
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  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-iota 5219  df-fv 5263  df-ov 5861  df-cvlat 29512  df-hlat 29541
  Copyright terms: Public domain W3C validator