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

Theorem hlhgt4 32673
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 2429 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 hlhgt4.s . . 3  |-  .<  =  ( lt `  K )
4 eqid 2429 . . 3  |-  ( join `  K )  =  (
join `  K )
5 hlhgt4.z . . 3  |-  .0.  =  ( 0. `  K )
6 hlhgt4.u . . 3  |-  .1.  =  ( 1. `  K )
7 eqid 2429 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat2 32639 . 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 764 . 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 198 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15160   ltcplt 16141   joincjn 16144   0.cp0 16238   1.cp1 16239   CLatccla 16308   OMLcoml 32461   Atomscatm 32549   AtLatcal 32550   HLchlt 32636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-iota 5565  df-fv 5609  df-ov 6308  df-cvlat 32608  df-hlat 32637
This theorem is referenced by:  hlhgt2  32674  athgt  32741
  Copyright terms: Public domain W3C validator