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Theorem hlhgt4 33037
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 2443 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 hlhgt4.s . . 3  |-  .<  =  ( lt `  K )
4 eqid 2443 . . 3  |-  ( join `  K )  =  (
join `  K )
5 hlhgt4.z . . 3  |-  .0.  =  ( 0. `  K )
6 hlhgt4.u . . 3  |-  .1.  =  ( 1. `  K )
7 eqid 2443 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat2 33003 . 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 756 . 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 195 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   ltcplt 15116   joincjn 15119   0.cp0 15212   1.cp1 15213   CLatccla 15282   OMLcoml 32825   Atomscatm 32913   AtLatcal 32914   HLchlt 33000
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
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-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431  df-ov 6099  df-cvlat 32972  df-hlat 33001
This theorem is referenced by:  hlhgt2  33038  athgt  33105
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