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Theorem hlhgt4 34202
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 2467 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 hlhgt4.s . . 3  |-  .<  =  ( lt `  K )
4 eqid 2467 . . 3  |-  ( join `  K )  =  (
join `  K )
5 hlhgt4.z . . 3  |-  .0.  =  ( 0. `  K )
6 hlhgt4.u . . 3  |-  .1.  =  ( 1. `  K )
7 eqid 2467 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat2 34168 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   lecple 14562   ltcplt 15428   joincjn 15431   0.cp0 15524   1.cp1 15525   CLatccla 15594   OMLcoml 33990   Atomscatm 34078   AtLatcal 34079   HLchlt 34165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6287  df-cvlat 34137  df-hlat 34166
This theorem is referenced by:  hlhgt2  34203  athgt  34270
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