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Theorem hlhgt2 32662
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 32661 . 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 32639 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Poset )
76ad3antrrr 734 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  K  e.  Poset )
8 hlop 32636 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
98ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  K  e.  OP )
101, 3op0cl 32458 . . . . . . . 8  |-  ( K  e.  OP  ->  .0.  e.  B )
119, 10syl 17 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  .0.  e.  B )
12 simpllr 767 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  y  e.  B )
13 simplr 760 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  x  e.  B )
141, 2plttr 16167 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (  .0.  e.  B  /\  y  e.  B  /\  x  e.  B ) )  -> 
( (  .0.  .<  y  /\  y  .<  x
)  ->  .0.  .<  x
) )
157, 11, 12, 13, 14syl13anc 1266 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  (
(  .0.  .<  y  /\  y  .<  x )  ->  .0.  .<  x ) )
16 simpr 462 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  z  e.  B )
171, 4op1cl 32459 . . . . . . . 8  |-  ( K  e.  OP  ->  .1.  e.  B )
189, 17syl 17 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  .1.  e.  B )
191, 2plttr 16167 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (
x  e.  B  /\  z  e.  B  /\  .1.  e.  B ) )  ->  ( ( x 
.<  z  /\  z  .<  .1.  )  ->  x  .<  .1.  ) )
207, 13, 16, 18, 19syl13anc 1266 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  (
( x  .<  z  /\  z  .<  .1.  )  ->  x  .<  .1.  )
)
2115, 20anim12d 565 . . . . 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 2924 . . . 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 2907 . . 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 2924 . 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 15 1  |-  ( K  e.  HL  ->  E. x  e.  B  (  .0.  .<  x  /\  x  .<  .1.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   class class class wbr 4426   ` cfv 5601   Basecbs 15084   Posetcpo 16136   ltcplt 16137   0.cp0 16234   1.cp1 16235   OPcops 32446   HLchlt 32624
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-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
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-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-p0 16236  df-p1 16237  df-lat 16243  df-oposet 32450  df-ol 32452  df-oml 32453  df-atl 32572  df-cvlat 32596  df-hlat 32625
This theorem is referenced by:  hl0lt1N  32663  hl2at  32678
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