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Theorem hlhgt2 33356
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 33355 . 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 33333 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Poset )
76ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  K  e.  Poset )
8 hlop 33330 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
98ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  K  e.  OP )
101, 3op0cl 33152 . . . . . . . 8  |-  ( K  e.  OP  ->  .0.  e.  B )
119, 10syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  .0.  e.  B )
12 simpllr 758 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  y  e.  B )
13 simplr 754 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  x  e.  B )
141, 2plttr 15258 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (  .0.  e.  B  /\  y  e.  B  /\  x  e.  B ) )  -> 
( (  .0.  .<  y  /\  y  .<  x
)  ->  .0.  .<  x
) )
157, 11, 12, 13, 14syl13anc 1221 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  (
(  .0.  .<  y  /\  y  .<  x )  ->  .0.  .<  x ) )
16 simpr 461 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  z  e.  B )
171, 4op1cl 33153 . . . . . . . 8  |-  ( K  e.  OP  ->  .1.  e.  B )
189, 17syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  .1.  e.  B )
191, 2plttr 15258 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (
x  e.  B  /\  z  e.  B  /\  .1.  e.  B ) )  ->  ( ( x 
.<  z  /\  z  .<  .1.  )  ->  x  .<  .1.  ) )
207, 13, 16, 18, 19syl13anc 1221 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  y  e.  B )  /\  x  e.  B )  /\  z  e.  B )  ->  (
( x  .<  z  /\  z  .<  .1.  )  ->  x  .<  .1.  )
)
2115, 20anim12d 563 . . . . 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 2945 . . . 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 2932 . . 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 2945 . 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 369    = wceq 1370    e. wcel 1758   E.wrex 2799   class class class wbr 4399   ` cfv 5525   Basecbs 14291   Posetcpo 15228   ltcplt 15229   0.cp0 15325   1.cp1 15326   OPcops 33140   HLchlt 33318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-p0 15327  df-p1 15328  df-lat 15334  df-oposet 33144  df-ol 33146  df-oml 33147  df-atl 33266  df-cvlat 33290  df-hlat 33319
This theorem is referenced by:  hl0lt1N  33357  hl2at  33372
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