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Theorem hlhgt2 32873
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 32872 . 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 32850 . . . . . . . 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 32847 . . . . . . . . 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 32669 . . . . . . . 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 15132 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (  .0.  e.  B  /\  y  e.  B  /\  x  e.  B ) )  -> 
( (  .0.  .<  y  /\  y  .<  x
)  ->  .0.  .<  x
) )
157, 11, 12, 13, 14syl13anc 1220 . . . . . 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 32670 . . . . . . . 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 15132 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (
x  e.  B  /\  z  e.  B  /\  .1.  e.  B ) )  ->  ( ( x 
.<  z  /\  z  .<  .1.  )  ->  x  .<  .1.  ) )
207, 13, 16, 18, 19syl13anc 1220 . . . . . 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 2836 . . . 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 2823 . . 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 2836 . 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 1369    e. wcel 1756   E.wrex 2711   class class class wbr 4287   ` cfv 5413   Basecbs 14166   Posetcpo 15102   ltcplt 15103   0.cp0 15199   1.cp1 15200   OPcops 32657   HLchlt 32835
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
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-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-p0 15201  df-p1 15202  df-lat 15208  df-oposet 32661  df-ol 32663  df-oml 32664  df-atl 32783  df-cvlat 32807  df-hlat 32836
This theorem is referenced by:  hl0lt1N  32874  hl2at  32889
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