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Theorem hlhgt2 34203
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 34202 . 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 34180 . . . . . . . 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 34177 . . . . . . . . 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 33999 . . . . . . . 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 15457 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (  .0.  e.  B  /\  y  e.  B  /\  x  e.  B ) )  -> 
( (  .0.  .<  y  /\  y  .<  x
)  ->  .0.  .<  x
) )
157, 11, 12, 13, 14syl13anc 1230 . . . . . 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 34000 . . . . . . . 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 15457 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (
x  e.  B  /\  z  e.  B  /\  .1.  e.  B ) )  ->  ( ( x 
.<  z  /\  z  .<  .1.  )  ->  x  .<  .1.  ) )
207, 13, 16, 18, 19syl13anc 1230 . . . . . 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 2955 . . . 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 2938 . . 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 2955 . 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 1379    e. wcel 1767   E.wrex 2815   class class class wbr 4447   ` cfv 5588   Basecbs 14490   Posetcpo 15427   ltcplt 15428   0.cp0 15524   1.cp1 15525   OPcops 33987   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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
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-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-p0 15526  df-p1 15527  df-lat 15533  df-oposet 33991  df-ol 33993  df-oml 33994  df-atl 34113  df-cvlat 34137  df-hlat 34166
This theorem is referenced by:  hl0lt1N  34204  hl2at  34219
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