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Theorem hl0lt1N 29579
Description: Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
hl0lt1.s  |-  .<  =  ( lt `  K )
hl0lt1.z  |-  .0.  =  ( 0. `  K )
hl0lt1.u  |-  .1.  =  ( 1. `  K )
Assertion
Ref Expression
hl0lt1N  |-  ( K  e.  HL  ->  .0.  .<  .1.  )

Proof of Theorem hl0lt1N
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 hl0lt1.s . . 3  |-  .<  =  ( lt `  K )
3 hl0lt1.z . . 3  |-  .0.  =  ( 0. `  K )
4 hl0lt1.u . . 3  |-  .1.  =  ( 1. `  K )
51, 2, 3, 4hlhgt2 29578 . 2  |-  ( K  e.  HL  ->  E. x  e.  ( Base `  K
) (  .0.  .<  x  /\  x  .<  .1.  )
)
6 hlpos 29555 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Poset )
76adantr 451 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  Poset )
8 hlop 29552 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
98adantr 451 . . . . 5  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  OP )
101, 3op0cl 29374 . . . . 5  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
119, 10syl 15 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  .0.  e.  ( Base `  K
) )
12 simpr 447 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  x  e.  ( Base `  K ) )
131, 4op1cl 29375 . . . . 5  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
149, 13syl 15 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  .1.  e.  ( Base `  K
) )
151, 2plttr 14104 . . . 4  |-  ( ( K  e.  Poset  /\  (  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )  /\  .1.  e.  ( Base `  K ) ) )  ->  ( (  .0. 
.<  x  /\  x  .<  .1.  )  ->  .0.  .<  .1.  ) )
167, 11, 12, 14, 15syl13anc 1184 . . 3  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( (  .0.  .<  x  /\  x  .<  .1.  )  ->  .0.  .<  .1.  )
)
1716rexlimdva 2667 . 2  |-  ( K  e.  HL  ->  ( E. x  e.  ( Base `  K ) (  .0.  .<  x  /\  x  .<  .1.  )  ->  .0. 
.<  .1.  ) )
185, 17mpd 14 1  |-  ( K  e.  HL  ->  .0.  .<  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   Posetcpo 14074   ltcplt 14075   0.cp0 14143   1.cp1 14144   OPcops 29362   HLchlt 29540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-poset 14080  df-plt 14092  df-lat 14152  df-oposet 29366  df-ol 29368  df-oml 29369  df-atl 29488  df-cvlat 29512  df-hlat 29541
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