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Theorem hl0lt1N 32407
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 2402 . . 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 32406 . 2  |-  ( K  e.  HL  ->  E. x  e.  ( Base `  K
) (  .0.  .<  x  /\  x  .<  .1.  )
)
6 hlpos 32383 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Poset )
76adantr 463 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  Poset )
8 hlop 32380 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
98adantr 463 . . . . 5  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  OP )
101, 3op0cl 32202 . . . . 5  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
119, 10syl 17 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  .0.  e.  ( Base `  K
) )
12 simpr 459 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  x  e.  ( Base `  K ) )
131, 4op1cl 32203 . . . . 5  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
149, 13syl 17 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  .1.  e.  ( Base `  K
) )
151, 2plttr 15924 . . . 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 1232 . . 3  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( (  .0.  .<  x  /\  x  .<  .1.  )  ->  .0.  .<  .1.  )
)
1716rexlimdva 2896 . 2  |-  ( K  e.  HL  ->  ( E. x  e.  ( Base `  K ) (  .0.  .<  x  /\  x  .<  .1.  )  ->  .0. 
.<  .1.  ) )
185, 17mpd 15 1  |-  ( K  e.  HL  ->  .0.  .<  .1.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2755   class class class wbr 4395   ` cfv 5569   Basecbs 14841   Posetcpo 15893   ltcplt 15894   0.cp0 15991   1.cp1 15992   OPcops 32190   HLchlt 32368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-p0 15993  df-p1 15994  df-lat 16000  df-oposet 32194  df-ol 32196  df-oml 32197  df-atl 32316  df-cvlat 32340  df-hlat 32369
This theorem is referenced by: (None)
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