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Theorem hl0lt1N 30124
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 2435 . . 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 30123 . 2  |-  ( K  e.  HL  ->  E. x  e.  ( Base `  K
) (  .0.  .<  x  /\  x  .<  .1.  )
)
6 hlpos 30100 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Poset )
76adantr 452 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  Poset )
8 hlop 30097 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
98adantr 452 . . . . 5  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  OP )
101, 3op0cl 29919 . . . . 5  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
119, 10syl 16 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  .0.  e.  ( Base `  K
) )
12 simpr 448 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  x  e.  ( Base `  K ) )
131, 4op1cl 29920 . . . . 5  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
149, 13syl 16 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  .1.  e.  ( Base `  K
) )
151, 2plttr 14419 . . . 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 1186 . . 3  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( (  .0.  .<  x  /\  x  .<  .1.  )  ->  .0.  .<  .1.  )
)
1716rexlimdva 2822 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   class class class wbr 4204   ` cfv 5446   Basecbs 13461   Posetcpo 14389   ltcplt 14390   0.cp0 14458   1.cp1 14459   OPcops 29907   HLchlt 30085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-poset 14395  df-plt 14407  df-lat 14467  df-oposet 29911  df-ol 29913  df-oml 29914  df-atl 30033  df-cvlat 30057  df-hlat 30086
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