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Theorem hl0lt1N 30201
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 2296 . . 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 30200 . 2  |-  ( K  e.  HL  ->  E. x  e.  ( Base `  K
) (  .0.  .<  x  /\  x  .<  .1.  )
)
6 hlpos 30177 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Poset )
76adantr 451 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  Poset )
8 hlop 30174 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
98adantr 451 . . . . 5  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  OP )
101, 3op0cl 29996 . . . . 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 29997 . . . . 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 14120 . . . 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 2680 . 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 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271   Basecbs 13164   Posetcpo 14090   ltcplt 14091   0.cp0 14159   1.cp1 14160   OPcops 29984   HLchlt 30162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-poset 14096  df-plt 14108  df-lat 14168  df-oposet 29988  df-ol 29990  df-oml 29991  df-atl 30110  df-cvlat 30134  df-hlat 30163
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