Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hl0lt1N Structured version   Unicode version

Theorem hl0lt1N 33130
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 2443 . . 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 33129 . 2  |-  ( K  e.  HL  ->  E. x  e.  ( Base `  K
) (  .0.  .<  x  /\  x  .<  .1.  )
)
6 hlpos 33106 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Poset )
76adantr 465 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  Poset )
8 hlop 33103 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
98adantr 465 . . . . 5  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  K  e.  OP )
101, 3op0cl 32925 . . . . 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 461 . . . 4  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  ->  x  e.  ( Base `  K ) )
131, 4op1cl 32926 . . . . 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 15161 . . . 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 1220 . . 3  |-  ( ( K  e.  HL  /\  x  e.  ( Base `  K ) )  -> 
( (  .0.  .<  x  /\  x  .<  .1.  )  ->  .0.  .<  .1.  )
)
1716rexlimdva 2862 . 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 369    = wceq 1369    e. wcel 1756   E.wrex 2737   class class class wbr 4313   ` cfv 5439   Basecbs 14195   Posetcpo 15131   ltcplt 15132   0.cp0 15228   1.cp1 15229   OPcops 32913   HLchlt 33091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-p0 15230  df-p1 15231  df-lat 15237  df-oposet 32917  df-ol 32919  df-oml 32920  df-atl 33039  df-cvlat 33063  df-hlat 33092
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator