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Theorem lhp0lt 30192
Description: A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
Hypotheses
Ref Expression
lhp0lt.s  |-  .<  =  ( lt `  K )
lhp0lt.z  |-  .0.  =  ( 0. `  K )
lhp0lt.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhp0lt  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  .<  W )

Proof of Theorem lhp0lt
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 lhp0lt.s . . 3  |-  .<  =  ( lt `  K )
2 eqid 2283 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 lhp0lt.h . . 3  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexlt 30191 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K ) p 
.<  W )
5 simp1l 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  K  e.  HL )
6 hlop 29552 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
7 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 lhp0lt.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
97, 8op0cl 29374 . . . . . 6  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
105, 6, 93syl 18 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  e.  ( Base `  K
) )
117, 2atbase 29479 . . . . . 6  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
12113ad2ant2 977 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  e.  ( Base `  K ) )
13 simp2 956 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  e.  ( Atoms `  K ) )
14 eqid 2283 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
158, 14, 2atcvr0 29478 . . . . . 6  |-  ( ( K  e.  HL  /\  p  e.  ( Atoms `  K ) )  ->  .0.  (  <o  `  K
) p )
165, 13, 15syl2anc 642 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  (  <o  `  K
) p )
177, 1, 14cvrlt 29460 . . . . 5  |-  ( ( ( K  e.  HL  /\  .0.  e.  ( Base `  K )  /\  p  e.  ( Base `  K
) )  /\  .0.  (  <o  `  K )
p )  ->  .0.  .<  p )
185, 10, 12, 16, 17syl31anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  .<  p )
19 simp3 957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  .<  W )
20 hlpos 29555 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Poset )
215, 20syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  K  e.  Poset )
22 simp1r 980 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  W  e.  H )
237, 3lhpbase 30187 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2422, 23syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  W  e.  ( Base `  K ) )
257, 1plttr 14104 . . . . 5  |-  ( ( K  e.  Poset  /\  (  .0.  e.  ( Base `  K
)  /\  p  e.  ( Base `  K )  /\  W  e.  ( Base `  K ) ) )  ->  ( (  .0.  .<  p  /\  p  .<  W )  ->  .0.  .<  W ) )
2621, 10, 12, 24, 25syl13anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  -> 
( (  .0.  .<  p  /\  p  .<  W )  ->  .0.  .<  W ) )
2718, 19, 26mp2and 660 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  .<  W )
2827rexlimdv3a 2669 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( E. p  e.  ( Atoms `  K )
p  .<  W  ->  .0.  .<  W ) )
294, 28mpd 14 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  .<  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   Posetcpo 14074   ltcplt 14075   0.cp0 14143   OPcops 29362    <o ccvr 29452   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  lhpn0  30193
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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