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Theorem lhp0lt 30814
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 2296 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 lhp0lt.h . . 3  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexlt 30813 . 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 30174 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
7 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 lhp0lt.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
97, 8op0cl 29996 . . . . . 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 30101 . . . . . 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 2296 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
158, 14, 2atcvr0 30100 . . . . . 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 30082 . . . . 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 30177 . . . . . 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 30809 . . . . . 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 14120 . . . . 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 2682 . 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 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271   Basecbs 13164   Posetcpo 14090   ltcplt 14091   0.cp0 14159   OPcops 29984    <o ccvr 30074   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  lhpn0  30815
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  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-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799
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