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Theorem lhp0lt 30800
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 2436 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 lhp0lt.h . . 3  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexlt 30799 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K ) p 
.<  W )
5 simp1l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  K  e.  HL )
6 hlop 30160 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
7 eqid 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 lhp0lt.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
97, 8op0cl 29982 . . . . . 6  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
105, 6, 93syl 19 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  e.  ( Base `  K
) )
117, 2atbase 30087 . . . . . 6  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
12113ad2ant2 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  e.  ( Base `  K ) )
13 simp2 958 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  e.  ( Atoms `  K ) )
14 eqid 2436 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
158, 14, 2atcvr0 30086 . . . . . 6  |-  ( ( K  e.  HL  /\  p  e.  ( Atoms `  K ) )  ->  .0.  (  <o  `  K
) p )
165, 13, 15syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  (  <o  `  K
) p )
177, 1, 14cvrlt 30068 . . . . 5  |-  ( ( ( K  e.  HL  /\  .0.  e.  ( Base `  K )  /\  p  e.  ( Base `  K
) )  /\  .0.  (  <o  `  K )
p )  ->  .0.  .<  p )
185, 10, 12, 16, 17syl31anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  .<  p )
19 simp3 959 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  .<  W )
20 hlpos 30163 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Poset )
215, 20syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  K  e.  Poset )
22 simp1r 982 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  W  e.  H )
237, 3lhpbase 30795 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2422, 23syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  W  e.  ( Base `  K ) )
257, 1plttr 14427 . . . . 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 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  -> 
( (  .0.  .<  p  /\  p  .<  W )  ->  .0.  .<  W ) )
2718, 19, 26mp2and 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  .<  W )
2827rexlimdv3a 2832 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( E. p  e.  ( Atoms `  K )
p  .<  W  ->  .0.  .<  W ) )
294, 28mpd 15 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  .<  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706   class class class wbr 4212   ` cfv 5454   Basecbs 13469   Posetcpo 14397   ltcplt 14398   0.cp0 14466   OPcops 29970    <o ccvr 30060   Atomscatm 30061   HLchlt 30148   LHypclh 30781
This theorem is referenced by:  lhpn0  30801
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785
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