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Theorem lhp0lt 33487
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 2438 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 lhp0lt.h . . 3  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexlt 33486 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K ) p 
.<  W )
5 simp1l 1012 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  K  e.  HL )
6 hlop 32847 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
7 eqid 2438 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 lhp0lt.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
97, 8op0cl 32669 . . . . . 6  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
105, 6, 93syl 20 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  e.  ( Base `  K
) )
117, 2atbase 32774 . . . . . 6  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
12113ad2ant2 1010 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  e.  ( Base `  K ) )
13 simp2 989 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  e.  ( Atoms `  K ) )
14 eqid 2438 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
158, 14, 2atcvr0 32773 . . . . . 6  |-  ( ( K  e.  HL  /\  p  e.  ( Atoms `  K ) )  ->  .0.  (  <o  `  K
) p )
165, 13, 15syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  (  <o  `  K
) p )
177, 1, 14cvrlt 32755 . . . . 5  |-  ( ( ( K  e.  HL  /\  .0.  e.  ( Base `  K )  /\  p  e.  ( Base `  K
) )  /\  .0.  (  <o  `  K )
p )  ->  .0.  .<  p )
185, 10, 12, 16, 17syl31anc 1221 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  .<  p )
19 simp3 990 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  .<  W )
20 hlpos 32850 . . . . . 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 1013 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  W  e.  H )
237, 3lhpbase 33482 . . . . . 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 15132 . . . . 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 1220 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  -> 
( (  .0.  .<  p  /\  p  .<  W )  ->  .0.  .<  W ) )
2718, 19, 26mp2and 679 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  .<  W )
2827rexlimdv3a 2838 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2711   class class class wbr 4287   ` cfv 5413   Basecbs 14166   Posetcpo 15102   ltcplt 15103   0.cp0 15199   OPcops 32657    <o ccvr 32747   Atomscatm 32748   HLchlt 32835   LHypclh 33468
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-lhyp 33472
This theorem is referenced by:  lhpn0  33488
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