Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhp0lt Structured version   Unicode version

Theorem lhp0lt 35849
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 2457 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 lhp0lt.h . . 3  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexlt 35848 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K ) p 
.<  W )
5 simp1l 1020 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  K  e.  HL )
6 hlop 35209 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
7 eqid 2457 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 lhp0lt.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
97, 8op0cl 35031 . . . . . 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 35136 . . . . . 6  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
12113ad2ant2 1018 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  e.  ( Base `  K ) )
13 simp2 997 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  e.  ( Atoms `  K ) )
14 eqid 2457 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
158, 14, 2atcvr0 35135 . . . . . 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 35117 . . . . 5  |-  ( ( ( K  e.  HL  /\  .0.  e.  ( Base `  K )  /\  p  e.  ( Base `  K
) )  /\  .0.  (  <o  `  K )
p )  ->  .0.  .<  p )
185, 10, 12, 16, 17syl31anc 1231 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  .0.  .<  p )
19 simp3 998 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  p  .<  W )
20 hlpos 35212 . . . . . 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 1021 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  p  e.  (
Atoms `  K )  /\  p  .<  W )  ->  W  e.  H )
237, 3lhpbase 35844 . . . . . 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 15727 . . . . 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 1230 . . . 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 2951 . 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 973    = wceq 1395    e. wcel 1819   E.wrex 2808   class class class wbr 4456   ` cfv 5594   Basecbs 14644   Posetcpo 15696   ltcplt 15697   0.cp0 15794   OPcops 35019    <o ccvr 35109   Atomscatm 35110   HLchlt 35197   LHypclh 35830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35023  df-ol 35025  df-oml 35026  df-covers 35113  df-ats 35114  df-atl 35145  df-cvlat 35169  df-hlat 35198  df-lhyp 35834
This theorem is referenced by:  lhpn0  35850
  Copyright terms: Public domain W3C validator