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

Theorem llnle 34281
Description: Any element greater than 0 and not an atom majorizes a lattice line. (Contributed by NM, 28-Jun-2012.)
Hypotheses
Ref Expression
llnle.b 𝐵 = (Base‘𝐾)
llnle.l = (le‘𝐾)
llnle.z 0 = (0.‘𝐾)
llnle.a 𝐴 = (Atoms‘𝐾)
llnle.n 𝑁 = (LLines‘𝐾)
Assertion
Ref Expression
llnle (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ∃𝑦𝑁 𝑦 𝑋)
Distinct variable groups:   𝑦,𝐾   𝑦,   𝑦,𝑁   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   0 (𝑦)

Proof of Theorem llnle
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 789 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → 𝐾 ∈ HL)
2 simplr 791 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → 𝑋𝐵)
3 simprl 793 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → 𝑋0 )
4 llnle.b . . . 4 𝐵 = (Base‘𝐾)
5 llnle.l . . . 4 = (le‘𝐾)
6 llnle.z . . . 4 0 = (0.‘𝐾)
7 llnle.a . . . 4 𝐴 = (Atoms‘𝐾)
84, 5, 6, 7atle 34199 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑋0 ) → ∃𝑝𝐴 𝑝 𝑋)
91, 2, 3, 8syl3anc 1323 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ∃𝑝𝐴 𝑝 𝑋)
10 simp1ll 1122 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝐾 ∈ HL)
114, 7atbase 34053 . . . . . . 7 (𝑝𝐴𝑝𝐵)
12113ad2ant2 1081 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝐵)
13 simp1lr 1123 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑋𝐵)
14 simp3 1061 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝 𝑋)
15 simp2 1060 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝐴)
16 simp1rr 1125 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ¬ 𝑋𝐴)
17 nelne2 2887 . . . . . . . 8 ((𝑝𝐴 ∧ ¬ 𝑋𝐴) → 𝑝𝑋)
1815, 16, 17syl2anc 692 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝑋)
19 eqid 2621 . . . . . . . . 9 (lt‘𝐾) = (lt‘𝐾)
205, 19pltval 16881 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑋𝐵) → (𝑝(lt‘𝐾)𝑋 ↔ (𝑝 𝑋𝑝𝑋)))
2110, 15, 13, 20syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → (𝑝(lt‘𝐾)𝑋 ↔ (𝑝 𝑋𝑝𝑋)))
2214, 18, 21mpbir2and 956 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝(lt‘𝐾)𝑋)
23 eqid 2621 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
24 eqid 2621 . . . . . . 7 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
254, 5, 19, 23, 24, 7hlrelat3 34175 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑝𝐵𝑋𝐵) ∧ 𝑝(lt‘𝐾)𝑋) → ∃𝑞𝐴 (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋))
2610, 12, 13, 22, 25syl31anc 1326 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ∃𝑞𝐴 (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋))
27 simp1ll 1122 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝐾 ∈ HL)
28 simp21 1092 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑝𝐴)
29 simp23 1094 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑞𝐴)
304, 23, 7hlatjcl 34130 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
3127, 28, 29, 30syl3anc 1323 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
32 simp3l 1087 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞))
33 llnle.n . . . . . . . . . . . 12 𝑁 = (LLines‘𝐾)
344, 24, 7, 33llni 34271 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ 𝐵𝑝𝐴) ∧ 𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞)) → (𝑝(join‘𝐾)𝑞) ∈ 𝑁)
3527, 31, 28, 32, 34syl31anc 1326 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → (𝑝(join‘𝐾)𝑞) ∈ 𝑁)
36 simp3r 1088 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → (𝑝(join‘𝐾)𝑞) 𝑋)
37 breq1 4616 . . . . . . . . . . 11 (𝑦 = (𝑝(join‘𝐾)𝑞) → (𝑦 𝑋 ↔ (𝑝(join‘𝐾)𝑞) 𝑋))
3837rspcev 3295 . . . . . . . . . 10 (((𝑝(join‘𝐾)𝑞) ∈ 𝑁 ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)
3935, 36, 38syl2anc 692 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → ∃𝑦𝑁 𝑦 𝑋)
40393exp 1261 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ((𝑝𝐴𝑝 𝑋𝑞𝐴) → ((𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)))
41403expd 1281 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → (𝑝𝐴 → (𝑝 𝑋 → (𝑞𝐴 → ((𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)))))
42413imp 1254 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → (𝑞𝐴 → ((𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)))
4342rexlimdv 3023 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → (∃𝑞𝐴 (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋))
4426, 43mpd 15 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ∃𝑦𝑁 𝑦 𝑋)
45443exp 1261 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → (𝑝𝐴 → (𝑝 𝑋 → ∃𝑦𝑁 𝑦 𝑋)))
4645rexlimdv 3023 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → (∃𝑝𝐴 𝑝 𝑋 → ∃𝑦𝑁 𝑦 𝑋))
479, 46mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ∃𝑦𝑁 𝑦 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  ltcplt 16862  joincjn 16865  0.cp0 16958  ccvr 34026  Atomscatm 34027  HLchlt 34114  LLinesclln 34254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-clat 17029  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115  df-llines 34261
This theorem is referenced by:  llnmlplnN  34302  lplnle  34303  llncvrlpln  34321
  Copyright terms: Public domain W3C validator