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Theorem llnle 35326
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 750 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → 𝐾 ∈ HL)
2 simplr 752 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → 𝑋𝐵)
3 simprl 754 . . 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 35244 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑋0 ) → ∃𝑝𝐴 𝑝 𝑋)
91, 2, 3, 8syl3anc 1476 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ∃𝑝𝐴 𝑝 𝑋)
10 simp1ll 1302 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝐾 ∈ HL)
114, 7atbase 35098 . . . . . . 7 (𝑝𝐴𝑝𝐵)
12113ad2ant2 1128 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝐵)
13 simp1lr 1303 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑋𝐵)
14 simp3 1132 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝 𝑋)
15 simp2 1131 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝐴)
16 simp1rr 1305 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ¬ 𝑋𝐴)
17 nelne2 3040 . . . . . . . 8 ((𝑝𝐴 ∧ ¬ 𝑋𝐴) → 𝑝𝑋)
1815, 16, 17syl2anc 573 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝑋)
19 eqid 2771 . . . . . . . . 9 (lt‘𝐾) = (lt‘𝐾)
205, 19pltval 17168 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑋𝐵) → (𝑝(lt‘𝐾)𝑋 ↔ (𝑝 𝑋𝑝𝑋)))
2110, 15, 13, 20syl3anc 1476 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → (𝑝(lt‘𝐾)𝑋 ↔ (𝑝 𝑋𝑝𝑋)))
2214, 18, 21mpbir2and 692 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝(lt‘𝐾)𝑋)
23 eqid 2771 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
24 eqid 2771 . . . . . . 7 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
254, 5, 19, 23, 24, 7hlrelat3 35220 . . . . . 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 1479 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ∃𝑞𝐴 (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋))
27 simp1ll 1302 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝐾 ∈ HL)
28 simp21 1248 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑝𝐴)
29 simp23 1250 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑞𝐴)
304, 23, 7hlatjcl 35175 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
3127, 28, 29, 30syl3anc 1476 . . . . . . . . . . 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 1243 . . . . . . . . . . 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 35316 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ 𝐵𝑝𝐴) ∧ 𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞)) → (𝑝(join‘𝐾)𝑞) ∈ 𝑁)
3527, 31, 28, 32, 34syl31anc 1479 . . . . . . . . . 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 1244 . . . . . . . . . 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 4789 . . . . . . . . . . 11 (𝑦 = (𝑝(join‘𝐾)𝑞) → (𝑦 𝑋 ↔ (𝑝(join‘𝐾)𝑞) 𝑋))
3837rspcev 3460 . . . . . . . . . 10 (((𝑝(join‘𝐾)𝑞) ∈ 𝑁 ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)
3935, 36, 38syl2anc 573 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → ∃𝑦𝑁 𝑦 𝑋)
40393exp 1112 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ((𝑝𝐴𝑝 𝑋𝑞𝐴) → ((𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)))
41403expd 1446 . . . . . . 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 1101 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → (𝑞𝐴 → ((𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)))
4342rexlimdv 3178 . . . . 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 1112 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → (𝑝𝐴 → (𝑝 𝑋 → ∃𝑦𝑁 𝑦 𝑋)))
4645rexlimdv 3178 . 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 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wrex 3062   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  ltcplt 17149  joincjn 17152  0.cp0 17245  ccvr 35071  Atomscatm 35072  HLchlt 35159  LLinesclln 35299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306
This theorem is referenced by:  llnmlplnN  35347  lplnle  35348  llncvrlpln  35366
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