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Theorem llnle 36690
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 765 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → 𝐾 ∈ HL)
2 simplr 767 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → 𝑋𝐵)
3 simprl 769 . . 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 36608 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑋0 ) → ∃𝑝𝐴 𝑝 𝑋)
91, 2, 3, 8syl3anc 1367 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ∃𝑝𝐴 𝑝 𝑋)
10 simp1ll 1232 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝐾 ∈ HL)
114, 7atbase 36461 . . . . . . 7 (𝑝𝐴𝑝𝐵)
12113ad2ant2 1130 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝐵)
13 simp1lr 1233 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑋𝐵)
14 simp3 1134 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝 𝑋)
15 simp2 1133 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝐴)
16 simp1rr 1235 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ¬ 𝑋𝐴)
17 nelne2 3103 . . . . . . . 8 ((𝑝𝐴 ∧ ¬ 𝑋𝐴) → 𝑝𝑋)
1815, 16, 17syl2anc 586 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝑋)
19 eqid 2820 . . . . . . . . 9 (lt‘𝐾) = (lt‘𝐾)
205, 19pltval 17549 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑋𝐵) → (𝑝(lt‘𝐾)𝑋 ↔ (𝑝 𝑋𝑝𝑋)))
2110, 15, 13, 20syl3anc 1367 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → (𝑝(lt‘𝐾)𝑋 ↔ (𝑝 𝑋𝑝𝑋)))
2214, 18, 21mpbir2and 711 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝(lt‘𝐾)𝑋)
23 eqid 2820 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
24 eqid 2820 . . . . . . 7 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
254, 5, 19, 23, 24, 7hlrelat3 36584 . . . . . 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 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ∃𝑞𝐴 (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋))
27 simp1ll 1232 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝐾 ∈ HL)
28 simp21 1202 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑝𝐴)
29 simp23 1204 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑞𝐴)
304, 23, 7hlatjcl 36539 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
3127, 28, 29, 30syl3anc 1367 . . . . . . . . . . 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 1197 . . . . . . . . . . 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 36680 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ 𝐵𝑝𝐴) ∧ 𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞)) → (𝑝(join‘𝐾)𝑞) ∈ 𝑁)
3527, 31, 28, 32, 34syl31anc 1369 . . . . . . . . . 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 1198 . . . . . . . . . 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 5045 . . . . . . . . . . 11 (𝑦 = (𝑝(join‘𝐾)𝑞) → (𝑦 𝑋 ↔ (𝑝(join‘𝐾)𝑞) 𝑋))
3837rspcev 3602 . . . . . . . . . 10 (((𝑝(join‘𝐾)𝑞) ∈ 𝑁 ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)
3935, 36, 38syl2anc 586 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → ∃𝑦𝑁 𝑦 𝑋)
40393exp 1115 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ((𝑝𝐴𝑝 𝑋𝑞𝐴) → ((𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)))
41403expd 1349 . . . . . . 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 1107 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → (𝑞𝐴 → ((𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)))
4342rexlimdv 3270 . . . . 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 1115 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → (𝑝𝐴 → (𝑝 𝑋 → ∃𝑦𝑁 𝑦 𝑋)))
4645rexlimdv 3270 . 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 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wne 3006  wrex 3126   class class class wbr 5042  cfv 6331  (class class class)co 7133  Basecbs 16462  lecple 16551  ltcplt 17530  joincjn 17533  0.cp0 17626  ccvr 36434  Atomscatm 36435  HLchlt 36522  LLinesclln 36663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-proset 17517  df-poset 17535  df-plt 17547  df-lub 17563  df-glb 17564  df-join 17565  df-meet 17566  df-p0 17628  df-lat 17635  df-clat 17697  df-oposet 36348  df-ol 36350  df-oml 36351  df-covers 36438  df-ats 36439  df-atl 36470  df-cvlat 36494  df-hlat 36523  df-llines 36670
This theorem is referenced by:  llnmlplnN  36711  lplnle  36712  llncvrlpln  36730
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