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Theorem llnle 37981
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 37899 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑋0 ) → ∃𝑝𝐴 𝑝 𝑋)
91, 2, 3, 8syl3anc 1371 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ∃𝑝𝐴 𝑝 𝑋)
10 simp1ll 1236 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝐾 ∈ HL)
114, 7atbase 37751 . . . . . . 7 (𝑝𝐴𝑝𝐵)
12113ad2ant2 1134 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝐵)
13 simp1lr 1237 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑋𝐵)
14 simp3 1138 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝 𝑋)
15 simp2 1137 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝐴)
16 simp1rr 1239 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ¬ 𝑋𝐴)
17 nelne2 3042 . . . . . . . 8 ((𝑝𝐴 ∧ ¬ 𝑋𝐴) → 𝑝𝑋)
1815, 16, 17syl2anc 584 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝑋)
19 eqid 2736 . . . . . . . . 9 (lt‘𝐾) = (lt‘𝐾)
205, 19pltval 18221 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑋𝐵) → (𝑝(lt‘𝐾)𝑋 ↔ (𝑝 𝑋𝑝𝑋)))
2110, 15, 13, 20syl3anc 1371 . . . . . . 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 2736 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
24 eqid 2736 . . . . . . 7 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
254, 5, 19, 23, 24, 7hlrelat3 37875 . . . . . 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 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ∃𝑞𝐴 (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋))
27 simp1ll 1236 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝐾 ∈ HL)
28 simp21 1206 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑝𝐴)
29 simp23 1208 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑞𝐴)
304, 23, 7hlatjcl 37829 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
3127, 28, 29, 30syl3anc 1371 . . . . . . . . . . 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 1201 . . . . . . . . . . 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 37971 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ 𝐵𝑝𝐴) ∧ 𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞)) → (𝑝(join‘𝐾)𝑞) ∈ 𝑁)
3527, 31, 28, 32, 34syl31anc 1373 . . . . . . . . . 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 1202 . . . . . . . . . 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 5108 . . . . . . . . . . 11 (𝑦 = (𝑝(join‘𝐾)𝑞) → (𝑦 𝑋 ↔ (𝑝(join‘𝐾)𝑞) 𝑋))
3837rspcev 3581 . . . . . . . . . 10 (((𝑝(join‘𝐾)𝑞) ∈ 𝑁 ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)
3935, 36, 38syl2anc 584 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → ∃𝑦𝑁 𝑦 𝑋)
40393exp 1119 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ((𝑝𝐴𝑝 𝑋𝑞𝐴) → ((𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)))
41403expd 1353 . . . . . . 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 1111 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → (𝑞𝐴 → ((𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)))
4342rexlimdv 3150 . . . . 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 1119 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → (𝑝𝐴 → (𝑝 𝑋 → ∃𝑦𝑁 𝑦 𝑋)))
4645rexlimdv 3150 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  ltcplt 18197  joincjn 18200  0.cp0 18312  ccvr 37724  Atomscatm 37725  HLchlt 37812  LLinesclln 37954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961
This theorem is referenced by:  llnmlplnN  38002  lplnle  38003  llncvrlpln  38021
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