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Theorem llnle 35299
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 807 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → 𝐾 ∈ HL)
2 simplr 809 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → 𝑋𝐵)
3 simprl 811 . . 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 35217 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑋0 ) → ∃𝑝𝐴 𝑝 𝑋)
91, 2, 3, 8syl3anc 1473 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) → ∃𝑝𝐴 𝑝 𝑋)
10 simp1ll 1300 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝐾 ∈ HL)
114, 7atbase 35071 . . . . . . 7 (𝑝𝐴𝑝𝐵)
12113ad2ant2 1128 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝐵)
13 simp1lr 1301 . . . . . 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 1303 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ¬ 𝑋𝐴)
17 nelne2 3021 . . . . . . . 8 ((𝑝𝐴 ∧ ¬ 𝑋𝐴) → 𝑝𝑋)
1815, 16, 17syl2anc 696 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝𝑋)
19 eqid 2752 . . . . . . . . 9 (lt‘𝐾) = (lt‘𝐾)
205, 19pltval 17153 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑋𝐵) → (𝑝(lt‘𝐾)𝑋 ↔ (𝑝 𝑋𝑝𝑋)))
2110, 15, 13, 20syl3anc 1473 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → (𝑝(lt‘𝐾)𝑋 ↔ (𝑝 𝑋𝑝𝑋)))
2214, 18, 21mpbir2and 995 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → 𝑝(lt‘𝐾)𝑋)
23 eqid 2752 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
24 eqid 2752 . . . . . . 7 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
254, 5, 19, 23, 24, 7hlrelat3 35193 . . . . . 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 1476 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ 𝑝𝐴𝑝 𝑋) → ∃𝑞𝐴 (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋))
27 simp1ll 1300 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝐾 ∈ HL)
28 simp21 1246 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑝𝐴)
29 simp23 1248 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑋0 ∧ ¬ 𝑋𝐴)) ∧ (𝑝𝐴𝑝 𝑋𝑞𝐴) ∧ (𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞) 𝑋)) → 𝑞𝐴)
304, 23, 7hlatjcl 35148 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
3127, 28, 29, 30syl3anc 1473 . . . . . . . . . . 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 1241 . . . . . . . . . . 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 35289 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ 𝐵𝑝𝐴) ∧ 𝑝( ⋖ ‘𝐾)(𝑝(join‘𝐾)𝑞)) → (𝑝(join‘𝐾)𝑞) ∈ 𝑁)
3527, 31, 28, 32, 34syl31anc 1476 . . . . . . . . . 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 1242 . . . . . . . . . 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 4799 . . . . . . . . . . 11 (𝑦 = (𝑝(join‘𝐾)𝑞) → (𝑦 𝑋 ↔ (𝑝(join‘𝐾)𝑞) 𝑋))
3837rspcev 3441 . . . . . . . . . 10 (((𝑝(join‘𝐾)𝑞) ∈ 𝑁 ∧ (𝑝(join‘𝐾)𝑞) 𝑋) → ∃𝑦𝑁 𝑦 𝑋)
3935, 36, 38syl2anc 696 . . . . . . . . 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 1442 . . . . . . 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 3160 . . . . 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 3160 . 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 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wrex 3043   class class class wbr 4796  cfv 6041  (class class class)co 6805  Basecbs 16051  lecple 16142  ltcplt 17134  joincjn 17137  0.cp0 17230  ccvr 35044  Atomscatm 35045  HLchlt 35132  LLinesclln 35272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-preset 17121  df-poset 17139  df-plt 17151  df-lub 17167  df-glb 17168  df-join 17169  df-meet 17170  df-p0 17232  df-lat 17239  df-clat 17301  df-oposet 34958  df-ol 34960  df-oml 34961  df-covers 35048  df-ats 35049  df-atl 35080  df-cvlat 35104  df-hlat 35133  df-llines 35279
This theorem is referenced by:  llnmlplnN  35320  lplnle  35321  llncvrlpln  35339
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