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Theorem 2llnma1 39790
Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)
Hypotheses
Ref Expression
2llnm.l = (le‘𝐾)
2llnm.j = (join‘𝐾)
2llnm.m = (meet‘𝐾)
2llnm.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
2llnma1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑄 𝑃) (𝑄 𝑅)) = 𝑄)

Proof of Theorem 2llnma1
StepHypRef Expression
1 simp1 1136 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝐾 ∈ HL)
2 simp21 1206 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃𝐴)
3 eqid 2736 . . . 4 (Base‘𝐾) = (Base‘𝐾)
4 2llnm.a . . . 4 𝐴 = (Atoms‘𝐾)
53, 4atbase 39291 . . 3 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
62, 5syl 17 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃 ∈ (Base‘𝐾))
7 simp22 1207 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑄𝐴)
8 simp23 1208 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅𝐴)
9 simp3 1138 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝑅 (𝑃 𝑄))
10 2llnm.j . . . . . 6 = (join‘𝐾)
1110, 4hlatjcom 39370 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
121, 2, 7, 11syl3anc 1372 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃 𝑄) = (𝑄 𝑃))
1312breq2d 5154 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 (𝑃 𝑄) ↔ 𝑅 (𝑄 𝑃)))
149, 13mtbid 324 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝑅 (𝑄 𝑃))
15 2llnm.l . . 3 = (le‘𝐾)
16 2llnm.m . . 3 = (meet‘𝐾)
173, 15, 10, 16, 42llnma1b 39789 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑄 𝑃)) → ((𝑄 𝑃) (𝑄 𝑅)) = 𝑄)
181, 6, 7, 8, 14, 17syl131anc 1384 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑄 𝑃) (𝑄 𝑅)) = 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1086   = wceq 1539  wcel 2107   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  joincjn 18358  meetcmee 18359  Atomscatm 39265  HLchlt 39352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-proset 18341  df-poset 18360  df-plt 18376  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-p0 18471  df-lat 18478  df-clat 18545  df-oposet 39178  df-ol 39180  df-oml 39181  df-covers 39268  df-ats 39269  df-atl 39300  df-cvlat 39324  df-hlat 39353
This theorem is referenced by:  2llnma3r  39791  2llnma2  39792  cdleme17c  40291
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