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Theorem dalem2 36905
 Description: Lemma for dath 36980. Show the lines 𝑃𝑄 and 𝑆𝑇 form a plane. (Contributed by NM, 11-Aug-2012.)
Hypotheses
Ref Expression
dalema.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
dalemc.l = (le‘𝐾)
dalemc.j = (join‘𝐾)
dalemc.a 𝐴 = (Atoms‘𝐾)
dalem1.o 𝑂 = (LPlanes‘𝐾)
dalem1.y 𝑌 = ((𝑃 𝑄) 𝑅)
Assertion
Ref Expression
dalem2 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝑂)

Proof of Theorem dalem2
StepHypRef Expression
1 dalema.ph . . . 4 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
21dalemkehl 36867 . . 3 (𝜑𝐾 ∈ HL)
31dalempea 36870 . . 3 (𝜑𝑃𝐴)
41dalemqea 36871 . . 3 (𝜑𝑄𝐴)
51dalemsea 36873 . . 3 (𝜑𝑆𝐴)
61dalemtea 36874 . . 3 (𝜑𝑇𝐴)
7 dalemc.j . . . 4 = (join‘𝐾)
8 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
97, 8hlatj4 36618 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑆) (𝑄 𝑇)))
102, 3, 4, 5, 6, 9syl122anc 1376 . 2 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑆) (𝑄 𝑇)))
11 dalemc.l . . . . 5 = (le‘𝐾)
12 dalem1.o . . . . 5 𝑂 = (LPlanes‘𝐾)
13 dalem1.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
141, 11, 7, 8, 12, 13dalempjsen 36897 . . . 4 (𝜑 → (𝑃 𝑆) ∈ (LLines‘𝐾))
151, 11, 7, 8, 12, 13dalemqnet 36896 . . . . 5 (𝜑𝑄𝑇)
16 eqid 2824 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
177, 8, 16llni2 36756 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) ∧ 𝑄𝑇) → (𝑄 𝑇) ∈ (LLines‘𝐾))
182, 4, 6, 15, 17syl31anc 1370 . . . 4 (𝜑 → (𝑄 𝑇) ∈ (LLines‘𝐾))
191, 11, 7, 8, 12, 13dalem1 36903 . . . 4 (𝜑 → (𝑃 𝑆) ≠ (𝑄 𝑇))
201, 11, 7, 8, 12, 13dalemcea 36904 . . . . 5 (𝜑𝐶𝐴)
211dalemclpjs 36878 . . . . 5 (𝜑𝐶 (𝑃 𝑆))
221dalemclqjt 36879 . . . . 5 (𝜑𝐶 (𝑄 𝑇))
23 eqid 2824 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
24 eqid 2824 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
2511, 23, 24, 8, 162llnm4 36814 . . . . 5 ((𝐾 ∈ HL ∧ (𝐶𝐴 ∧ (𝑃 𝑆) ∈ (LLines‘𝐾) ∧ (𝑄 𝑇) ∈ (LLines‘𝐾)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇))) → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ≠ (0.‘𝐾))
262, 20, 14, 18, 21, 22, 25syl132anc 1385 . . . 4 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ≠ (0.‘𝐾))
2723, 24, 8, 162llnmat 36768 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 𝑆) ∈ (LLines‘𝐾) ∧ (𝑄 𝑇) ∈ (LLines‘𝐾)) ∧ ((𝑃 𝑆) ≠ (𝑄 𝑇) ∧ ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ≠ (0.‘𝐾))) → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ 𝐴)
282, 14, 18, 19, 26, 27syl32anc 1375 . . 3 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ 𝐴)
297, 23, 8, 16, 122llnmj 36804 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 𝑆) ∈ (LLines‘𝐾) ∧ (𝑄 𝑇) ∈ (LLines‘𝐾)) → (((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝑂))
302, 14, 18, 29syl3anc 1368 . . 3 (𝜑 → (((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝑂))
3128, 30mpbid 235 . 2 (𝜑 → ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝑂)
3210, 31eqeltrd 2916 1 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝑂)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3014   class class class wbr 5052  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  0.cp0 17647  Atomscatm 36507  HLchlt 36594  LLinesclln 36735  LPlanesclpl 36736 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-lat 17656  df-clat 17718  df-oposet 36420  df-ol 36422  df-oml 36423  df-covers 36510  df-ats 36511  df-atl 36542  df-cvlat 36566  df-hlat 36595  df-llines 36742  df-lplanes 36743 This theorem is referenced by:  dalemdea  36906
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