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Theorem dalem2 35815
Description: Lemma for dath 35890. 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 35777 . . 3 (𝜑𝐾 ∈ HL)
31dalempea 35780 . . 3 (𝜑𝑃𝐴)
41dalemqea 35781 . . 3 (𝜑𝑄𝐴)
51dalemsea 35783 . . 3 (𝜑𝑆𝐴)
61dalemtea 35784 . . 3 (𝜑𝑇𝐴)
7 dalemc.j . . . 4 = (join‘𝐾)
8 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
97, 8hlatj4 35528 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑆) (𝑄 𝑇)))
102, 3, 4, 5, 6, 9syl122anc 1447 . 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 35807 . . . 4 (𝜑 → (𝑃 𝑆) ∈ (LLines‘𝐾))
151, 11, 7, 8, 12, 13dalemqnet 35806 . . . . 5 (𝜑𝑄𝑇)
16 eqid 2778 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
177, 8, 16llni2 35666 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) ∧ 𝑄𝑇) → (𝑄 𝑇) ∈ (LLines‘𝐾))
182, 4, 6, 15, 17syl31anc 1441 . . . 4 (𝜑 → (𝑄 𝑇) ∈ (LLines‘𝐾))
191, 11, 7, 8, 12, 13dalem1 35813 . . . 4 (𝜑 → (𝑃 𝑆) ≠ (𝑄 𝑇))
201, 11, 7, 8, 12, 13dalemcea 35814 . . . . 5 (𝜑𝐶𝐴)
211dalemclpjs 35788 . . . . 5 (𝜑𝐶 (𝑃 𝑆))
221dalemclqjt 35789 . . . . 5 (𝜑𝐶 (𝑄 𝑇))
23 eqid 2778 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
24 eqid 2778 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
2511, 23, 24, 8, 162llnm4 35724 . . . . 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 1456 . . . 4 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ≠ (0.‘𝐾))
2723, 24, 8, 162llnmat 35678 . . . 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 1446 . . 3 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ 𝐴)
297, 23, 8, 16, 122llnmj 35714 . . . 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 1439 . . 3 (𝜑 → (((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝑂))
3128, 30mpbid 224 . 2 (𝜑 → ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝑂)
3210, 31eqeltrd 2859 1 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝑂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969   class class class wbr 4886  cfv 6135  (class class class)co 6922  Basecbs 16255  lecple 16345  joincjn 17330  meetcmee 17331  0.cp0 17423  Atomscatm 35417  HLchlt 35504  LLinesclln 35645  LPlanesclpl 35646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-proset 17314  df-poset 17332  df-plt 17344  df-lub 17360  df-glb 17361  df-join 17362  df-meet 17363  df-p0 17425  df-lat 17432  df-clat 17494  df-oposet 35330  df-ol 35332  df-oml 35333  df-covers 35420  df-ats 35421  df-atl 35452  df-cvlat 35476  df-hlat 35505  df-llines 35652  df-lplanes 35653
This theorem is referenced by:  dalemdea  35816
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