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Theorem dalemdea 39664
Description: Lemma for dath 39738. Frequently-used utility lemma. (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)
dalemdea.m = (meet‘𝐾)
dalemdea.o 𝑂 = (LPlanes‘𝐾)
dalemdea.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalemdea.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalemdea.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
Assertion
Ref Expression
dalemdea (𝜑𝐷𝐴)

Proof of Theorem dalemdea
StepHypRef Expression
1 dalemdea.d . 2 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
2 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)))))
3 dalemc.l . . . 4 = (le‘𝐾)
4 dalemc.j . . . 4 = (join‘𝐾)
5 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
6 dalemdea.o . . . 4 𝑂 = (LPlanes‘𝐾)
7 dalemdea.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
82, 3, 4, 5, 6, 7dalem2 39663 . . 3 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝑂)
92dalemkehl 39625 . . . 4 (𝜑𝐾 ∈ HL)
102dalempea 39628 . . . . 5 (𝜑𝑃𝐴)
112dalemqea 39629 . . . . 5 (𝜑𝑄𝐴)
122dalemrea 39630 . . . . . 6 (𝜑𝑅𝐴)
132dalemyeo 39634 . . . . . 6 (𝜑𝑌𝑂)
144, 5, 6, 7lplnri1 39555 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑌𝑂) → 𝑃𝑄)
159, 10, 11, 12, 13, 14syl131anc 1385 . . . . 5 (𝜑𝑃𝑄)
16 eqid 2737 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
174, 5, 16llni2 39514 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 𝑄) ∈ (LLines‘𝐾))
189, 10, 11, 15, 17syl31anc 1375 . . . 4 (𝜑 → (𝑃 𝑄) ∈ (LLines‘𝐾))
192dalemsea 39631 . . . . 5 (𝜑𝑆𝐴)
202dalemtea 39632 . . . . 5 (𝜑𝑇𝐴)
212dalemuea 39633 . . . . . 6 (𝜑𝑈𝐴)
222dalemzeo 39635 . . . . . 6 (𝜑𝑍𝑂)
23 dalemdea.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
244, 5, 6, 23lplnri1 39555 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ 𝑍𝑂) → 𝑆𝑇)
259, 19, 20, 21, 22, 24syl131anc 1385 . . . . 5 (𝜑𝑆𝑇)
264, 5, 16llni2 39514 . . . . 5 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
279, 19, 20, 25, 26syl31anc 1375 . . . 4 (𝜑 → (𝑆 𝑇) ∈ (LLines‘𝐾))
28 dalemdea.m . . . . 5 = (meet‘𝐾)
294, 28, 5, 16, 62llnmj 39562 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 𝑄) ∈ (LLines‘𝐾) ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → (((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝑂))
309, 18, 27, 29syl3anc 1373 . . 3 (𝜑 → (((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝑂))
318, 30mpbird 257 . 2 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴)
321, 31eqeltrid 2845 1 (𝜑𝐷𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  Atomscatm 39264  HLchlt 39351  LLinesclln 39493  LPlanesclpl 39494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501
This theorem is referenced by:  dalemeea  39665  dalem3  39666  dalem16  39681  dalem52  39726  dalem57  39731  dalem60  39734
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