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Theorem dalemdea 33764
Description: Lemma for dath 33838. 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 33763 . . 3 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝑂)
92dalemkehl 33725 . . . 4 (𝜑𝐾 ∈ HL)
102dalempea 33728 . . . . 5 (𝜑𝑃𝐴)
112dalemqea 33729 . . . . 5 (𝜑𝑄𝐴)
122dalemrea 33730 . . . . . 6 (𝜑𝑅𝐴)
132dalemyeo 33734 . . . . . 6 (𝜑𝑌𝑂)
144, 5, 6, 7lplnri1 33655 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑌𝑂) → 𝑃𝑄)
159, 10, 11, 12, 13, 14syl131anc 1330 . . . . 5 (𝜑𝑃𝑄)
16 eqid 2604 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
174, 5, 16llni2 33614 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 𝑄) ∈ (LLines‘𝐾))
189, 10, 11, 15, 17syl31anc 1320 . . . 4 (𝜑 → (𝑃 𝑄) ∈ (LLines‘𝐾))
192dalemsea 33731 . . . . 5 (𝜑𝑆𝐴)
202dalemtea 33732 . . . . 5 (𝜑𝑇𝐴)
212dalemuea 33733 . . . . . 6 (𝜑𝑈𝐴)
222dalemzeo 33735 . . . . . 6 (𝜑𝑍𝑂)
23 dalemdea.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
244, 5, 6, 23lplnri1 33655 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ 𝑍𝑂) → 𝑆𝑇)
259, 19, 20, 21, 22, 24syl131anc 1330 . . . . 5 (𝜑𝑆𝑇)
264, 5, 16llni2 33614 . . . . 5 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
279, 19, 20, 25, 26syl31anc 1320 . . . 4 (𝜑 → (𝑆 𝑇) ∈ (LLines‘𝐾))
28 dalemdea.m . . . . 5 = (meet‘𝐾)
294, 28, 5, 16, 62llnmj 33662 . . . 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 1317 . . 3 (𝜑 → (((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝑂))
318, 30mpbird 245 . 2 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴)
321, 31syl5eqel 2686 1 (𝜑𝐷𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774   class class class wbr 4572  cfv 5785  (class class class)co 6522  Basecbs 15636  lecple 15716  joincjn 16708  meetcmee 16709  Atomscatm 33366  HLchlt 33453  LLinesclln 33593  LPlanesclpl 33594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-lat 16810  df-clat 16872  df-oposet 33279  df-ol 33281  df-oml 33282  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454  df-llines 33600  df-lplanes 33601
This theorem is referenced by:  dalemeea  33765  dalem3  33766  dalem16  33781  dalem52  33826  dalem57  33831  dalem60  33834
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