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Theorem dalem52 34517
Description: Lemma for dath 34529. Lines 𝐺𝐻 and 𝑃𝑄 intersect at an atom. (Contributed by NM, 8-Aug-2012.)
Hypotheses
Ref Expression
dalem.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)))))
dalem.l = (le‘𝐾)
dalem.j = (join‘𝐾)
dalem.a 𝐴 = (Atoms‘𝐾)
dalem.ps (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
dalem44.m = (meet‘𝐾)
dalem44.o 𝑂 = (LPlanes‘𝐾)
dalem44.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem44.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem44.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem44.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem44.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
Assertion
Ref Expression
dalem52 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)

Proof of Theorem dalem52
StepHypRef Expression
1 dalem.ph . . . . 5 (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 34416 . . . 4 (𝜑𝐾 ∈ HL)
323ad2ant1 1080 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
4 dalem.ps . . . . 5 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
5 dalem.a . . . . 5 𝐴 = (Atoms‘𝐾)
64, 5dalemcceb 34482 . . . 4 (𝜓𝑐 ∈ (Base‘𝐾))
763ad2ant3 1082 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
83, 7jca 554 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)))
9 dalem.l . . . 4 = (le‘𝐾)
10 dalem.j . . . 4 = (join‘𝐾)
11 dalem44.m . . . 4 = (meet‘𝐾)
12 dalem44.o . . . 4 𝑂 = (LPlanes‘𝐾)
13 dalem44.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
14 dalem44.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
15 dalem44.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
161, 9, 10, 5, 4, 11, 12, 13, 14, 15dalem23 34489 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
17 dalem44.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
181, 9, 10, 5, 4, 11, 12, 13, 14, 17dalem29 34494 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
19 dalem44.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
201, 9, 10, 5, 4, 11, 12, 13, 14, 19dalem34 34499 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2116, 18, 203jca 1240 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺𝐴𝐻𝐴𝐼𝐴))
221dalempea 34419 . . . 4 (𝜑𝑃𝐴)
231dalemqea 34420 . . . 4 (𝜑𝑄𝐴)
241dalemrea 34421 . . . 4 (𝜑𝑅𝐴)
2522, 23, 243jca 1240 . . 3 (𝜑 → (𝑃𝐴𝑄𝐴𝑅𝐴))
26253ad2ant1 1080 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑃𝐴𝑄𝐴𝑅𝐴))
271, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem42 34507 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
281dalemyeo 34425 . . 3 (𝜑𝑌𝑂)
29283ad2ant1 1080 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
301, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem45 34510 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐺 𝐻))
311, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem46 34511 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐻 𝐼))
321, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem47 34512 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐼 𝐺))
3330, 31, 323jca 1240 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)))
341, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem48 34513 . . . 4 ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))
351, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem49 34514 . . . 4 ((𝜑𝜓) → ¬ 𝑐 (𝑄 𝑅))
361, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem50 34515 . . . 4 ((𝜑𝜓) → ¬ 𝑐 (𝑅 𝑃))
3734, 35, 363jca 1240 . . 3 ((𝜑𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
38373adant2 1078 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
391, 9, 10, 5, 4, 11, 12, 13, 14, 15dalem27 34492 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐺 𝑃))
401, 9, 10, 5, 4, 11, 12, 13, 14, 17dalem32 34497 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐻 𝑄))
411, 9, 10, 5, 4, 11, 12, 13, 14, 19dalem36 34501 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐼 𝑅))
4239, 40, 413jca 1240 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))
43 biid 251 . . 3 ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂) ∧ ((¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)) ∧ (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)) ∧ (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))) ↔ (((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂) ∧ ((¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)) ∧ (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)) ∧ (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))))
44 eqid 2621 . . 3 ((𝐺 𝐻) 𝐼) = ((𝐺 𝐻) 𝐼)
45 eqid 2621 . . 3 ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) (𝑃 𝑄))
4643, 9, 10, 5, 11, 12, 44, 13, 45dalemdea 34455 . 2 ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂) ∧ ((¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)) ∧ (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)) ∧ (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
478, 21, 26, 27, 29, 33, 38, 42, 46syl323anc 1353 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15788  lecple 15876  joincjn 16872  meetcmee 16873  Atomscatm 34057  HLchlt 34144  LPlanesclpl 34285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-preset 16856  df-poset 16874  df-plt 16886  df-lub 16902  df-glb 16903  df-join 16904  df-meet 16905  df-p0 16967  df-lat 16974  df-clat 17036  df-oposet 33970  df-ol 33972  df-oml 33973  df-covers 34060  df-ats 34061  df-atl 34092  df-cvlat 34116  df-hlat 34145  df-llines 34291  df-lplanes 34292  df-lvols 34293
This theorem is referenced by:  dalem54  34519  dalem55  34520
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