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Theorem dalem52 36729
Description: Lemma for dath 36741. 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 36628 . . . 4 (𝜑𝐾 ∈ HL)
323ad2ant1 1127 . . 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 36694 . . . 4 (𝜓𝑐 ∈ (Base‘𝐾))
763ad2ant3 1129 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
83, 7jca 512 . 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 36701 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
17 dalem44.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
181, 9, 10, 5, 4, 11, 12, 13, 14, 17dalem29 36706 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
19 dalem44.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
201, 9, 10, 5, 4, 11, 12, 13, 14, 19dalem34 36711 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2116, 18, 203jca 1122 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺𝐴𝐻𝐴𝐼𝐴))
221dalempea 36631 . . . 4 (𝜑𝑃𝐴)
231dalemqea 36632 . . . 4 (𝜑𝑄𝐴)
241dalemrea 36633 . . . 4 (𝜑𝑅𝐴)
2522, 23, 243jca 1122 . . 3 (𝜑 → (𝑃𝐴𝑄𝐴𝑅𝐴))
26253ad2ant1 1127 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑃𝐴𝑄𝐴𝑅𝐴))
271, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem42 36719 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
281dalemyeo 36637 . . 3 (𝜑𝑌𝑂)
29283ad2ant1 1127 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
301, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem45 36722 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐺 𝐻))
311, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem46 36723 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐻 𝐼))
321, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem47 36724 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐼 𝐺))
3330, 31, 323jca 1122 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)))
341, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem48 36725 . . . 4 ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))
351, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem49 36726 . . . 4 ((𝜑𝜓) → ¬ 𝑐 (𝑄 𝑅))
361, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem50 36727 . . . 4 ((𝜑𝜓) → ¬ 𝑐 (𝑅 𝑃))
3734, 35, 363jca 1122 . . 3 ((𝜑𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
38373adant2 1125 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
391, 9, 10, 5, 4, 11, 12, 13, 14, 15dalem27 36704 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐺 𝑃))
401, 9, 10, 5, 4, 11, 12, 13, 14, 17dalem32 36709 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐻 𝑄))
411, 9, 10, 5, 4, 11, 12, 13, 14, 19dalem36 36713 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐼 𝑅))
4239, 40, 413jca 1122 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))
43 biid 262 . . 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 2825 . . 3 ((𝐺 𝐻) 𝐼) = ((𝐺 𝐻) 𝐼)
45 eqid 2825 . . 3 ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) (𝑃 𝑄))
4643, 9, 10, 5, 11, 12, 44, 13, 45dalemdea 36667 . 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 1394 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3020   class class class wbr 5062  cfv 6351  (class class class)co 7151  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Atomscatm 36268  HLchlt 36355  LPlanesclpl 36497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36181  df-ol 36183  df-oml 36184  df-covers 36271  df-ats 36272  df-atl 36303  df-cvlat 36327  df-hlat 36356  df-llines 36503  df-lplanes 36504  df-lvols 36505
This theorem is referenced by:  dalem54  36731  dalem55  36732
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