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Theorem dalem59 36736
 Description: Lemma for dath 36741. Analogue of dalem57 36734 for 𝐹. (Contributed by NM, 10-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))))
dalem59.m = (meet‘𝐾)
dalem59.o 𝑂 = (LPlanes‘𝐾)
dalem59.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem59.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem59.f 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
dalem59.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem59.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem59.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
dalem59.b1 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
Assertion
Ref Expression
dalem59 ((𝜑𝑌 = 𝑍𝜓) → 𝐹 𝐵)

Proof of Theorem dalem59
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)))))
2 dalem.l . . . . 5 = (le‘𝐾)
3 dalem.j . . . . 5 = (join‘𝐾)
4 dalem.a . . . . 5 𝐴 = (Atoms‘𝐾)
5 dalem59.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
6 dalem59.z . . . . 5 𝑍 = ((𝑆 𝑇) 𝑈)
71, 2, 3, 4, 5, 6dalemrot 36662 . . . 4 (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))
873ad2ant1 1127 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))
91, 2, 3, 4, 5, 6dalemrotyz 36663 . . . 4 ((𝜑𝑌 = 𝑍) → ((𝑄 𝑅) 𝑃) = ((𝑇 𝑈) 𝑆))
1093adant3 1126 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑄 𝑅) 𝑃) = ((𝑇 𝑈) 𝑆))
11 dalem.ps . . . . 5 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
121, 2, 3, 4, 11, 5dalemrotps 36696 . . . 4 ((𝜑𝜓) → ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
13123adant2 1125 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
14 biid 262 . . . 4 ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))) ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))
15 biid 262 . . . 4 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))) ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
16 dalem59.m . . . 4 = (meet‘𝐾)
17 dalem59.o . . . 4 𝑂 = (LPlanes‘𝐾)
18 eqid 2825 . . . 4 ((𝑄 𝑅) 𝑃) = ((𝑄 𝑅) 𝑃)
19 eqid 2825 . . . 4 ((𝑇 𝑈) 𝑆) = ((𝑇 𝑈) 𝑆)
20 dalem59.f . . . 4 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
21 dalem59.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
22 dalem59.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
23 dalem59.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
24 eqid 2825 . . . 4 (((𝐻 𝐼) 𝐺) ((𝑄 𝑅) 𝑃)) = (((𝐻 𝐼) 𝐺) ((𝑄 𝑅) 𝑃))
2514, 2, 3, 4, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24dalem58 36735 . . 3 (((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))) ∧ ((𝑄 𝑅) 𝑃) = ((𝑇 𝑈) 𝑆) ∧ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑)))) → 𝐹 (((𝐻 𝐼) 𝐺) ((𝑄 𝑅) 𝑃)))
268, 10, 13, 25syl3anc 1365 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐹 (((𝐻 𝐼) 𝐺) ((𝑄 𝑅) 𝑃)))
271dalemkehl 36628 . . . . . 6 (𝜑𝐾 ∈ HL)
28273ad2ant1 1127 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
291, 2, 3, 4, 11, 16, 17, 5, 6, 21dalem29 36706 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
301, 2, 3, 4, 11, 16, 17, 5, 6, 22dalem34 36711 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
311, 2, 3, 4, 11, 16, 17, 5, 6, 23dalem23 36701 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
323, 4hlatjrot 36378 . . . . 5 ((𝐾 ∈ HL ∧ (𝐻𝐴𝐼𝐴𝐺𝐴)) → ((𝐻 𝐼) 𝐺) = ((𝐺 𝐻) 𝐼))
3328, 29, 30, 31, 32syl13anc 1366 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐻 𝐼) 𝐺) = ((𝐺 𝐻) 𝐼))
341, 3, 4dalemqrprot 36653 . . . . . 6 (𝜑 → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
3534, 5syl6eqr 2878 . . . . 5 (𝜑 → ((𝑄 𝑅) 𝑃) = 𝑌)
36353ad2ant1 1127 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑄 𝑅) 𝑃) = 𝑌)
3733, 36oveq12d 7169 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐻 𝐼) 𝐺) ((𝑄 𝑅) 𝑃)) = (((𝐺 𝐻) 𝐼) 𝑌))
38 dalem59.b1 . . 3 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
3937, 38syl6eqr 2878 . 2 ((𝜑𝑌 = 𝑍𝜓) → (((𝐻 𝐼) 𝐺) ((𝑄 𝑅) 𝑃)) = 𝐵)
4026, 39breqtrd 5088 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐹 𝐵)
 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-3or 1082  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:  dalem61  36738
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