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Theorem dalem59 39334
Description: Lemma for dath 39339. Analogue of dalem57 39332 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 39260 . . . 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 1130 . . 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 39261 . . . 4 ((𝜑𝑌 = 𝑍) → ((𝑄 𝑅) 𝑃) = ((𝑇 𝑈) 𝑆))
1093adant3 1129 . . 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 39294 . . . 4 ((𝜑𝜓) → ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
13123adant2 1128 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 ((𝑄 𝑅) 𝑃) ∧ (𝑑𝑐 ∧ ¬ 𝑑 ((𝑄 𝑅) 𝑃) ∧ 𝐶 (𝑐 𝑑))))
14 biid 260 . . . 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 260 . . . 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 2725 . . . 4 ((𝑄 𝑅) 𝑃) = ((𝑄 𝑅) 𝑃)
19 eqid 2725 . . . 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 2725 . . . 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 39333 . . 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 1368 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐹 (((𝐻 𝐼) 𝐺) ((𝑄 𝑅) 𝑃)))
271dalemkehl 39226 . . . . . 6 (𝜑𝐾 ∈ HL)
28273ad2ant1 1130 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
291, 2, 3, 4, 11, 16, 17, 5, 6, 21dalem29 39304 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
301, 2, 3, 4, 11, 16, 17, 5, 6, 22dalem34 39309 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
311, 2, 3, 4, 11, 16, 17, 5, 6, 23dalem23 39299 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
323, 4hlatjrot 38975 . . . . 5 ((𝐾 ∈ HL ∧ (𝐻𝐴𝐼𝐴𝐺𝐴)) → ((𝐻 𝐼) 𝐺) = ((𝐺 𝐻) 𝐼))
3328, 29, 30, 31, 32syl13anc 1369 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐻 𝐼) 𝐺) = ((𝐺 𝐻) 𝐼))
341, 3, 4dalemqrprot 39251 . . . . . 6 (𝜑 → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
3534, 5eqtr4di 2783 . . . . 5 (𝜑 → ((𝑄 𝑅) 𝑃) = 𝑌)
36353ad2ant1 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑄 𝑅) 𝑃) = 𝑌)
3733, 36oveq12d 7437 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐻 𝐼) 𝐺) ((𝑄 𝑅) 𝑃)) = (((𝐺 𝐻) 𝐼) 𝑌))
38 dalem59.b1 . . 3 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
3937, 38eqtr4di 2783 . 2 ((𝜑𝑌 = 𝑍𝜓) → (((𝐻 𝐼) 𝐺) ((𝑄 𝑅) 𝑃)) = 𝐵)
4026, 39breqtrd 5175 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐹 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  lecple 17243  joincjn 18306  meetcmee 18307  Atomscatm 38865  HLchlt 38952  LPlanesclpl 39095
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-proset 18290  df-poset 18308  df-plt 18325  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-lat 18427  df-clat 18494  df-oposet 38778  df-ol 38780  df-oml 38781  df-covers 38868  df-ats 38869  df-atl 38900  df-cvlat 38924  df-hlat 38953  df-llines 39101  df-lplanes 39102  df-lvols 39103
This theorem is referenced by:  dalem61  39336
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