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Theorem dalem60 34525
Description: Lemma for dath 34529. 𝐵 is an axis of perspectivity (almost). (Contributed by NM, 11-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))))
dalem60.m = (meet‘𝐾)
dalem60.o 𝑂 = (LPlanes‘𝐾)
dalem60.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem60.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem60.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
dalem60.e 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
dalem60.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem60.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem60.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
dalem60.b1 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
Assertion
Ref Expression
dalem60 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) = 𝐵)

Proof of Theorem dalem60
StepHypRef Expression
1 dalem.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)))))
2 dalem.l . . . 4 = (le‘𝐾)
3 dalem.j . . . 4 = (join‘𝐾)
4 dalem.a . . . 4 𝐴 = (Atoms‘𝐾)
5 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
6 dalem60.m . . . 4 = (meet‘𝐾)
7 dalem60.o . . . 4 𝑂 = (LPlanes‘𝐾)
8 dalem60.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
9 dalem60.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
10 dalem60.d . . . 4 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
11 dalem60.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
12 dalem60.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
13 dalem60.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
14 dalem60.b1 . . . 4 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem57 34522 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 𝐵)
16 dalem60.e . . . 4 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
171, 2, 3, 4, 5, 6, 7, 8, 9, 16, 11, 12, 13, 14dalem58 34523 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐸 𝐵)
181dalemkelat 34417 . . . . 5 (𝜑𝐾 ∈ Lat)
19183ad2ant1 1080 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
201, 2, 3, 4, 6, 7, 8, 9, 10dalemdea 34455 . . . . . 6 (𝜑𝐷𝐴)
21 eqid 2621 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2221, 4atbase 34083 . . . . . 6 (𝐷𝐴𝐷 ∈ (Base‘𝐾))
2320, 22syl 17 . . . . 5 (𝜑𝐷 ∈ (Base‘𝐾))
24233ad2ant1 1080 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 ∈ (Base‘𝐾))
251, 2, 3, 4, 6, 7, 8, 9, 16dalemeea 34456 . . . . . 6 (𝜑𝐸𝐴)
2621, 4atbase 34083 . . . . . 6 (𝐸𝐴𝐸 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . 5 (𝜑𝐸 ∈ (Base‘𝐾))
28273ad2ant1 1080 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐸 ∈ (Base‘𝐾))
29 eqid 2621 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
301, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14dalem53 34518 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
3121, 29llnbase 34302 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
3230, 31syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
3321, 2, 3latjle12 16990 . . . 4 ((𝐾 ∈ Lat ∧ (𝐷 ∈ (Base‘𝐾) ∧ 𝐸 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾))) → ((𝐷 𝐵𝐸 𝐵) ↔ (𝐷 𝐸) 𝐵))
3419, 24, 28, 32, 33syl13anc 1325 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐷 𝐵𝐸 𝐵) ↔ (𝐷 𝐸) 𝐵))
3515, 17, 34mpbi2and 955 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) 𝐵)
361dalemkehl 34416 . . . 4 (𝜑𝐾 ∈ HL)
37363ad2ant1 1080 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
381, 2, 3, 4, 6, 7, 8, 9, 10, 16dalemdnee 34459 . . . . 5 (𝜑𝐷𝐸)
393, 4, 29llni2 34305 . . . . 5 (((𝐾 ∈ HL ∧ 𝐷𝐴𝐸𝐴) ∧ 𝐷𝐸) → (𝐷 𝐸) ∈ (LLines‘𝐾))
4036, 20, 25, 38, 39syl31anc 1326 . . . 4 (𝜑 → (𝐷 𝐸) ∈ (LLines‘𝐾))
41403ad2ant1 1080 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) ∈ (LLines‘𝐾))
422, 29llncmp 34315 . . 3 ((𝐾 ∈ HL ∧ (𝐷 𝐸) ∈ (LLines‘𝐾) ∧ 𝐵 ∈ (LLines‘𝐾)) → ((𝐷 𝐸) 𝐵 ↔ (𝐷 𝐸) = 𝐵))
4337, 41, 30, 42syl3anc 1323 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐷 𝐸) 𝐵 ↔ (𝐷 𝐸) = 𝐵))
4435, 43mpbid 222 1 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) = 𝐵)
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  Latclat 16973  Atomscatm 34057  HLchlt 34144  LLinesclln 34284  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-3or 1037  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:  dalem61  34526
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