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Theorem dalem51 34524
Description: Lemma for dath 34537. Construct the condition 𝜑 with 𝑐, 𝐺𝐻𝐼, and 𝑌 in place of 𝐶, 𝑌, and 𝑍 respectively. This lets us reuse the special case of Desargues' Theorem where 𝑌𝑍, to eventually prove the case where 𝑌 = 𝑍. (Contributed by NM, 16-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
dalem51 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (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) 𝐼) ≠ 𝑌))

Proof of Theorem dalem51
StepHypRef Expression
1 dalem.ph . . . . . . 7 (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 34424 . . . . . 6 (𝜑𝐾 ∈ HL)
323ad2ant1 1080 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
4 dalem.ps . . . . . . 7 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
54dalemccea 34484 . . . . . 6 (𝜓𝑐𝐴)
653ad2ant3 1082 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
73, 6jca 554 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝐾 ∈ HL ∧ 𝑐𝐴))
8 dalem.l . . . . . 6 = (le‘𝐾)
9 dalem.j . . . . . 6 = (join‘𝐾)
10 dalem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
11 dalem44.m . . . . . 6 = (meet‘𝐾)
12 dalem44.o . . . . . 6 𝑂 = (LPlanes‘𝐾)
13 dalem44.y . . . . . 6 𝑌 = ((𝑃 𝑄) 𝑅)
14 dalem44.z . . . . . 6 𝑍 = ((𝑆 𝑇) 𝑈)
15 dalem44.g . . . . . 6 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
161, 8, 9, 10, 4, 11, 12, 13, 14, 15dalem23 34497 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
17 dalem44.h . . . . . 6 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
181, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem29 34502 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
19 dalem44.i . . . . . 6 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
201, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem34 34507 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2116, 18, 203jca 1240 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝐺𝐴𝐻𝐴𝐼𝐴))
221dalempea 34427 . . . . . 6 (𝜑𝑃𝐴)
231dalemqea 34428 . . . . . 6 (𝜑𝑄𝐴)
241dalemrea 34429 . . . . . 6 (𝜑𝑅𝐴)
2522, 23, 243jca 1240 . . . . 5 (𝜑 → (𝑃𝐴𝑄𝐴𝑅𝐴))
26253ad2ant1 1080 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃𝐴𝑄𝐴𝑅𝐴))
277, 21, 263jca 1240 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)))
281, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem42 34515 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
291dalemyeo 34433 . . . . 5 (𝜑𝑌𝑂)
30293ad2ant1 1080 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
3128, 30jca 554 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂))
321, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem45 34518 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐺 𝐻))
331, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem46 34519 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐻 𝐼))
341, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem47 34520 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐼 𝐺))
3532, 33, 343jca 1240 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)))
361, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem48 34521 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))
371, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem49 34522 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑄 𝑅))
381, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem50 34523 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑅 𝑃))
3936, 37, 383jca 1240 . . . . 5 ((𝜑𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
40393adant2 1078 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
411, 8, 9, 10, 4, 11, 12, 13, 14, 15dalem27 34500 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐺 𝑃))
421, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem32 34505 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐻 𝑄))
431, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem36 34509 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐼 𝑅))
4441, 42, 433jca 1240 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))
4535, 40, 443jca 1240 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((¬ 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))))
4627, 31, 453jca 1240 . 2 ((𝜑𝑌 = 𝑍𝜓) → (((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (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)))))
471, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem43 34516 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)
4846, 47jca 554 1 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (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) 𝐼) ≠ 𝑌))
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 15792  lecple 15880  joincjn 16876  meetcmee 16877  Atomscatm 34065  HLchlt 34152  LPlanesclpl 34293
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 16860  df-poset 16878  df-plt 16890  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-p0 16971  df-lat 16978  df-clat 17040  df-oposet 33978  df-ol 33980  df-oml 33981  df-covers 34068  df-ats 34069  df-atl 34100  df-cvlat 34124  df-hlat 34153  df-llines 34299  df-lplanes 34300  df-lvols 34301
This theorem is referenced by:  dalem53  34526  dalem54  34527
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