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Theorem dalem51 39326
Description: Lemma for dath 39339. 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's 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 39226 . . . . . 6 (𝜑𝐾 ∈ HL)
323ad2ant1 1130 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
4 dalem.ps . . . . . . 7 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
54dalemccea 39286 . . . . . 6 (𝜓𝑐𝐴)
653ad2ant3 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
73, 6jca 510 . . . 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 39299 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
17 dalem44.h . . . . . 6 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
181, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem29 39304 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
19 dalem44.i . . . . . 6 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
201, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem34 39309 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2116, 18, 203jca 1125 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝐺𝐴𝐻𝐴𝐼𝐴))
221dalempea 39229 . . . . . 6 (𝜑𝑃𝐴)
231dalemqea 39230 . . . . . 6 (𝜑𝑄𝐴)
241dalemrea 39231 . . . . . 6 (𝜑𝑅𝐴)
2522, 23, 243jca 1125 . . . . 5 (𝜑 → (𝑃𝐴𝑄𝐴𝑅𝐴))
26253ad2ant1 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃𝐴𝑄𝐴𝑅𝐴))
277, 21, 263jca 1125 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)))
281, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem42 39317 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
291dalemyeo 39235 . . . . 5 (𝜑𝑌𝑂)
30293ad2ant1 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
3128, 30jca 510 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂))
321, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem45 39320 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐺 𝐻))
331, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem46 39321 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐻 𝐼))
341, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem47 39322 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐼 𝐺))
3532, 33, 343jca 1125 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)))
361, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem48 39323 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))
371, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem49 39324 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑄 𝑅))
381, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem50 39325 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑅 𝑃))
3936, 37, 383jca 1125 . . . . 5 ((𝜑𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
40393adant2 1128 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
411, 8, 9, 10, 4, 11, 12, 13, 14, 15dalem27 39302 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐺 𝑃))
421, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem32 39307 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐻 𝑄))
431, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem36 39311 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐼 𝑅))
4441, 42, 433jca 1125 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))
4535, 40, 443jca 1125 . . 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 1125 . 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 39318 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)
4846, 47jca 510 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 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-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:  dalem53  39328  dalem54  39329
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