Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem51 Structured version   Visualization version   GIF version

Theorem dalem51 35531
Description: Lemma for dath 35544. 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 35431 . . . . . 6 (𝜑𝐾 ∈ HL)
323ad2ant1 1127 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
4 dalem.ps . . . . . . 7 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
54dalemccea 35491 . . . . . 6 (𝜓𝑐𝐴)
653ad2ant3 1129 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
73, 6jca 501 . . . 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 35504 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
17 dalem44.h . . . . . 6 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
181, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem29 35509 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
19 dalem44.i . . . . . 6 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
201, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem34 35514 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2116, 18, 203jca 1122 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝐺𝐴𝐻𝐴𝐼𝐴))
221dalempea 35434 . . . . . 6 (𝜑𝑃𝐴)
231dalemqea 35435 . . . . . 6 (𝜑𝑄𝐴)
241dalemrea 35436 . . . . . 6 (𝜑𝑅𝐴)
2522, 23, 243jca 1122 . . . . 5 (𝜑 → (𝑃𝐴𝑄𝐴𝑅𝐴))
26253ad2ant1 1127 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃𝐴𝑄𝐴𝑅𝐴))
277, 21, 263jca 1122 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)))
281, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem42 35522 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
291dalemyeo 35440 . . . . 5 (𝜑𝑌𝑂)
30293ad2ant1 1127 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
3128, 30jca 501 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂))
321, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem45 35525 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐺 𝐻))
331, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem46 35526 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐻 𝐼))
341, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem47 35527 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐼 𝐺))
3532, 33, 343jca 1122 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)))
361, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem48 35528 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))
371, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem49 35529 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑄 𝑅))
381, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem50 35530 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑅 𝑃))
3936, 37, 383jca 1122 . . . . 5 ((𝜑𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
40393adant2 1125 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
411, 8, 9, 10, 4, 11, 12, 13, 14, 15dalem27 35507 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐺 𝑃))
421, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem32 35512 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐻 𝑄))
431, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem36 35516 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐼 𝑅))
4441, 42, 433jca 1122 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))
4535, 40, 443jca 1122 . . 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 1122 . 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 35523 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)
4846, 47jca 501 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 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4787  cfv 6030  (class class class)co 6795  Basecbs 16063  lecple 16155  joincjn 17151  meetcmee 17152  Atomscatm 35071  HLchlt 35158  LPlanesclpl 35300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6756  df-ov 6798  df-oprab 6799  df-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-lat 17253  df-clat 17315  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-llines 35306  df-lplanes 35307  df-lvols 35308
This theorem is referenced by:  dalem53  35533  dalem54  35534
  Copyright terms: Public domain W3C validator