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 36739
Description: Lemma for dath 36752. 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 36639 . . . . . 6 (𝜑𝐾 ∈ HL)
323ad2ant1 1125 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
4 dalem.ps . . . . . . 7 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
54dalemccea 36699 . . . . . 6 (𝜓𝑐𝐴)
653ad2ant3 1127 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
73, 6jca 512 . . . 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 36712 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
17 dalem44.h . . . . . 6 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
181, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem29 36717 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
19 dalem44.i . . . . . 6 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
201, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem34 36722 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2116, 18, 203jca 1120 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝐺𝐴𝐻𝐴𝐼𝐴))
221dalempea 36642 . . . . . 6 (𝜑𝑃𝐴)
231dalemqea 36643 . . . . . 6 (𝜑𝑄𝐴)
241dalemrea 36644 . . . . . 6 (𝜑𝑅𝐴)
2522, 23, 243jca 1120 . . . . 5 (𝜑 → (𝑃𝐴𝑄𝐴𝑅𝐴))
26253ad2ant1 1125 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃𝐴𝑄𝐴𝑅𝐴))
277, 21, 263jca 1120 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)))
281, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem42 36730 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
291dalemyeo 36648 . . . . 5 (𝜑𝑌𝑂)
30293ad2ant1 1125 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
3128, 30jca 512 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂))
321, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem45 36733 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐺 𝐻))
331, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem46 36734 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐻 𝐼))
341, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem47 36735 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐼 𝐺))
3532, 33, 343jca 1120 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)))
361, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem48 36736 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))
371, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem49 36737 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑄 𝑅))
381, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem50 36738 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 (𝑅 𝑃))
3936, 37, 383jca 1120 . . . . 5 ((𝜑𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
40393adant2 1123 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
411, 8, 9, 10, 4, 11, 12, 13, 14, 15dalem27 36715 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐺 𝑃))
421, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem32 36720 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐻 𝑄))
431, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem36 36724 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐼 𝑅))
4441, 42, 433jca 1120 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))
4535, 40, 443jca 1120 . . 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 1120 . 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 36731 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)
4846, 47jca 512 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  meetcmee 17543  Atomscatm 36279  HLchlt 36366  LPlanesclpl 36508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-llines 36514  df-lplanes 36515  df-lvols 36516
This theorem is referenced by:  dalem53  36741  dalem54  36742
  Copyright terms: Public domain W3C validator