Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem52 Structured version   Visualization version   GIF version

Theorem dalem52 35749
Description: Lemma for dath 35761. Lines 𝐺𝐻 and 𝑃𝑄 intersect at an atom. (Contributed by NM, 8-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
dalem52 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)

Proof of Theorem dalem52
StepHypRef Expression
1 dalem.ph . . . . 5 (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 35648 . . . 4 (𝜑𝐾 ∈ HL)
323ad2ant1 1164 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
4 dalem.ps . . . . 5 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
5 dalem.a . . . . 5 𝐴 = (Atoms‘𝐾)
64, 5dalemcceb 35714 . . . 4 (𝜓𝑐 ∈ (Base‘𝐾))
763ad2ant3 1166 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
83, 7jca 508 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)))
9 dalem.l . . . 4 = (le‘𝐾)
10 dalem.j . . . 4 = (join‘𝐾)
11 dalem44.m . . . 4 = (meet‘𝐾)
12 dalem44.o . . . 4 𝑂 = (LPlanes‘𝐾)
13 dalem44.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
14 dalem44.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
15 dalem44.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
161, 9, 10, 5, 4, 11, 12, 13, 14, 15dalem23 35721 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
17 dalem44.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
181, 9, 10, 5, 4, 11, 12, 13, 14, 17dalem29 35726 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
19 dalem44.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
201, 9, 10, 5, 4, 11, 12, 13, 14, 19dalem34 35731 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2116, 18, 203jca 1159 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺𝐴𝐻𝐴𝐼𝐴))
221dalempea 35651 . . . 4 (𝜑𝑃𝐴)
231dalemqea 35652 . . . 4 (𝜑𝑄𝐴)
241dalemrea 35653 . . . 4 (𝜑𝑅𝐴)
2522, 23, 243jca 1159 . . 3 (𝜑 → (𝑃𝐴𝑄𝐴𝑅𝐴))
26253ad2ant1 1164 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑃𝐴𝑄𝐴𝑅𝐴))
271, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem42 35739 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
281dalemyeo 35657 . . 3 (𝜑𝑌𝑂)
29283ad2ant1 1164 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
301, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem45 35742 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐺 𝐻))
311, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem46 35743 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐻 𝐼))
321, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem47 35744 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 (𝐼 𝐺))
3330, 31, 323jca 1159 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝐺 𝐻) ∧ ¬ 𝑐 (𝐻 𝐼) ∧ ¬ 𝑐 (𝐼 𝐺)))
341, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem48 35745 . . . 4 ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))
351, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem49 35746 . . . 4 ((𝜑𝜓) → ¬ 𝑐 (𝑄 𝑅))
361, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem50 35747 . . . 4 ((𝜑𝜓) → ¬ 𝑐 (𝑅 𝑃))
3734, 35, 363jca 1159 . . 3 ((𝜑𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
38373adant2 1162 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝑐 (𝑃 𝑄) ∧ ¬ 𝑐 (𝑄 𝑅) ∧ ¬ 𝑐 (𝑅 𝑃)))
391, 9, 10, 5, 4, 11, 12, 13, 14, 15dalem27 35724 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐺 𝑃))
401, 9, 10, 5, 4, 11, 12, 13, 14, 17dalem32 35729 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐻 𝑄))
411, 9, 10, 5, 4, 11, 12, 13, 14, 19dalem36 35733 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐼 𝑅))
4239, 40, 413jca 1159 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝑃) ∧ 𝑐 (𝐻 𝑄) ∧ 𝑐 (𝐼 𝑅)))
43 biid 253 . . 3 ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (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)))) ↔ (((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (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)))))
44 eqid 2803 . . 3 ((𝐺 𝐻) 𝐼) = ((𝐺 𝐻) 𝐼)
45 eqid 2803 . . 3 ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) (𝑃 𝑄))
4643, 9, 10, 5, 11, 12, 44, 13, 45dalemdea 35687 . 2 ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (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) (𝑃 𝑄)) ∈ 𝐴)
478, 21, 26, 27, 29, 33, 38, 42, 46syl323anc 1520 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2975   class class class wbr 4847  cfv 6105  (class class class)co 6882  Basecbs 16188  lecple 16278  joincjn 17263  meetcmee 17264  Atomscatm 35288  HLchlt 35375  LPlanesclpl 35517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-rep 4968  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-reu 3100  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-pw 4355  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-iun 4716  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-res 5328  df-ima 5329  df-iota 6068  df-fun 6107  df-fn 6108  df-f 6109  df-f1 6110  df-fo 6111  df-f1o 6112  df-fv 6113  df-riota 6843  df-ov 6885  df-oprab 6886  df-proset 17247  df-poset 17265  df-plt 17277  df-lub 17293  df-glb 17294  df-join 17295  df-meet 17296  df-p0 17358  df-lat 17365  df-clat 17427  df-oposet 35201  df-ol 35203  df-oml 35204  df-covers 35291  df-ats 35292  df-atl 35323  df-cvlat 35347  df-hlat 35376  df-llines 35523  df-lplanes 35524  df-lvols 35525
This theorem is referenced by:  dalem54  35751  dalem55  35752
  Copyright terms: Public domain W3C validator