Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem24 Structured version   Visualization version   GIF version

Theorem dalem24 39699
Description: Lemma for dath 39738. Show that auxiliary atom 𝐺 is outside of plane 𝑌. (Contributed by NM, 2-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))))
dalem23.m = (meet‘𝐾)
dalem23.o 𝑂 = (LPlanes‘𝐾)
dalem23.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem23.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem23.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
Assertion
Ref Expression
dalem24 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)

Proof of Theorem dalem24
StepHypRef Expression
1 dalem23.g . . . . 5 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
21oveq1i 7441 . . . 4 (𝐺 𝑌) = (((𝑐 𝑃) (𝑑 𝑆)) 𝑌)
3 dalem.ph . . . . . . . 8 (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)))))
43dalemkehl 39625 . . . . . . 7 (𝜑𝐾 ∈ HL)
5 hlol 39362 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OL)
64, 5syl 17 . . . . . 6 (𝜑𝐾 ∈ OL)
763ad2ant1 1134 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ OL)
843ad2ant1 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
9 dalem.ps . . . . . . . 8 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
109dalemccea 39685 . . . . . . 7 (𝜓𝑐𝐴)
11103ad2ant3 1136 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
123dalempea 39628 . . . . . . 7 (𝜑𝑃𝐴)
13123ad2ant1 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
14 eqid 2737 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
15 dalem.j . . . . . . 7 = (join‘𝐾)
16 dalem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
1714, 15, 16hlatjcl 39368 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
188, 11, 13, 17syl3anc 1373 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
199dalemddea 39686 . . . . . . 7 (𝜓𝑑𝐴)
20193ad2ant3 1136 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
213dalemsea 39631 . . . . . . 7 (𝜑𝑆𝐴)
22213ad2ant1 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2314, 15, 16hlatjcl 39368 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
248, 20, 22, 23syl3anc 1373 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
25 dalem23.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
263, 25dalemyeb 39651 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐾))
27263ad2ant1 1134 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
28 dalem23.m . . . . . 6 = (meet‘𝐾)
2914, 28latmmdir 39236 . . . . 5 ((𝐾 ∈ OL ∧ ((𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → (((𝑐 𝑃) (𝑑 𝑆)) 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
307, 18, 24, 27, 29syl13anc 1374 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) (𝑑 𝑆)) 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
312, 30eqtrid 2789 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
3215, 16hlatjcom 39369 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) = (𝑃 𝑐))
338, 11, 13, 32syl3anc 1373 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) = (𝑃 𝑐))
3433oveq1d 7446 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) 𝑌) = ((𝑃 𝑐) 𝑌))
35 dalem.l . . . . . . . 8 = (le‘𝐾)
36 dalem23.y . . . . . . . 8 𝑌 = ((𝑃 𝑄) 𝑅)
373, 35, 15, 16, 25, 36dalemply 39656 . . . . . . 7 (𝜑𝑃 𝑌)
38373ad2ant1 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 𝑌)
399dalem-ccly 39687 . . . . . . 7 (𝜓 → ¬ 𝑐 𝑌)
40393ad2ant3 1136 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑌)
4114, 35, 15, 28, 162atjm 39447 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑐𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑃 𝑌 ∧ ¬ 𝑐 𝑌)) → ((𝑃 𝑐) 𝑌) = 𝑃)
428, 13, 11, 27, 38, 40, 41syl132anc 1390 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑐) 𝑌) = 𝑃)
4334, 42eqtrd 2777 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) 𝑌) = 𝑃)
4415, 16hlatjcom 39369 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) = (𝑆 𝑑))
458, 20, 22, 44syl3anc 1373 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) = (𝑆 𝑑))
4645oveq1d 7446 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑 𝑆) 𝑌) = ((𝑆 𝑑) 𝑌))
47 dalem23.z . . . . . . . 8 𝑍 = ((𝑆 𝑇) 𝑈)
483, 35, 15, 16, 47dalemsly 39657 . . . . . . 7 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
49483adant3 1133 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
509dalem-ddly 39688 . . . . . . 7 (𝜓 → ¬ 𝑑 𝑌)
51503ad2ant3 1136 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
5214, 35, 15, 28, 162atjm 39447 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑑𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 𝑌 ∧ ¬ 𝑑 𝑌)) → ((𝑆 𝑑) 𝑌) = 𝑆)
538, 22, 20, 27, 49, 51, 52syl132anc 1390 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑆 𝑑) 𝑌) = 𝑆)
5446, 53eqtrd 2777 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑 𝑆) 𝑌) = 𝑆)
5543, 54oveq12d 7449 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)) = (𝑃 𝑆))
563, 35, 15, 16, 25, 36dalempnes 39653 . . . . 5 (𝜑𝑃𝑆)
57 hlatl 39361 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
584, 57syl 17 . . . . . 6 (𝜑𝐾 ∈ AtLat)
59 eqid 2737 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
6028, 59, 16atnem0 39319 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑆𝐴) → (𝑃𝑆 ↔ (𝑃 𝑆) = (0.‘𝐾)))
6158, 12, 21, 60syl3anc 1373 . . . . 5 (𝜑 → (𝑃𝑆 ↔ (𝑃 𝑆) = (0.‘𝐾)))
6256, 61mpbid 232 . . . 4 (𝜑 → (𝑃 𝑆) = (0.‘𝐾))
63623ad2ant1 1134 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) = (0.‘𝐾))
6431, 55, 633eqtrd 2781 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑌) = (0.‘𝐾))
65583ad2ant1 1134 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
663, 35, 15, 16, 9, 28, 25, 36, 47, 1dalem23 39698 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
6714, 35, 28, 59, 16atnle 39318 . . 3 ((𝐾 ∈ AtLat ∧ 𝐺𝐴𝑌 ∈ (Base‘𝐾)) → (¬ 𝐺 𝑌 ↔ (𝐺 𝑌) = (0.‘𝐾)))
6865, 66, 27, 67syl3anc 1373 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝐺 𝑌 ↔ (𝐺 𝑌) = (0.‘𝐾)))
6964, 68mpbird 257 1 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  0.cp0 18468  OLcol 39175  Atomscatm 39264  AtLatcal 39265  HLchlt 39351  LPlanesclpl 39494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501
This theorem is referenced by:  dalem27  39701  dalem30  39704  dalem54  39728
  Copyright terms: Public domain W3C validator