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

Theorem dalemswapyzps 39692
Description: Lemma for dath 39738. Swap the 𝑌 and 𝑍 planes, along with dummy concurrency (center of perspectivity) atoms 𝑐 and 𝑑, to allow reuse of analogous proofs. (Contributed by NM, 17-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))))
Assertion
Ref Expression
dalemswapyzps ((𝜑𝑌 = 𝑍𝜓) → ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))))

Proof of Theorem dalemswapyzps
StepHypRef Expression
1 dalem.ps . . . . 5 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
21dalemddea 39686 . . . 4 (𝜓𝑑𝐴)
31dalemccea 39685 . . . 4 (𝜓𝑐𝐴)
42, 3jca 511 . . 3 (𝜓 → (𝑑𝐴𝑐𝐴))
543ad2ant3 1136 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑑𝐴𝑐𝐴))
61dalem-ddly 39688 . . . 4 (𝜓 → ¬ 𝑑 𝑌)
763ad2ant3 1136 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
8 simp2 1138 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 = 𝑍)
98breq2d 5155 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑌𝑑 𝑍))
107, 9mtbid 324 . 2 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑍)
111dalemccnedd 39689 . . . 4 (𝜓𝑐𝑑)
12113ad2ant3 1136 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝑑)
131dalem-ccly 39687 . . . . 5 (𝜓 → ¬ 𝑐 𝑌)
14133ad2ant3 1136 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑌)
158breq2d 5155 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑌𝑐 𝑍))
1614, 15mtbid 324 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑍)
171dalemclccjdd 39690 . . . . 5 (𝜓𝐶 (𝑐 𝑑))
18173ad2ant3 1136 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 (𝑐 𝑑))
19 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)))))
2019dalemkehl 39625 . . . . . 6 (𝜑𝐾 ∈ HL)
21203ad2ant1 1134 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
2233ad2ant3 1136 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
2323ad2ant3 1136 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
24 dalem.j . . . . . 6 = (join‘𝐾)
25 dalem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
2624, 25hlatjcom 39369 . . . . 5 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑑𝐴) → (𝑐 𝑑) = (𝑑 𝑐))
2721, 22, 23, 26syl3anc 1373 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) = (𝑑 𝑐))
2818, 27breqtrd 5169 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 (𝑑 𝑐))
2912, 16, 283jca 1129 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐)))
305, 10, 293jca 1129 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))))
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  Atomscatm 39264  HLchlt 39351
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-lub 18391  df-join 18393  df-lat 18477  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352
This theorem is referenced by:  dalem56  39730
  Copyright terms: Public domain W3C validator