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Theorem dalemswapyzps 36934
 Description: Lemma for dath 36980. 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 36928 . . . 4 (𝜓𝑑𝐴)
31dalemccea 36927 . . . 4 (𝜓𝑐𝐴)
42, 3jca 515 . . 3 (𝜓 → (𝑑𝐴𝑐𝐴))
543ad2ant3 1132 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑑𝐴𝑐𝐴))
61dalem-ddly 36930 . . . 4 (𝜓 → ¬ 𝑑 𝑌)
763ad2ant3 1132 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
8 simp2 1134 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 = 𝑍)
98breq2d 5064 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑌𝑑 𝑍))
107, 9mtbid 327 . 2 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑍)
111dalemccnedd 36931 . . . 4 (𝜓𝑐𝑑)
12113ad2ant3 1132 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝑑)
131dalem-ccly 36929 . . . . 5 (𝜓 → ¬ 𝑐 𝑌)
14133ad2ant3 1132 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑌)
158breq2d 5064 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑌𝑐 𝑍))
1614, 15mtbid 327 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑍)
171dalemclccjdd 36932 . . . . 5 (𝜓𝐶 (𝑐 𝑑))
18173ad2ant3 1132 . . . 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 36867 . . . . . 6 (𝜑𝐾 ∈ HL)
21203ad2ant1 1130 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
2233ad2ant3 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
2323ad2ant3 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
24 dalem.j . . . . . 6 = (join‘𝐾)
25 dalem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
2624, 25hlatjcom 36612 . . . . 5 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑑𝐴) → (𝑐 𝑑) = (𝑑 𝑐))
2721, 22, 23, 26syl3anc 1368 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) = (𝑑 𝑐))
2818, 27breqtrd 5078 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 (𝑑 𝑐))
2912, 16, 283jca 1125 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐)))
305, 10, 293jca 1125 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3014   class class class wbr 5052  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  Atomscatm 36507  HLchlt 36594 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-lub 17584  df-join 17586  df-lat 17656  df-ats 36511  df-atl 36542  df-cvlat 36566  df-hlat 36595 This theorem is referenced by:  dalem56  36972
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