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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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