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Theorem dalemswapyzps 35764
Description: Lemma for dath 35810. 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 35758 . . . 4 (𝜓𝑑𝐴)
31dalemccea 35757 . . . 4 (𝜓𝑐𝐴)
42, 3jca 507 . . 3 (𝜓 → (𝑑𝐴𝑐𝐴))
543ad2ant3 1169 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑑𝐴𝑐𝐴))
61dalem-ddly 35760 . . . 4 (𝜓 → ¬ 𝑑 𝑌)
763ad2ant3 1169 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
8 simp2 1171 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 = 𝑍)
98breq2d 4887 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑌𝑑 𝑍))
107, 9mtbid 316 . 2 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑍)
111dalemccnedd 35761 . . . 4 (𝜓𝑐𝑑)
12113ad2ant3 1169 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝑑)
131dalem-ccly 35759 . . . . 5 (𝜓 → ¬ 𝑐 𝑌)
14133ad2ant3 1169 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑌)
158breq2d 4887 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑌𝑐 𝑍))
1614, 15mtbid 316 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑍)
171dalemclccjdd 35762 . . . . 5 (𝜓𝐶 (𝑐 𝑑))
18173ad2ant3 1169 . . . 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 35697 . . . . . 6 (𝜑𝐾 ∈ HL)
21203ad2ant1 1167 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
2233ad2ant3 1169 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
2323ad2ant3 1169 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
24 dalem.j . . . . . 6 = (join‘𝐾)
25 dalem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
2624, 25hlatjcom 35442 . . . . 5 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑑𝐴) → (𝑐 𝑑) = (𝑑 𝑐))
2721, 22, 23, 26syl3anc 1494 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) = (𝑑 𝑐))
2818, 27breqtrd 4901 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 (𝑑 𝑐))
2912, 16, 283jca 1162 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐)))
305, 10, 293jca 1162 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑𝐴𝑐𝐴) ∧ ¬ 𝑑 𝑍 ∧ (𝑐𝑑 ∧ ¬ 𝑐 𝑍𝐶 (𝑑 𝑐))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999   class class class wbr 4875  cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  Atomscatm 35337  HLchlt 35424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-lub 17334  df-join 17336  df-lat 17406  df-ats 35341  df-atl 35372  df-cvlat 35396  df-hlat 35425
This theorem is referenced by:  dalem56  35802
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