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Theorem dalemrot 36797
Description: Lemma for dath 36876. Rotate triangles 𝑌 = 𝑃𝑄𝑅 and 𝑍 = 𝑆𝑇𝑈 to allow reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.)
Hypotheses
Ref Expression
dalema.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)))))
dalemc.l = (le‘𝐾)
dalemc.j = (join‘𝐾)
dalemc.a 𝐴 = (Atoms‘𝐾)
dalemrot.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalemrot.z 𝑍 = ((𝑆 𝑇) 𝑈)
Assertion
Ref Expression
dalemrot (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))

Proof of Theorem dalemrot
StepHypRef Expression
1 dalema.ph . . . . 5 (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)))))
21dalemkehl 36763 . . . 4 (𝜑𝐾 ∈ HL)
3 dalemc.a . . . . 5 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 36778 . . . 4 (𝜑𝐶 ∈ (Base‘𝐾))
52, 4jca 514 . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)))
61dalemqea 36767 . . . 4 (𝜑𝑄𝐴)
71dalemrea 36768 . . . 4 (𝜑𝑅𝐴)
81dalempea 36766 . . . 4 (𝜑𝑃𝐴)
96, 7, 83jca 1124 . . 3 (𝜑 → (𝑄𝐴𝑅𝐴𝑃𝐴))
101dalemtea 36770 . . . 4 (𝜑𝑇𝐴)
111dalemuea 36771 . . . 4 (𝜑𝑈𝐴)
121dalemsea 36769 . . . 4 (𝜑𝑆𝐴)
1310, 11, 123jca 1124 . . 3 (𝜑 → (𝑇𝐴𝑈𝐴𝑆𝐴))
145, 9, 133jca 1124 . 2 (𝜑 → ((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)))
15 dalemc.j . . . . 5 = (join‘𝐾)
161, 15, 3dalemqrprot 36788 . . . 4 (𝜑 → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
17 dalemrot.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
181dalemyeo 36772 . . . . 5 (𝜑𝑌𝑂)
1917, 18eqeltrrid 2921 . . . 4 (𝜑 → ((𝑃 𝑄) 𝑅) ∈ 𝑂)
2016, 19eqeltrd 2916 . . 3 (𝜑 → ((𝑄 𝑅) 𝑃) ∈ 𝑂)
2115, 3hlatjrot 36513 . . . . 5 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) → ((𝑇 𝑈) 𝑆) = ((𝑆 𝑇) 𝑈))
222, 10, 11, 12, 21syl13anc 1368 . . . 4 (𝜑 → ((𝑇 𝑈) 𝑆) = ((𝑆 𝑇) 𝑈))
23 dalemrot.z . . . . 5 𝑍 = ((𝑆 𝑇) 𝑈)
241dalemzeo 36773 . . . . 5 (𝜑𝑍𝑂)
2523, 24eqeltrrid 2921 . . . 4 (𝜑 → ((𝑆 𝑇) 𝑈) ∈ 𝑂)
2622, 25eqeltrd 2916 . . 3 (𝜑 → ((𝑇 𝑈) 𝑆) ∈ 𝑂)
2720, 26jca 514 . 2 (𝜑 → (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂))
28 simp312 1317 . . . . 5 ((((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)))) → ¬ 𝐶 (𝑄 𝑅))
291, 28sylbi 219 . . . 4 (𝜑 → ¬ 𝐶 (𝑄 𝑅))
30 simp313 1318 . . . . 5 ((((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)))) → ¬ 𝐶 (𝑅 𝑃))
311, 30sylbi 219 . . . 4 (𝜑 → ¬ 𝐶 (𝑅 𝑃))
321dalem-clpjq 36777 . . . 4 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
3329, 31, 323jca 1124 . . 3 (𝜑 → (¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)))
34 simp322 1320 . . . . 5 ((((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)))) → ¬ 𝐶 (𝑇 𝑈))
351, 34sylbi 219 . . . 4 (𝜑 → ¬ 𝐶 (𝑇 𝑈))
36 simp323 1321 . . . . 5 ((((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)))) → ¬ 𝐶 (𝑈 𝑆))
371, 36sylbi 219 . . . 4 (𝜑 → ¬ 𝐶 (𝑈 𝑆))
38 simp321 1319 . . . . 5 ((((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)))) → ¬ 𝐶 (𝑆 𝑇))
391, 38sylbi 219 . . . 4 (𝜑 → ¬ 𝐶 (𝑆 𝑇))
4035, 37, 393jca 1124 . . 3 (𝜑 → (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)))
411dalemclqjt 36775 . . . 4 (𝜑𝐶 (𝑄 𝑇))
421dalemclrju 36776 . . . 4 (𝜑𝐶 (𝑅 𝑈))
431dalemclpjs 36774 . . . 4 (𝜑𝐶 (𝑃 𝑆))
4441, 42, 433jca 1124 . . 3 (𝜑 → (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))
4533, 40, 443jca 1124 . 2 (𝜑 → ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆))))
4614, 27, 453jca 1124 1 (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1536  wcel 2113   class class class wbr 5069  cfv 6358  (class class class)co 7159  Basecbs 16486  lecple 16575  joincjn 17557  Atomscatm 36403  HLchlt 36490
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-proset 17541  df-poset 17559  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-lat 17659  df-ats 36407  df-atl 36438  df-cvlat 36462  df-hlat 36491
This theorem is referenced by:  dalemeea  36803  dalem6  36808  dalem7  36809  dalem11  36814  dalem12  36815  dalem29  36841  dalem30  36842  dalem31N  36843  dalem32  36844  dalem33  36845  dalem34  36846  dalem35  36847  dalem36  36848  dalem37  36849  dalem40  36852  dalem46  36858  dalem47  36859  dalem49  36861  dalem50  36862  dalem58  36870  dalem59  36871
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