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Theorem dalemrot 35733
Description: Lemma for dath 35812. 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 35699 . . . 4 (𝜑𝐾 ∈ HL)
3 dalemc.a . . . . 5 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 35714 . . . 4 (𝜑𝐶 ∈ (Base‘𝐾))
52, 4jca 509 . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)))
61dalemqea 35703 . . . 4 (𝜑𝑄𝐴)
71dalemrea 35704 . . . 4 (𝜑𝑅𝐴)
81dalempea 35702 . . . 4 (𝜑𝑃𝐴)
96, 7, 83jca 1164 . . 3 (𝜑 → (𝑄𝐴𝑅𝐴𝑃𝐴))
101dalemtea 35706 . . . 4 (𝜑𝑇𝐴)
111dalemuea 35707 . . . 4 (𝜑𝑈𝐴)
121dalemsea 35705 . . . 4 (𝜑𝑆𝐴)
1310, 11, 123jca 1164 . . 3 (𝜑 → (𝑇𝐴𝑈𝐴𝑆𝐴))
145, 9, 133jca 1164 . 2 (𝜑 → ((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)))
15 dalemc.j . . . . 5 = (join‘𝐾)
161, 15, 3dalemqrprot 35724 . . . 4 (𝜑 → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
17 dalemrot.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
181dalemyeo 35708 . . . . 5 (𝜑𝑌𝑂)
1917, 18syl5eqelr 2912 . . . 4 (𝜑 → ((𝑃 𝑄) 𝑅) ∈ 𝑂)
2016, 19eqeltrd 2907 . . 3 (𝜑 → ((𝑄 𝑅) 𝑃) ∈ 𝑂)
2115, 3hlatjrot 35449 . . . . 5 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) → ((𝑇 𝑈) 𝑆) = ((𝑆 𝑇) 𝑈))
222, 10, 11, 12, 21syl13anc 1497 . . . 4 (𝜑 → ((𝑇 𝑈) 𝑆) = ((𝑆 𝑇) 𝑈))
23 dalemrot.z . . . . 5 𝑍 = ((𝑆 𝑇) 𝑈)
241dalemzeo 35709 . . . . 5 (𝜑𝑍𝑂)
2523, 24syl5eqelr 2912 . . . 4 (𝜑 → ((𝑆 𝑇) 𝑈) ∈ 𝑂)
2622, 25eqeltrd 2907 . . 3 (𝜑 → ((𝑇 𝑈) 𝑆) ∈ 𝑂)
2720, 26jca 509 . 2 (𝜑 → (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂))
28 simp312 1426 . . . . 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 209 . . . 4 (𝜑 → ¬ 𝐶 (𝑄 𝑅))
30 simp313 1427 . . . . 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 209 . . . 4 (𝜑 → ¬ 𝐶 (𝑅 𝑃))
321dalem-clpjq 35713 . . . 4 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
3329, 31, 323jca 1164 . . 3 (𝜑 → (¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)))
34 simp322 1429 . . . . 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 209 . . . 4 (𝜑 → ¬ 𝐶 (𝑇 𝑈))
36 simp323 1430 . . . . 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 209 . . . 4 (𝜑 → ¬ 𝐶 (𝑈 𝑆))
38 simp321 1428 . . . . 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 209 . . . 4 (𝜑 → ¬ 𝐶 (𝑆 𝑇))
4035, 37, 393jca 1164 . . 3 (𝜑 → (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)))
411dalemclqjt 35711 . . . 4 (𝜑𝐶 (𝑄 𝑇))
421dalemclrju 35712 . . . 4 (𝜑𝐶 (𝑅 𝑈))
431dalemclpjs 35710 . . . 4 (𝜑𝐶 (𝑃 𝑆))
4441, 42, 433jca 1164 . . 3 (𝜑 → (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))
4533, 40, 443jca 1164 . 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 1164 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 198  wa 386  w3a 1113   = wceq 1658  wcel 2166   class class class wbr 4874  cfv 6124  (class class class)co 6906  Basecbs 16223  lecple 16313  joincjn 17298  Atomscatm 35339  HLchlt 35426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-proset 17282  df-poset 17300  df-lub 17328  df-glb 17329  df-join 17330  df-meet 17331  df-lat 17400  df-ats 35343  df-atl 35374  df-cvlat 35398  df-hlat 35427
This theorem is referenced by:  dalemeea  35739  dalem6  35744  dalem7  35745  dalem11  35750  dalem12  35751  dalem29  35777  dalem30  35778  dalem31N  35779  dalem32  35780  dalem33  35781  dalem34  35782  dalem35  35783  dalem36  35784  dalem37  35785  dalem40  35788  dalem46  35794  dalem47  35795  dalem49  35797  dalem50  35798  dalem58  35806  dalem59  35807
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