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Theorem dalemrot 39640
Description: Lemma for dath 39719. 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 39606 . . . 4 (𝜑𝐾 ∈ HL)
3 dalemc.a . . . . 5 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 39621 . . . 4 (𝜑𝐶 ∈ (Base‘𝐾))
52, 4jca 511 . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)))
61dalemqea 39610 . . . 4 (𝜑𝑄𝐴)
71dalemrea 39611 . . . 4 (𝜑𝑅𝐴)
81dalempea 39609 . . . 4 (𝜑𝑃𝐴)
96, 7, 83jca 1127 . . 3 (𝜑 → (𝑄𝐴𝑅𝐴𝑃𝐴))
101dalemtea 39613 . . . 4 (𝜑𝑇𝐴)
111dalemuea 39614 . . . 4 (𝜑𝑈𝐴)
121dalemsea 39612 . . . 4 (𝜑𝑆𝐴)
1310, 11, 123jca 1127 . . 3 (𝜑 → (𝑇𝐴𝑈𝐴𝑆𝐴))
145, 9, 133jca 1127 . 2 (𝜑 → ((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)))
15 dalemc.j . . . . 5 = (join‘𝐾)
161, 15, 3dalemqrprot 39631 . . . 4 (𝜑 → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
17 dalemrot.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
181dalemyeo 39615 . . . . 5 (𝜑𝑌𝑂)
1917, 18eqeltrrid 2844 . . . 4 (𝜑 → ((𝑃 𝑄) 𝑅) ∈ 𝑂)
2016, 19eqeltrd 2839 . . 3 (𝜑 → ((𝑄 𝑅) 𝑃) ∈ 𝑂)
2115, 3hlatjrot 39355 . . . . 5 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) → ((𝑇 𝑈) 𝑆) = ((𝑆 𝑇) 𝑈))
222, 10, 11, 12, 21syl13anc 1371 . . . 4 (𝜑 → ((𝑇 𝑈) 𝑆) = ((𝑆 𝑇) 𝑈))
23 dalemrot.z . . . . 5 𝑍 = ((𝑆 𝑇) 𝑈)
241dalemzeo 39616 . . . . 5 (𝜑𝑍𝑂)
2523, 24eqeltrrid 2844 . . . 4 (𝜑 → ((𝑆 𝑇) 𝑈) ∈ 𝑂)
2622, 25eqeltrd 2839 . . 3 (𝜑 → ((𝑇 𝑈) 𝑆) ∈ 𝑂)
2720, 26jca 511 . 2 (𝜑 → (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂))
28 simp312 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 (𝑄 𝑅))
291, 28sylbi 217 . . . 4 (𝜑 → ¬ 𝐶 (𝑄 𝑅))
30 simp313 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 (𝑅 𝑃))
311, 30sylbi 217 . . . 4 (𝜑 → ¬ 𝐶 (𝑅 𝑃))
321dalem-clpjq 39620 . . . 4 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
3329, 31, 323jca 1127 . . 3 (𝜑 → (¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)))
34 simp322 1323 . . . . 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 217 . . . 4 (𝜑 → ¬ 𝐶 (𝑇 𝑈))
36 simp323 1324 . . . . 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 217 . . . 4 (𝜑 → ¬ 𝐶 (𝑈 𝑆))
38 simp321 1322 . . . . 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 217 . . . 4 (𝜑 → ¬ 𝐶 (𝑆 𝑇))
4035, 37, 393jca 1127 . . 3 (𝜑 → (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)))
411dalemclqjt 39618 . . . 4 (𝜑𝐶 (𝑄 𝑇))
421dalemclrju 39619 . . . 4 (𝜑𝐶 (𝑅 𝑈))
431dalemclpjs 39617 . . . 4 (𝜑𝐶 (𝑃 𝑆))
4441, 42, 433jca 1127 . . 3 (𝜑 → (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))
4533, 40, 443jca 1127 . 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 1127 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Atomscatm 39245  HLchlt 39332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-lat 18490  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333
This theorem is referenced by:  dalemeea  39646  dalem6  39651  dalem7  39652  dalem11  39657  dalem12  39658  dalem29  39684  dalem30  39685  dalem31N  39686  dalem32  39687  dalem33  39688  dalem34  39689  dalem35  39690  dalem36  39691  dalem37  39692  dalem40  39695  dalem46  39701  dalem47  39702  dalem49  39704  dalem50  39705  dalem58  39713  dalem59  39714
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