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Theorem dalem17 33778
Description: Lemma for dath 33834. When planes 𝑌 and 𝑍 are equal, the center of perspectivity 𝐶 is in 𝑌. (Contributed by NM, 1-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)
dalem17.o 𝑂 = (LPlanes‘𝐾)
dalem17.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem17.z 𝑍 = ((𝑆 𝑇) 𝑈)
Assertion
Ref Expression
dalem17 ((𝜑𝑌 = 𝑍) → 𝐶 𝑌)

Proof of Theorem dalem17
StepHypRef Expression
1 dalema.ph . . . 4 (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)))))
21dalemclrju 33734 . . 3 (𝜑𝐶 (𝑅 𝑈))
32adantr 480 . 2 ((𝜑𝑌 = 𝑍) → 𝐶 (𝑅 𝑈))
41dalemkelat 33722 . . . . . 6 (𝜑𝐾 ∈ Lat)
5 dalemc.j . . . . . . 7 = (join‘𝐾)
6 dalemc.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
71, 5, 6dalempjqeb 33743 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
81, 6dalemreb 33739 . . . . . 6 (𝜑𝑅 ∈ (Base‘𝐾))
9 eqid 2610 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
10 dalemc.l . . . . . . 7 = (le‘𝐾)
119, 10, 5latlej2 16833 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 ((𝑃 𝑄) 𝑅))
124, 7, 8, 11syl3anc 1318 . . . . 5 (𝜑𝑅 ((𝑃 𝑄) 𝑅))
13 dalem17.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
1412, 13syl6breqr 4620 . . . 4 (𝜑𝑅 𝑌)
1514adantr 480 . . 3 ((𝜑𝑌 = 𝑍) → 𝑅 𝑌)
161, 5, 6dalemsjteb 33744 . . . . . . 7 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
171, 6dalemueb 33742 . . . . . . 7 (𝜑𝑈 ∈ (Base‘𝐾))
189, 10, 5latlej2 16833 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑈 ((𝑆 𝑇) 𝑈))
194, 16, 17, 18syl3anc 1318 . . . . . 6 (𝜑𝑈 ((𝑆 𝑇) 𝑈))
20 dalem17.z . . . . . 6 𝑍 = ((𝑆 𝑇) 𝑈)
2119, 20syl6breqr 4620 . . . . 5 (𝜑𝑈 𝑍)
2221adantr 480 . . . 4 ((𝜑𝑌 = 𝑍) → 𝑈 𝑍)
23 simpr 476 . . . 4 ((𝜑𝑌 = 𝑍) → 𝑌 = 𝑍)
2422, 23breqtrrd 4606 . . 3 ((𝜑𝑌 = 𝑍) → 𝑈 𝑌)
25 dalem17.o . . . . . 6 𝑂 = (LPlanes‘𝐾)
261, 25dalemyeb 33747 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐾))
279, 10, 5latjle12 16834 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑅 𝑌𝑈 𝑌) ↔ (𝑅 𝑈) 𝑌))
284, 8, 17, 26, 27syl13anc 1320 . . . 4 (𝜑 → ((𝑅 𝑌𝑈 𝑌) ↔ (𝑅 𝑈) 𝑌))
2928adantr 480 . . 3 ((𝜑𝑌 = 𝑍) → ((𝑅 𝑌𝑈 𝑌) ↔ (𝑅 𝑈) 𝑌))
3015, 24, 29mpbi2and 958 . 2 ((𝜑𝑌 = 𝑍) → (𝑅 𝑈) 𝑌)
311, 6dalemceb 33736 . . . 4 (𝜑𝐶 ∈ (Base‘𝐾))
321dalemkehl 33721 . . . . 5 (𝜑𝐾 ∈ HL)
331dalemrea 33726 . . . . 5 (𝜑𝑅𝐴)
341dalemuea 33729 . . . . 5 (𝜑𝑈𝐴)
359, 5, 6hlatjcl 33465 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴) → (𝑅 𝑈) ∈ (Base‘𝐾))
3632, 33, 34, 35syl3anc 1318 . . . 4 (𝜑 → (𝑅 𝑈) ∈ (Base‘𝐾))
379, 10lattr 16828 . . . 4 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑅 𝑈) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝐶 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑌) → 𝐶 𝑌))
384, 31, 36, 26, 37syl13anc 1320 . . 3 (𝜑 → ((𝐶 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑌) → 𝐶 𝑌))
3938adantr 480 . 2 ((𝜑𝑌 = 𝑍) → ((𝐶 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑌) → 𝐶 𝑌))
403, 30, 39mp2and 711 1 ((𝜑𝑌 = 𝑍) → 𝐶 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977   class class class wbr 4578  cfv 5790  (class class class)co 6527  Basecbs 15644  lecple 15724  joincjn 16716  Latclat 16817  Atomscatm 33362  HLchlt 33449  LPlanesclpl 33590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-poset 16718  df-lub 16746  df-glb 16747  df-join 16748  df-meet 16749  df-lat 16818  df-ats 33366  df-atl 33397  df-cvlat 33421  df-hlat 33450  df-lplanes 33597
This theorem is referenced by:  dalem19  33780  dalem25  33796
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