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Theorem dalem17 36810
Description: Lemma for dath 36866. 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 36766 . . 3 (𝜑𝐶 (𝑅 𝑈))
32adantr 483 . 2 ((𝜑𝑌 = 𝑍) → 𝐶 (𝑅 𝑈))
41dalemkelat 36754 . . . . . 6 (𝜑𝐾 ∈ Lat)
5 dalemc.j . . . . . . 7 = (join‘𝐾)
6 dalemc.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
71, 5, 6dalempjqeb 36775 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
81, 6dalemreb 36771 . . . . . 6 (𝜑𝑅 ∈ (Base‘𝐾))
9 eqid 2821 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
10 dalemc.l . . . . . . 7 = (le‘𝐾)
119, 10, 5latlej2 17665 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 ((𝑃 𝑄) 𝑅))
124, 7, 8, 11syl3anc 1367 . . . . 5 (𝜑𝑅 ((𝑃 𝑄) 𝑅))
13 dalem17.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
1412, 13breqtrrdi 5101 . . . 4 (𝜑𝑅 𝑌)
1514adantr 483 . . 3 ((𝜑𝑌 = 𝑍) → 𝑅 𝑌)
161, 5, 6dalemsjteb 36776 . . . . . . 7 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
171, 6dalemueb 36774 . . . . . . 7 (𝜑𝑈 ∈ (Base‘𝐾))
189, 10, 5latlej2 17665 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑈 ((𝑆 𝑇) 𝑈))
194, 16, 17, 18syl3anc 1367 . . . . . 6 (𝜑𝑈 ((𝑆 𝑇) 𝑈))
20 dalem17.z . . . . . 6 𝑍 = ((𝑆 𝑇) 𝑈)
2119, 20breqtrrdi 5101 . . . . 5 (𝜑𝑈 𝑍)
2221adantr 483 . . . 4 ((𝜑𝑌 = 𝑍) → 𝑈 𝑍)
23 simpr 487 . . . 4 ((𝜑𝑌 = 𝑍) → 𝑌 = 𝑍)
2422, 23breqtrrd 5087 . . 3 ((𝜑𝑌 = 𝑍) → 𝑈 𝑌)
25 dalem17.o . . . . . 6 𝑂 = (LPlanes‘𝐾)
261, 25dalemyeb 36779 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐾))
279, 10, 5latjle12 17666 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑅 𝑌𝑈 𝑌) ↔ (𝑅 𝑈) 𝑌))
284, 8, 17, 26, 27syl13anc 1368 . . . 4 (𝜑 → ((𝑅 𝑌𝑈 𝑌) ↔ (𝑅 𝑈) 𝑌))
2928adantr 483 . . 3 ((𝜑𝑌 = 𝑍) → ((𝑅 𝑌𝑈 𝑌) ↔ (𝑅 𝑈) 𝑌))
3015, 24, 29mpbi2and 710 . 2 ((𝜑𝑌 = 𝑍) → (𝑅 𝑈) 𝑌)
311, 6dalemceb 36768 . . . 4 (𝜑𝐶 ∈ (Base‘𝐾))
321dalemkehl 36753 . . . . 5 (𝜑𝐾 ∈ HL)
331dalemrea 36758 . . . . 5 (𝜑𝑅𝐴)
341dalemuea 36761 . . . . 5 (𝜑𝑈𝐴)
359, 5, 6hlatjcl 36497 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴) → (𝑅 𝑈) ∈ (Base‘𝐾))
3632, 33, 34, 35syl3anc 1367 . . . 4 (𝜑 → (𝑅 𝑈) ∈ (Base‘𝐾))
379, 10lattr 17660 . . . 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 1368 . . 3 (𝜑 → ((𝐶 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑌) → 𝐶 𝑌))
3938adantr 483 . 2 ((𝜑𝑌 = 𝑍) → ((𝐶 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑌) → 𝐶 𝑌))
403, 30, 39mp2and 697 1 ((𝜑𝑌 = 𝑍) → 𝐶 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110   class class class wbr 5059  cfv 6350  (class class class)co 7150  Basecbs 16477  lecple 16566  joincjn 17548  Latclat 17649  Atomscatm 36393  HLchlt 36480  LPlanesclpl 36622
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-poset 17550  df-lub 17578  df-glb 17579  df-join 17580  df-meet 17581  df-lat 17650  df-ats 36397  df-atl 36428  df-cvlat 36452  df-hlat 36481  df-lplanes 36629
This theorem is referenced by:  dalem19  36812  dalem25  36828
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