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Theorem dalem17 35489
 Description: Lemma for dath 35545. 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 35445 . . 3 (𝜑𝐶 (𝑅 𝑈))
32adantr 466 . 2 ((𝜑𝑌 = 𝑍) → 𝐶 (𝑅 𝑈))
41dalemkelat 35433 . . . . . 6 (𝜑𝐾 ∈ Lat)
5 dalemc.j . . . . . . 7 = (join‘𝐾)
6 dalemc.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
71, 5, 6dalempjqeb 35454 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
81, 6dalemreb 35450 . . . . . 6 (𝜑𝑅 ∈ (Base‘𝐾))
9 eqid 2771 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
10 dalemc.l . . . . . . 7 = (le‘𝐾)
119, 10, 5latlej2 17269 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 ((𝑃 𝑄) 𝑅))
124, 7, 8, 11syl3anc 1476 . . . . 5 (𝜑𝑅 ((𝑃 𝑄) 𝑅))
13 dalem17.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
1412, 13syl6breqr 4829 . . . 4 (𝜑𝑅 𝑌)
1514adantr 466 . . 3 ((𝜑𝑌 = 𝑍) → 𝑅 𝑌)
161, 5, 6dalemsjteb 35455 . . . . . . 7 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
171, 6dalemueb 35453 . . . . . . 7 (𝜑𝑈 ∈ (Base‘𝐾))
189, 10, 5latlej2 17269 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑈 ((𝑆 𝑇) 𝑈))
194, 16, 17, 18syl3anc 1476 . . . . . 6 (𝜑𝑈 ((𝑆 𝑇) 𝑈))
20 dalem17.z . . . . . 6 𝑍 = ((𝑆 𝑇) 𝑈)
2119, 20syl6breqr 4829 . . . . 5 (𝜑𝑈 𝑍)
2221adantr 466 . . . 4 ((𝜑𝑌 = 𝑍) → 𝑈 𝑍)
23 simpr 471 . . . 4 ((𝜑𝑌 = 𝑍) → 𝑌 = 𝑍)
2422, 23breqtrrd 4815 . . 3 ((𝜑𝑌 = 𝑍) → 𝑈 𝑌)
25 dalem17.o . . . . . 6 𝑂 = (LPlanes‘𝐾)
261, 25dalemyeb 35458 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐾))
279, 10, 5latjle12 17270 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑅 𝑌𝑈 𝑌) ↔ (𝑅 𝑈) 𝑌))
284, 8, 17, 26, 27syl13anc 1478 . . . 4 (𝜑 → ((𝑅 𝑌𝑈 𝑌) ↔ (𝑅 𝑈) 𝑌))
2928adantr 466 . . 3 ((𝜑𝑌 = 𝑍) → ((𝑅 𝑌𝑈 𝑌) ↔ (𝑅 𝑈) 𝑌))
3015, 24, 29mpbi2and 691 . 2 ((𝜑𝑌 = 𝑍) → (𝑅 𝑈) 𝑌)
311, 6dalemceb 35447 . . . 4 (𝜑𝐶 ∈ (Base‘𝐾))
321dalemkehl 35432 . . . . 5 (𝜑𝐾 ∈ HL)
331dalemrea 35437 . . . . 5 (𝜑𝑅𝐴)
341dalemuea 35440 . . . . 5 (𝜑𝑈𝐴)
359, 5, 6hlatjcl 35176 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴) → (𝑅 𝑈) ∈ (Base‘𝐾))
3632, 33, 34, 35syl3anc 1476 . . . 4 (𝜑 → (𝑅 𝑈) ∈ (Base‘𝐾))
379, 10lattr 17264 . . . 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 1478 . . 3 (𝜑 → ((𝐶 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑌) → 𝐶 𝑌))
3938adantr 466 . 2 ((𝜑𝑌 = 𝑍) → ((𝐶 (𝑅 𝑈) ∧ (𝑅 𝑈) 𝑌) → 𝐶 𝑌))
403, 30, 39mp2and 679 1 ((𝜑𝑌 = 𝑍) → 𝐶 𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   class class class wbr 4787  ‘cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  Latclat 17253  Atomscatm 35072  HLchlt 35159  LPlanesclpl 35301 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-poset 17154  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-lat 17254  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-lplanes 35308 This theorem is referenced by:  dalem19  35491  dalem25  35507
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