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Theorem dalem-cly 39382
Description: Lemma for dalem9 39383. Center of perspectivity 𝐶 is not in plane 𝑌 (when 𝑌 and 𝑍 are different planes). (Contributed by NM, 13-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)
dalem-cly.o 𝑂 = (LPlanes‘𝐾)
dalem-cly.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem-cly.z 𝑍 = ((𝑆 𝑇) 𝑈)
Assertion
Ref Expression
dalem-cly ((𝜑𝑌𝑍) → ¬ 𝐶 𝑌)

Proof of Theorem dalem-cly
StepHypRef Expression
1 dalema.ph . . . . . . 7 (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)))))
21dalemkelat 39335 . . . . . 6 (𝜑𝐾 ∈ Lat)
3 dalemc.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 39349 . . . . . 6 (𝜑𝐶 ∈ (Base‘𝐾))
5 dalem-cly.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
61, 5dalemyeb 39360 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐾))
7 eqid 2726 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 dalemc.l . . . . . . 7 = (le‘𝐾)
9 dalemc.j . . . . . . 7 = (join‘𝐾)
107, 8, 9latleeqj1 18470 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝑌))
112, 4, 6, 10syl3anc 1368 . . . . 5 (𝜑 → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝑌))
121dalemclpjs 39345 . . . . . . . . . . . . 13 (𝜑𝐶 (𝑃 𝑆))
131dalemkehl 39334 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ HL)
14 dalem-cly.y . . . . . . . . . . . . . . 15 𝑌 = ((𝑃 𝑄) 𝑅)
151, 8, 9, 3, 5, 14dalemcea 39371 . . . . . . . . . . . . . 14 (𝜑𝐶𝐴)
161dalemsea 39340 . . . . . . . . . . . . . 14 (𝜑𝑆𝐴)
171dalempea 39337 . . . . . . . . . . . . . 14 (𝜑𝑃𝐴)
181dalemqea 39338 . . . . . . . . . . . . . . 15 (𝜑𝑄𝐴)
191dalem-clpjq 39348 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
208, 9, 3atnlej1 39090 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝐶 (𝑃 𝑄)) → 𝐶𝑃)
2113, 15, 17, 18, 19, 20syl131anc 1380 . . . . . . . . . . . . . 14 (𝜑𝐶𝑃)
228, 9, 3hlatexch1 39106 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑆𝐴𝑃𝐴) ∧ 𝐶𝑃) → (𝐶 (𝑃 𝑆) → 𝑆 (𝑃 𝐶)))
2313, 15, 16, 17, 21, 22syl131anc 1380 . . . . . . . . . . . . 13 (𝜑 → (𝐶 (𝑃 𝑆) → 𝑆 (𝑃 𝐶)))
2412, 23mpd 15 . . . . . . . . . . . 12 (𝜑𝑆 (𝑃 𝐶))
259, 3hlatjcom 39078 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑃𝐴) → (𝐶 𝑃) = (𝑃 𝐶))
2613, 15, 17, 25syl3anc 1368 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑃) = (𝑃 𝐶))
2724, 26breqtrrd 5173 . . . . . . . . . . 11 (𝜑𝑆 (𝐶 𝑃))
281dalemclqjt 39346 . . . . . . . . . . . . 13 (𝜑𝐶 (𝑄 𝑇))
291dalemtea 39341 . . . . . . . . . . . . . 14 (𝜑𝑇𝐴)
301dalemrea 39339 . . . . . . . . . . . . . . 15 (𝜑𝑅𝐴)
31 simp312 1318 . . . . . . . . . . . . . . . 16 ((((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))
321, 31sylbi 216 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝐶 (𝑄 𝑅))
338, 9, 3atnlej1 39090 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝐶 (𝑄 𝑅)) → 𝐶𝑄)
3413, 15, 18, 30, 32, 33syl131anc 1380 . . . . . . . . . . . . . 14 (𝜑𝐶𝑄)
358, 9, 3hlatexch1 39106 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑇𝐴𝑄𝐴) ∧ 𝐶𝑄) → (𝐶 (𝑄 𝑇) → 𝑇 (𝑄 𝐶)))
3613, 15, 29, 18, 34, 35syl131anc 1380 . . . . . . . . . . . . 13 (𝜑 → (𝐶 (𝑄 𝑇) → 𝑇 (𝑄 𝐶)))
3728, 36mpd 15 . . . . . . . . . . . 12 (𝜑𝑇 (𝑄 𝐶))
389, 3hlatjcom 39078 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑄𝐴) → (𝐶 𝑄) = (𝑄 𝐶))
3913, 15, 18, 38syl3anc 1368 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑄) = (𝑄 𝐶))
4037, 39breqtrrd 5173 . . . . . . . . . . 11 (𝜑𝑇 (𝐶 𝑄))
411, 3dalemseb 39353 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (Base‘𝐾))
427, 9, 3hlatjcl 39077 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑃𝐴) → (𝐶 𝑃) ∈ (Base‘𝐾))
4313, 15, 17, 42syl3anc 1368 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑃) ∈ (Base‘𝐾))
441, 3dalemteb 39354 . . . . . . . . . . . 12 (𝜑𝑇 ∈ (Base‘𝐾))
457, 9, 3hlatjcl 39077 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑄𝐴) → (𝐶 𝑄) ∈ (Base‘𝐾))
4613, 15, 18, 45syl3anc 1368 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑄) ∈ (Base‘𝐾))
477, 8, 9latjlej12 18474 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ (𝐶 𝑃) ∈ (Base‘𝐾)) ∧ (𝑇 ∈ (Base‘𝐾) ∧ (𝐶 𝑄) ∈ (Base‘𝐾))) → ((𝑆 (𝐶 𝑃) ∧ 𝑇 (𝐶 𝑄)) → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄))))
482, 41, 43, 44, 46, 47syl122anc 1376 . . . . . . . . . . 11 (𝜑 → ((𝑆 (𝐶 𝑃) ∧ 𝑇 (𝐶 𝑄)) → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄))))
4927, 40, 48mp2and 697 . . . . . . . . . 10 (𝜑 → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄)))
501, 3dalempeb 39350 . . . . . . . . . . 11 (𝜑𝑃 ∈ (Base‘𝐾))
511, 3dalemqeb 39351 . . . . . . . . . . 11 (𝜑𝑄 ∈ (Base‘𝐾))
527, 9latjjdi 18510 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (𝐶 (𝑃 𝑄)) = ((𝐶 𝑃) (𝐶 𝑄)))
532, 4, 50, 51, 52syl13anc 1369 . . . . . . . . . 10 (𝜑 → (𝐶 (𝑃 𝑄)) = ((𝐶 𝑃) (𝐶 𝑄)))
5449, 53breqtrrd 5173 . . . . . . . . 9 (𝜑 → (𝑆 𝑇) (𝐶 (𝑃 𝑄)))
551dalemclrju 39347 . . . . . . . . . . 11 (𝜑𝐶 (𝑅 𝑈))
561dalemuea 39342 . . . . . . . . . . . 12 (𝜑𝑈𝐴)
57 simp313 1319 . . . . . . . . . . . . . 14 ((((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))
581, 57sylbi 216 . . . . . . . . . . . . 13 (𝜑 → ¬ 𝐶 (𝑅 𝑃))
598, 9, 3atnlej1 39090 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑅𝐴𝑃𝐴) ∧ ¬ 𝐶 (𝑅 𝑃)) → 𝐶𝑅)
6013, 15, 30, 17, 58, 59syl131anc 1380 . . . . . . . . . . . 12 (𝜑𝐶𝑅)
618, 9, 3hlatexch1 39106 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑈𝐴𝑅𝐴) ∧ 𝐶𝑅) → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
6213, 15, 56, 30, 60, 61syl131anc 1380 . . . . . . . . . . 11 (𝜑 → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
6355, 62mpd 15 . . . . . . . . . 10 (𝜑𝑈 (𝑅 𝐶))
649, 3hlatjcom 39078 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑅𝐴) → (𝐶 𝑅) = (𝑅 𝐶))
6513, 15, 30, 64syl3anc 1368 . . . . . . . . . 10 (𝜑 → (𝐶 𝑅) = (𝑅 𝐶))
6663, 65breqtrrd 5173 . . . . . . . . 9 (𝜑𝑈 (𝐶 𝑅))
671, 9, 3dalemsjteb 39357 . . . . . . . . . 10 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
681, 9, 3dalempjqeb 39356 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
697, 9latjcl 18458 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾))
702, 4, 68, 69syl3anc 1368 . . . . . . . . . 10 (𝜑 → (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾))
711, 3dalemueb 39355 . . . . . . . . . 10 (𝜑𝑈 ∈ (Base‘𝐾))
727, 9, 3hlatjcl 39077 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑅𝐴) → (𝐶 𝑅) ∈ (Base‘𝐾))
7313, 15, 30, 72syl3anc 1368 . . . . . . . . . 10 (𝜑 → (𝐶 𝑅) ∈ (Base‘𝐾))
747, 8, 9latjlej12 18474 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ ((𝑆 𝑇) ∈ (Base‘𝐾) ∧ (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾)) ∧ (𝑈 ∈ (Base‘𝐾) ∧ (𝐶 𝑅) ∈ (Base‘𝐾))) → (((𝑆 𝑇) (𝐶 (𝑃 𝑄)) ∧ 𝑈 (𝐶 𝑅)) → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅))))
752, 67, 70, 71, 73, 74syl122anc 1376 . . . . . . . . 9 (𝜑 → (((𝑆 𝑇) (𝐶 (𝑃 𝑄)) ∧ 𝑈 (𝐶 𝑅)) → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅))))
7654, 66, 75mp2and 697 . . . . . . . 8 (𝜑 → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
771, 3dalemreb 39352 . . . . . . . . 9 (𝜑𝑅 ∈ (Base‘𝐾))
787, 9latjjdi 18510 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → (𝐶 ((𝑃 𝑄) 𝑅)) = ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
792, 4, 68, 77, 78syl13anc 1369 . . . . . . . 8 (𝜑 → (𝐶 ((𝑃 𝑄) 𝑅)) = ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
8076, 79breqtrrd 5173 . . . . . . 7 (𝜑 → ((𝑆 𝑇) 𝑈) (𝐶 ((𝑃 𝑄) 𝑅)))
81 dalem-cly.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
8214oveq2i 7426 . . . . . . 7 (𝐶 𝑌) = (𝐶 ((𝑃 𝑄) 𝑅))
8380, 81, 823brtr4g 5179 . . . . . 6 (𝜑𝑍 (𝐶 𝑌))
84 breq2 5149 . . . . . 6 ((𝐶 𝑌) = 𝑌 → (𝑍 (𝐶 𝑌) ↔ 𝑍 𝑌))
8583, 84syl5ibcom 244 . . . . 5 (𝜑 → ((𝐶 𝑌) = 𝑌𝑍 𝑌))
8611, 85sylbid 239 . . . 4 (𝜑 → (𝐶 𝑌𝑍 𝑌))
871dalemzeo 39344 . . . . . 6 (𝜑𝑍𝑂)
881dalemyeo 39343 . . . . . 6 (𝜑𝑌𝑂)
898, 5lplncmp 39273 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑍𝑂𝑌𝑂) → (𝑍 𝑌𝑍 = 𝑌))
9013, 87, 88, 89syl3anc 1368 . . . . 5 (𝜑 → (𝑍 𝑌𝑍 = 𝑌))
91 eqcom 2733 . . . . 5 (𝑍 = 𝑌𝑌 = 𝑍)
9290, 91bitrdi 286 . . . 4 (𝜑 → (𝑍 𝑌𝑌 = 𝑍))
9386, 92sylibd 238 . . 3 (𝜑 → (𝐶 𝑌𝑌 = 𝑍))
9493necon3ad 2943 . 2 (𝜑 → (𝑌𝑍 → ¬ 𝐶 𝑌))
9594imp 405 1 ((𝜑𝑌𝑍) → ¬ 𝐶 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930   class class class wbr 5145  cfv 6545  (class class class)co 7415  Basecbs 17207  lecple 17267  joincjn 18330  Latclat 18450  Atomscatm 38973  HLchlt 39060  LPlanesclpl 39203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-iun 4997  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-riota 7371  df-ov 7418  df-oprab 7419  df-proset 18314  df-poset 18332  df-plt 18349  df-lub 18365  df-glb 18366  df-join 18367  df-meet 18368  df-p0 18444  df-lat 18451  df-clat 18518  df-oposet 38886  df-ol 38888  df-oml 38889  df-covers 38976  df-ats 38977  df-atl 39008  df-cvlat 39032  df-hlat 39061  df-llines 39209  df-lplanes 39210
This theorem is referenced by:  dalem9  39383
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