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Theorem dalem-cly 36357
Description: Lemma for dalem9 36358. 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 36310 . . . . . 6 (𝜑𝐾 ∈ Lat)
3 dalemc.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 36324 . . . . . 6 (𝜑𝐶 ∈ (Base‘𝐾))
5 dalem-cly.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
61, 5dalemyeb 36335 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐾))
7 eqid 2795 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 dalemc.l . . . . . . 7 = (le‘𝐾)
9 dalemc.j . . . . . . 7 = (join‘𝐾)
107, 8, 9latleeqj1 17502 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝑌))
112, 4, 6, 10syl3anc 1364 . . . . 5 (𝜑 → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝑌))
121dalemclpjs 36320 . . . . . . . . . . . . 13 (𝜑𝐶 (𝑃 𝑆))
131dalemkehl 36309 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ HL)
14 dalem-cly.y . . . . . . . . . . . . . . 15 𝑌 = ((𝑃 𝑄) 𝑅)
151, 8, 9, 3, 5, 14dalemcea 36346 . . . . . . . . . . . . . 14 (𝜑𝐶𝐴)
161dalemsea 36315 . . . . . . . . . . . . . 14 (𝜑𝑆𝐴)
171dalempea 36312 . . . . . . . . . . . . . 14 (𝜑𝑃𝐴)
181dalemqea 36313 . . . . . . . . . . . . . . 15 (𝜑𝑄𝐴)
191dalem-clpjq 36323 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
208, 9, 3atnlej1 36065 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝐶 (𝑃 𝑄)) → 𝐶𝑃)
2113, 15, 17, 18, 19, 20syl131anc 1376 . . . . . . . . . . . . . 14 (𝜑𝐶𝑃)
228, 9, 3hlatexch1 36081 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑆𝐴𝑃𝐴) ∧ 𝐶𝑃) → (𝐶 (𝑃 𝑆) → 𝑆 (𝑃 𝐶)))
2313, 15, 16, 17, 21, 22syl131anc 1376 . . . . . . . . . . . . 13 (𝜑 → (𝐶 (𝑃 𝑆) → 𝑆 (𝑃 𝐶)))
2412, 23mpd 15 . . . . . . . . . . . 12 (𝜑𝑆 (𝑃 𝐶))
259, 3hlatjcom 36054 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑃𝐴) → (𝐶 𝑃) = (𝑃 𝐶))
2613, 15, 17, 25syl3anc 1364 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑃) = (𝑃 𝐶))
2724, 26breqtrrd 4990 . . . . . . . . . . 11 (𝜑𝑆 (𝐶 𝑃))
281dalemclqjt 36321 . . . . . . . . . . . . 13 (𝜑𝐶 (𝑄 𝑇))
291dalemtea 36316 . . . . . . . . . . . . . 14 (𝜑𝑇𝐴)
301dalemrea 36314 . . . . . . . . . . . . . . 15 (𝜑𝑅𝐴)
31 simp312 1314 . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝐶 (𝑄 𝑅))
338, 9, 3atnlej1 36065 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝐶 (𝑄 𝑅)) → 𝐶𝑄)
3413, 15, 18, 30, 32, 33syl131anc 1376 . . . . . . . . . . . . . 14 (𝜑𝐶𝑄)
358, 9, 3hlatexch1 36081 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑇𝐴𝑄𝐴) ∧ 𝐶𝑄) → (𝐶 (𝑄 𝑇) → 𝑇 (𝑄 𝐶)))
3613, 15, 29, 18, 34, 35syl131anc 1376 . . . . . . . . . . . . 13 (𝜑 → (𝐶 (𝑄 𝑇) → 𝑇 (𝑄 𝐶)))
3728, 36mpd 15 . . . . . . . . . . . 12 (𝜑𝑇 (𝑄 𝐶))
389, 3hlatjcom 36054 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑄𝐴) → (𝐶 𝑄) = (𝑄 𝐶))
3913, 15, 18, 38syl3anc 1364 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑄) = (𝑄 𝐶))
4037, 39breqtrrd 4990 . . . . . . . . . . 11 (𝜑𝑇 (𝐶 𝑄))
411, 3dalemseb 36328 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (Base‘𝐾))
427, 9, 3hlatjcl 36053 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑃𝐴) → (𝐶 𝑃) ∈ (Base‘𝐾))
4313, 15, 17, 42syl3anc 1364 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑃) ∈ (Base‘𝐾))
441, 3dalemteb 36329 . . . . . . . . . . . 12 (𝜑𝑇 ∈ (Base‘𝐾))
457, 9, 3hlatjcl 36053 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑄𝐴) → (𝐶 𝑄) ∈ (Base‘𝐾))
4613, 15, 18, 45syl3anc 1364 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑄) ∈ (Base‘𝐾))
477, 8, 9latjlej12 17506 . . . . . . . . . . . 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 1372 . . . . . . . . . . 11 (𝜑 → ((𝑆 (𝐶 𝑃) ∧ 𝑇 (𝐶 𝑄)) → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄))))
4927, 40, 48mp2and 695 . . . . . . . . . 10 (𝜑 → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄)))
501, 3dalempeb 36325 . . . . . . . . . . 11 (𝜑𝑃 ∈ (Base‘𝐾))
511, 3dalemqeb 36326 . . . . . . . . . . 11 (𝜑𝑄 ∈ (Base‘𝐾))
527, 9latjjdi 17542 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (𝐶 (𝑃 𝑄)) = ((𝐶 𝑃) (𝐶 𝑄)))
532, 4, 50, 51, 52syl13anc 1365 . . . . . . . . . 10 (𝜑 → (𝐶 (𝑃 𝑄)) = ((𝐶 𝑃) (𝐶 𝑄)))
5449, 53breqtrrd 4990 . . . . . . . . 9 (𝜑 → (𝑆 𝑇) (𝐶 (𝑃 𝑄)))
551dalemclrju 36322 . . . . . . . . . . 11 (𝜑𝐶 (𝑅 𝑈))
561dalemuea 36317 . . . . . . . . . . . 12 (𝜑𝑈𝐴)
57 simp313 1315 . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . 13 (𝜑 → ¬ 𝐶 (𝑅 𝑃))
598, 9, 3atnlej1 36065 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑅𝐴𝑃𝐴) ∧ ¬ 𝐶 (𝑅 𝑃)) → 𝐶𝑅)
6013, 15, 30, 17, 58, 59syl131anc 1376 . . . . . . . . . . . 12 (𝜑𝐶𝑅)
618, 9, 3hlatexch1 36081 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑈𝐴𝑅𝐴) ∧ 𝐶𝑅) → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
6213, 15, 56, 30, 60, 61syl131anc 1376 . . . . . . . . . . 11 (𝜑 → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
6355, 62mpd 15 . . . . . . . . . 10 (𝜑𝑈 (𝑅 𝐶))
649, 3hlatjcom 36054 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑅𝐴) → (𝐶 𝑅) = (𝑅 𝐶))
6513, 15, 30, 64syl3anc 1364 . . . . . . . . . 10 (𝜑 → (𝐶 𝑅) = (𝑅 𝐶))
6663, 65breqtrrd 4990 . . . . . . . . 9 (𝜑𝑈 (𝐶 𝑅))
671, 9, 3dalemsjteb 36332 . . . . . . . . . 10 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
681, 9, 3dalempjqeb 36331 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
697, 9latjcl 17490 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾))
702, 4, 68, 69syl3anc 1364 . . . . . . . . . 10 (𝜑 → (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾))
711, 3dalemueb 36330 . . . . . . . . . 10 (𝜑𝑈 ∈ (Base‘𝐾))
727, 9, 3hlatjcl 36053 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑅𝐴) → (𝐶 𝑅) ∈ (Base‘𝐾))
7313, 15, 30, 72syl3anc 1364 . . . . . . . . . 10 (𝜑 → (𝐶 𝑅) ∈ (Base‘𝐾))
747, 8, 9latjlej12 17506 . . . . . . . . . 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 1372 . . . . . . . . 9 (𝜑 → (((𝑆 𝑇) (𝐶 (𝑃 𝑄)) ∧ 𝑈 (𝐶 𝑅)) → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅))))
7654, 66, 75mp2and 695 . . . . . . . 8 (𝜑 → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
771, 3dalemreb 36327 . . . . . . . . 9 (𝜑𝑅 ∈ (Base‘𝐾))
787, 9latjjdi 17542 . . . . . . . . 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 1365 . . . . . . . 8 (𝜑 → (𝐶 ((𝑃 𝑄) 𝑅)) = ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
8076, 79breqtrrd 4990 . . . . . . 7 (𝜑 → ((𝑆 𝑇) 𝑈) (𝐶 ((𝑃 𝑄) 𝑅)))
81 dalem-cly.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
8214oveq2i 7027 . . . . . . 7 (𝐶 𝑌) = (𝐶 ((𝑃 𝑄) 𝑅))
8380, 81, 823brtr4g 4996 . . . . . 6 (𝜑𝑍 (𝐶 𝑌))
84 breq2 4966 . . . . . 6 ((𝐶 𝑌) = 𝑌 → (𝑍 (𝐶 𝑌) ↔ 𝑍 𝑌))
8583, 84syl5ibcom 246 . . . . 5 (𝜑 → ((𝐶 𝑌) = 𝑌𝑍 𝑌))
8611, 85sylbid 241 . . . 4 (𝜑 → (𝐶 𝑌𝑍 𝑌))
871dalemzeo 36319 . . . . . 6 (𝜑𝑍𝑂)
881dalemyeo 36318 . . . . . 6 (𝜑𝑌𝑂)
898, 5lplncmp 36248 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑍𝑂𝑌𝑂) → (𝑍 𝑌𝑍 = 𝑌))
9013, 87, 88, 89syl3anc 1364 . . . . 5 (𝜑 → (𝑍 𝑌𝑍 = 𝑌))
91 eqcom 2802 . . . . 5 (𝑍 = 𝑌𝑌 = 𝑍)
9290, 91syl6bb 288 . . . 4 (𝜑 → (𝑍 𝑌𝑌 = 𝑍))
9386, 92sylibd 240 . . 3 (𝜑 → (𝐶 𝑌𝑌 = 𝑍))
9493necon3ad 2997 . 2 (𝜑 → (𝑌𝑍 → ¬ 𝐶 𝑌))
9594imp 407 1 ((𝜑𝑌𝑍) → ¬ 𝐶 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984   class class class wbr 4962  cfv 6225  (class class class)co 7016  Basecbs 16312  lecple 16401  joincjn 17383  Latclat 17484  Atomscatm 35949  HLchlt 36036  LPlanesclpl 36178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-proset 17367  df-poset 17385  df-plt 17397  df-lub 17413  df-glb 17414  df-join 17415  df-meet 17416  df-p0 17478  df-lat 17485  df-clat 17547  df-oposet 35862  df-ol 35864  df-oml 35865  df-covers 35952  df-ats 35953  df-atl 35984  df-cvlat 36008  df-hlat 36037  df-llines 36184  df-lplanes 36185
This theorem is referenced by:  dalem9  36358
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