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Theorem dalem-cly 33778
Description: Lemma for dalem9 33779. 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 33731 . . . . . 6 (𝜑𝐾 ∈ Lat)
3 dalemc.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 33745 . . . . . 6 (𝜑𝐶 ∈ (Base‘𝐾))
5 dalem-cly.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
61, 5dalemyeb 33756 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐾))
7 eqid 2609 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 dalemc.l . . . . . . 7 = (le‘𝐾)
9 dalemc.j . . . . . . 7 = (join‘𝐾)
107, 8, 9latleeqj1 16832 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝑌))
112, 4, 6, 10syl3anc 1317 . . . . 5 (𝜑 → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝑌))
121dalemclpjs 33741 . . . . . . . . . . . . 13 (𝜑𝐶 (𝑃 𝑆))
131dalemkehl 33730 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ HL)
14 dalem-cly.y . . . . . . . . . . . . . . 15 𝑌 = ((𝑃 𝑄) 𝑅)
151, 8, 9, 3, 5, 14dalemcea 33767 . . . . . . . . . . . . . 14 (𝜑𝐶𝐴)
161dalemsea 33736 . . . . . . . . . . . . . 14 (𝜑𝑆𝐴)
171dalempea 33733 . . . . . . . . . . . . . 14 (𝜑𝑃𝐴)
181dalemqea 33734 . . . . . . . . . . . . . . 15 (𝜑𝑄𝐴)
191dalem-clpjq 33744 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
208, 9, 3atnlej1 33486 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝐶 (𝑃 𝑄)) → 𝐶𝑃)
2113, 15, 17, 18, 19, 20syl131anc 1330 . . . . . . . . . . . . . 14 (𝜑𝐶𝑃)
228, 9, 3hlatexch1 33502 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑆𝐴𝑃𝐴) ∧ 𝐶𝑃) → (𝐶 (𝑃 𝑆) → 𝑆 (𝑃 𝐶)))
2313, 15, 16, 17, 21, 22syl131anc 1330 . . . . . . . . . . . . 13 (𝜑 → (𝐶 (𝑃 𝑆) → 𝑆 (𝑃 𝐶)))
2412, 23mpd 15 . . . . . . . . . . . 12 (𝜑𝑆 (𝑃 𝐶))
259, 3hlatjcom 33475 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑃𝐴) → (𝐶 𝑃) = (𝑃 𝐶))
2613, 15, 17, 25syl3anc 1317 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑃) = (𝑃 𝐶))
2724, 26breqtrrd 4605 . . . . . . . . . . 11 (𝜑𝑆 (𝐶 𝑃))
281dalemclqjt 33742 . . . . . . . . . . . . 13 (𝜑𝐶 (𝑄 𝑇))
291dalemtea 33737 . . . . . . . . . . . . . 14 (𝜑𝑇𝐴)
301dalemrea 33735 . . . . . . . . . . . . . . 15 (𝜑𝑅𝐴)
31 simp312 1201 . . . . . . . . . . . . . . . 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 205 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝐶 (𝑄 𝑅))
338, 9, 3atnlej1 33486 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝐶 (𝑄 𝑅)) → 𝐶𝑄)
3413, 15, 18, 30, 32, 33syl131anc 1330 . . . . . . . . . . . . . 14 (𝜑𝐶𝑄)
358, 9, 3hlatexch1 33502 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑇𝐴𝑄𝐴) ∧ 𝐶𝑄) → (𝐶 (𝑄 𝑇) → 𝑇 (𝑄 𝐶)))
3613, 15, 29, 18, 34, 35syl131anc 1330 . . . . . . . . . . . . 13 (𝜑 → (𝐶 (𝑄 𝑇) → 𝑇 (𝑄 𝐶)))
3728, 36mpd 15 . . . . . . . . . . . 12 (𝜑𝑇 (𝑄 𝐶))
389, 3hlatjcom 33475 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑄𝐴) → (𝐶 𝑄) = (𝑄 𝐶))
3913, 15, 18, 38syl3anc 1317 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑄) = (𝑄 𝐶))
4037, 39breqtrrd 4605 . . . . . . . . . . 11 (𝜑𝑇 (𝐶 𝑄))
411, 3dalemseb 33749 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (Base‘𝐾))
427, 9, 3hlatjcl 33474 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑃𝐴) → (𝐶 𝑃) ∈ (Base‘𝐾))
4313, 15, 17, 42syl3anc 1317 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑃) ∈ (Base‘𝐾))
441, 3dalemteb 33750 . . . . . . . . . . . 12 (𝜑𝑇 ∈ (Base‘𝐾))
457, 9, 3hlatjcl 33474 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑄𝐴) → (𝐶 𝑄) ∈ (Base‘𝐾))
4613, 15, 18, 45syl3anc 1317 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑄) ∈ (Base‘𝐾))
477, 8, 9latjlej12 16836 . . . . . . . . . . . 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 1326 . . . . . . . . . . 11 (𝜑 → ((𝑆 (𝐶 𝑃) ∧ 𝑇 (𝐶 𝑄)) → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄))))
4927, 40, 48mp2and 710 . . . . . . . . . 10 (𝜑 → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄)))
501, 3dalempeb 33746 . . . . . . . . . . 11 (𝜑𝑃 ∈ (Base‘𝐾))
511, 3dalemqeb 33747 . . . . . . . . . . 11 (𝜑𝑄 ∈ (Base‘𝐾))
527, 9latjjdi 16872 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (𝐶 (𝑃 𝑄)) = ((𝐶 𝑃) (𝐶 𝑄)))
532, 4, 50, 51, 52syl13anc 1319 . . . . . . . . . 10 (𝜑 → (𝐶 (𝑃 𝑄)) = ((𝐶 𝑃) (𝐶 𝑄)))
5449, 53breqtrrd 4605 . . . . . . . . 9 (𝜑 → (𝑆 𝑇) (𝐶 (𝑃 𝑄)))
551dalemclrju 33743 . . . . . . . . . . 11 (𝜑𝐶 (𝑅 𝑈))
561dalemuea 33738 . . . . . . . . . . . 12 (𝜑𝑈𝐴)
57 simp313 1202 . . . . . . . . . . . . . 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 205 . . . . . . . . . . . . 13 (𝜑 → ¬ 𝐶 (𝑅 𝑃))
598, 9, 3atnlej1 33486 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑅𝐴𝑃𝐴) ∧ ¬ 𝐶 (𝑅 𝑃)) → 𝐶𝑅)
6013, 15, 30, 17, 58, 59syl131anc 1330 . . . . . . . . . . . 12 (𝜑𝐶𝑅)
618, 9, 3hlatexch1 33502 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑈𝐴𝑅𝐴) ∧ 𝐶𝑅) → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
6213, 15, 56, 30, 60, 61syl131anc 1330 . . . . . . . . . . 11 (𝜑 → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
6355, 62mpd 15 . . . . . . . . . 10 (𝜑𝑈 (𝑅 𝐶))
649, 3hlatjcom 33475 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑅𝐴) → (𝐶 𝑅) = (𝑅 𝐶))
6513, 15, 30, 64syl3anc 1317 . . . . . . . . . 10 (𝜑 → (𝐶 𝑅) = (𝑅 𝐶))
6663, 65breqtrrd 4605 . . . . . . . . 9 (𝜑𝑈 (𝐶 𝑅))
671, 9, 3dalemsjteb 33753 . . . . . . . . . 10 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
681, 9, 3dalempjqeb 33752 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
697, 9latjcl 16820 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾))
702, 4, 68, 69syl3anc 1317 . . . . . . . . . 10 (𝜑 → (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾))
711, 3dalemueb 33751 . . . . . . . . . 10 (𝜑𝑈 ∈ (Base‘𝐾))
727, 9, 3hlatjcl 33474 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑅𝐴) → (𝐶 𝑅) ∈ (Base‘𝐾))
7313, 15, 30, 72syl3anc 1317 . . . . . . . . . 10 (𝜑 → (𝐶 𝑅) ∈ (Base‘𝐾))
747, 8, 9latjlej12 16836 . . . . . . . . . 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 1326 . . . . . . . . 9 (𝜑 → (((𝑆 𝑇) (𝐶 (𝑃 𝑄)) ∧ 𝑈 (𝐶 𝑅)) → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅))))
7654, 66, 75mp2and 710 . . . . . . . 8 (𝜑 → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
771, 3dalemreb 33748 . . . . . . . . 9 (𝜑𝑅 ∈ (Base‘𝐾))
787, 9latjjdi 16872 . . . . . . . . 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 1319 . . . . . . . 8 (𝜑 → (𝐶 ((𝑃 𝑄) 𝑅)) = ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
8076, 79breqtrrd 4605 . . . . . . 7 (𝜑 → ((𝑆 𝑇) 𝑈) (𝐶 ((𝑃 𝑄) 𝑅)))
81 dalem-cly.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
8214oveq2i 6538 . . . . . . 7 (𝐶 𝑌) = (𝐶 ((𝑃 𝑄) 𝑅))
8380, 81, 823brtr4g 4611 . . . . . 6 (𝜑𝑍 (𝐶 𝑌))
84 breq2 4581 . . . . . 6 ((𝐶 𝑌) = 𝑌 → (𝑍 (𝐶 𝑌) ↔ 𝑍 𝑌))
8583, 84syl5ibcom 233 . . . . 5 (𝜑 → ((𝐶 𝑌) = 𝑌𝑍 𝑌))
8611, 85sylbid 228 . . . 4 (𝜑 → (𝐶 𝑌𝑍 𝑌))
871dalemzeo 33740 . . . . . 6 (𝜑𝑍𝑂)
881dalemyeo 33739 . . . . . 6 (𝜑𝑌𝑂)
898, 5lplncmp 33669 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑍𝑂𝑌𝑂) → (𝑍 𝑌𝑍 = 𝑌))
9013, 87, 88, 89syl3anc 1317 . . . . 5 (𝜑 → (𝑍 𝑌𝑍 = 𝑌))
91 eqcom 2616 . . . . 5 (𝑍 = 𝑌𝑌 = 𝑍)
9290, 91syl6bb 274 . . . 4 (𝜑 → (𝑍 𝑌𝑌 = 𝑍))
9386, 92sylibd 227 . . 3 (𝜑 → (𝐶 𝑌𝑌 = 𝑍))
9493necon3ad 2794 . 2 (𝜑 → (𝑌𝑍 → ¬ 𝐶 𝑌))
9594imp 443 1 ((𝜑𝑌𝑍) → ¬ 𝐶 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  Latclat 16814  Atomscatm 33371  HLchlt 33458  LPlanesclpl 33599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  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-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459  df-llines 33605  df-lplanes 33606
This theorem is referenced by:  dalem9  33779
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