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Theorem dalem-cly 35460
Description: Lemma for dalem9 35461. 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 35413 . . . . . 6 (𝜑𝐾 ∈ Lat)
3 dalemc.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 35427 . . . . . 6 (𝜑𝐶 ∈ (Base‘𝐾))
5 dalem-cly.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
61, 5dalemyeb 35438 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐾))
7 eqid 2760 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 dalemc.l . . . . . . 7 = (le‘𝐾)
9 dalemc.j . . . . . . 7 = (join‘𝐾)
107, 8, 9latleeqj1 17264 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝑌))
112, 4, 6, 10syl3anc 1477 . . . . 5 (𝜑 → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝑌))
121dalemclpjs 35423 . . . . . . . . . . . . 13 (𝜑𝐶 (𝑃 𝑆))
131dalemkehl 35412 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ HL)
14 dalem-cly.y . . . . . . . . . . . . . . 15 𝑌 = ((𝑃 𝑄) 𝑅)
151, 8, 9, 3, 5, 14dalemcea 35449 . . . . . . . . . . . . . 14 (𝜑𝐶𝐴)
161dalemsea 35418 . . . . . . . . . . . . . 14 (𝜑𝑆𝐴)
171dalempea 35415 . . . . . . . . . . . . . 14 (𝜑𝑃𝐴)
181dalemqea 35416 . . . . . . . . . . . . . . 15 (𝜑𝑄𝐴)
191dalem-clpjq 35426 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
208, 9, 3atnlej1 35168 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝐶 (𝑃 𝑄)) → 𝐶𝑃)
2113, 15, 17, 18, 19, 20syl131anc 1490 . . . . . . . . . . . . . 14 (𝜑𝐶𝑃)
228, 9, 3hlatexch1 35184 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑆𝐴𝑃𝐴) ∧ 𝐶𝑃) → (𝐶 (𝑃 𝑆) → 𝑆 (𝑃 𝐶)))
2313, 15, 16, 17, 21, 22syl131anc 1490 . . . . . . . . . . . . 13 (𝜑 → (𝐶 (𝑃 𝑆) → 𝑆 (𝑃 𝐶)))
2412, 23mpd 15 . . . . . . . . . . . 12 (𝜑𝑆 (𝑃 𝐶))
259, 3hlatjcom 35157 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑃𝐴) → (𝐶 𝑃) = (𝑃 𝐶))
2613, 15, 17, 25syl3anc 1477 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑃) = (𝑃 𝐶))
2724, 26breqtrrd 4832 . . . . . . . . . . 11 (𝜑𝑆 (𝐶 𝑃))
281dalemclqjt 35424 . . . . . . . . . . . . 13 (𝜑𝐶 (𝑄 𝑇))
291dalemtea 35419 . . . . . . . . . . . . . 14 (𝜑𝑇𝐴)
301dalemrea 35417 . . . . . . . . . . . . . . 15 (𝜑𝑅𝐴)
31 simp312 1406 . . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝐶 (𝑄 𝑅))
338, 9, 3atnlej1 35168 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝐶 (𝑄 𝑅)) → 𝐶𝑄)
3413, 15, 18, 30, 32, 33syl131anc 1490 . . . . . . . . . . . . . 14 (𝜑𝐶𝑄)
358, 9, 3hlatexch1 35184 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑇𝐴𝑄𝐴) ∧ 𝐶𝑄) → (𝐶 (𝑄 𝑇) → 𝑇 (𝑄 𝐶)))
3613, 15, 29, 18, 34, 35syl131anc 1490 . . . . . . . . . . . . 13 (𝜑 → (𝐶 (𝑄 𝑇) → 𝑇 (𝑄 𝐶)))
3728, 36mpd 15 . . . . . . . . . . . 12 (𝜑𝑇 (𝑄 𝐶))
389, 3hlatjcom 35157 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑄𝐴) → (𝐶 𝑄) = (𝑄 𝐶))
3913, 15, 18, 38syl3anc 1477 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑄) = (𝑄 𝐶))
4037, 39breqtrrd 4832 . . . . . . . . . . 11 (𝜑𝑇 (𝐶 𝑄))
411, 3dalemseb 35431 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (Base‘𝐾))
427, 9, 3hlatjcl 35156 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑃𝐴) → (𝐶 𝑃) ∈ (Base‘𝐾))
4313, 15, 17, 42syl3anc 1477 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑃) ∈ (Base‘𝐾))
441, 3dalemteb 35432 . . . . . . . . . . . 12 (𝜑𝑇 ∈ (Base‘𝐾))
457, 9, 3hlatjcl 35156 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑄𝐴) → (𝐶 𝑄) ∈ (Base‘𝐾))
4613, 15, 18, 45syl3anc 1477 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑄) ∈ (Base‘𝐾))
477, 8, 9latjlej12 17268 . . . . . . . . . . . 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 1486 . . . . . . . . . . 11 (𝜑 → ((𝑆 (𝐶 𝑃) ∧ 𝑇 (𝐶 𝑄)) → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄))))
4927, 40, 48mp2and 717 . . . . . . . . . 10 (𝜑 → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄)))
501, 3dalempeb 35428 . . . . . . . . . . 11 (𝜑𝑃 ∈ (Base‘𝐾))
511, 3dalemqeb 35429 . . . . . . . . . . 11 (𝜑𝑄 ∈ (Base‘𝐾))
527, 9latjjdi 17304 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (𝐶 (𝑃 𝑄)) = ((𝐶 𝑃) (𝐶 𝑄)))
532, 4, 50, 51, 52syl13anc 1479 . . . . . . . . . 10 (𝜑 → (𝐶 (𝑃 𝑄)) = ((𝐶 𝑃) (𝐶 𝑄)))
5449, 53breqtrrd 4832 . . . . . . . . 9 (𝜑 → (𝑆 𝑇) (𝐶 (𝑃 𝑄)))
551dalemclrju 35425 . . . . . . . . . . 11 (𝜑𝐶 (𝑅 𝑈))
561dalemuea 35420 . . . . . . . . . . . 12 (𝜑𝑈𝐴)
57 simp313 1407 . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . 13 (𝜑 → ¬ 𝐶 (𝑅 𝑃))
598, 9, 3atnlej1 35168 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑅𝐴𝑃𝐴) ∧ ¬ 𝐶 (𝑅 𝑃)) → 𝐶𝑅)
6013, 15, 30, 17, 58, 59syl131anc 1490 . . . . . . . . . . . 12 (𝜑𝐶𝑅)
618, 9, 3hlatexch1 35184 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑈𝐴𝑅𝐴) ∧ 𝐶𝑅) → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
6213, 15, 56, 30, 60, 61syl131anc 1490 . . . . . . . . . . 11 (𝜑 → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
6355, 62mpd 15 . . . . . . . . . 10 (𝜑𝑈 (𝑅 𝐶))
649, 3hlatjcom 35157 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑅𝐴) → (𝐶 𝑅) = (𝑅 𝐶))
6513, 15, 30, 64syl3anc 1477 . . . . . . . . . 10 (𝜑 → (𝐶 𝑅) = (𝑅 𝐶))
6663, 65breqtrrd 4832 . . . . . . . . 9 (𝜑𝑈 (𝐶 𝑅))
671, 9, 3dalemsjteb 35435 . . . . . . . . . 10 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
681, 9, 3dalempjqeb 35434 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
697, 9latjcl 17252 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾))
702, 4, 68, 69syl3anc 1477 . . . . . . . . . 10 (𝜑 → (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾))
711, 3dalemueb 35433 . . . . . . . . . 10 (𝜑𝑈 ∈ (Base‘𝐾))
727, 9, 3hlatjcl 35156 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑅𝐴) → (𝐶 𝑅) ∈ (Base‘𝐾))
7313, 15, 30, 72syl3anc 1477 . . . . . . . . . 10 (𝜑 → (𝐶 𝑅) ∈ (Base‘𝐾))
747, 8, 9latjlej12 17268 . . . . . . . . . 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 1486 . . . . . . . . 9 (𝜑 → (((𝑆 𝑇) (𝐶 (𝑃 𝑄)) ∧ 𝑈 (𝐶 𝑅)) → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅))))
7654, 66, 75mp2and 717 . . . . . . . 8 (𝜑 → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
771, 3dalemreb 35430 . . . . . . . . 9 (𝜑𝑅 ∈ (Base‘𝐾))
787, 9latjjdi 17304 . . . . . . . . 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 1479 . . . . . . . 8 (𝜑 → (𝐶 ((𝑃 𝑄) 𝑅)) = ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
8076, 79breqtrrd 4832 . . . . . . 7 (𝜑 → ((𝑆 𝑇) 𝑈) (𝐶 ((𝑃 𝑄) 𝑅)))
81 dalem-cly.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
8214oveq2i 6824 . . . . . . 7 (𝐶 𝑌) = (𝐶 ((𝑃 𝑄) 𝑅))
8380, 81, 823brtr4g 4838 . . . . . 6 (𝜑𝑍 (𝐶 𝑌))
84 breq2 4808 . . . . . 6 ((𝐶 𝑌) = 𝑌 → (𝑍 (𝐶 𝑌) ↔ 𝑍 𝑌))
8583, 84syl5ibcom 235 . . . . 5 (𝜑 → ((𝐶 𝑌) = 𝑌𝑍 𝑌))
8611, 85sylbid 230 . . . 4 (𝜑 → (𝐶 𝑌𝑍 𝑌))
871dalemzeo 35422 . . . . . 6 (𝜑𝑍𝑂)
881dalemyeo 35421 . . . . . 6 (𝜑𝑌𝑂)
898, 5lplncmp 35351 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑍𝑂𝑌𝑂) → (𝑍 𝑌𝑍 = 𝑌))
9013, 87, 88, 89syl3anc 1477 . . . . 5 (𝜑 → (𝑍 𝑌𝑍 = 𝑌))
91 eqcom 2767 . . . . 5 (𝑍 = 𝑌𝑌 = 𝑍)
9290, 91syl6bb 276 . . . 4 (𝜑 → (𝑍 𝑌𝑌 = 𝑍))
9386, 92sylibd 229 . . 3 (𝜑 → (𝐶 𝑌𝑌 = 𝑍))
9493necon3ad 2945 . 2 (𝜑 → (𝑌𝑍 → ¬ 𝐶 𝑌))
9594imp 444 1 ((𝜑𝑌𝑍) → ¬ 𝐶 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  Latclat 17246  Atomscatm 35053  HLchlt 35140  LPlanesclpl 35281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-lat 17247  df-clat 17309  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-llines 35287  df-lplanes 35288
This theorem is referenced by:  dalem9  35461
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