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Theorem dalem-cly 37990
Description: Lemma for dalem9 37991. 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 37943 . . . . . 6 (𝜑𝐾 ∈ Lat)
3 dalemc.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 37957 . . . . . 6 (𝜑𝐶 ∈ (Base‘𝐾))
5 dalem-cly.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
61, 5dalemyeb 37968 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐾))
7 eqid 2736 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 dalemc.l . . . . . . 7 = (le‘𝐾)
9 dalemc.j . . . . . . 7 = (join‘𝐾)
107, 8, 9latleeqj1 18267 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝑌))
112, 4, 6, 10syl3anc 1370 . . . . 5 (𝜑 → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝑌))
121dalemclpjs 37953 . . . . . . . . . . . . 13 (𝜑𝐶 (𝑃 𝑆))
131dalemkehl 37942 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ HL)
14 dalem-cly.y . . . . . . . . . . . . . . 15 𝑌 = ((𝑃 𝑄) 𝑅)
151, 8, 9, 3, 5, 14dalemcea 37979 . . . . . . . . . . . . . 14 (𝜑𝐶𝐴)
161dalemsea 37948 . . . . . . . . . . . . . 14 (𝜑𝑆𝐴)
171dalempea 37945 . . . . . . . . . . . . . 14 (𝜑𝑃𝐴)
181dalemqea 37946 . . . . . . . . . . . . . . 15 (𝜑𝑄𝐴)
191dalem-clpjq 37956 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
208, 9, 3atnlej1 37698 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝐶 (𝑃 𝑄)) → 𝐶𝑃)
2113, 15, 17, 18, 19, 20syl131anc 1382 . . . . . . . . . . . . . 14 (𝜑𝐶𝑃)
228, 9, 3hlatexch1 37714 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑆𝐴𝑃𝐴) ∧ 𝐶𝑃) → (𝐶 (𝑃 𝑆) → 𝑆 (𝑃 𝐶)))
2313, 15, 16, 17, 21, 22syl131anc 1382 . . . . . . . . . . . . 13 (𝜑 → (𝐶 (𝑃 𝑆) → 𝑆 (𝑃 𝐶)))
2412, 23mpd 15 . . . . . . . . . . . 12 (𝜑𝑆 (𝑃 𝐶))
259, 3hlatjcom 37686 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑃𝐴) → (𝐶 𝑃) = (𝑃 𝐶))
2613, 15, 17, 25syl3anc 1370 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑃) = (𝑃 𝐶))
2724, 26breqtrrd 5121 . . . . . . . . . . 11 (𝜑𝑆 (𝐶 𝑃))
281dalemclqjt 37954 . . . . . . . . . . . . 13 (𝜑𝐶 (𝑄 𝑇))
291dalemtea 37949 . . . . . . . . . . . . . 14 (𝜑𝑇𝐴)
301dalemrea 37947 . . . . . . . . . . . . . . 15 (𝜑𝑅𝐴)
31 simp312 1320 . . . . . . . . . . . . . . . 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 37698 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝐶 (𝑄 𝑅)) → 𝐶𝑄)
3413, 15, 18, 30, 32, 33syl131anc 1382 . . . . . . . . . . . . . 14 (𝜑𝐶𝑄)
358, 9, 3hlatexch1 37714 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑇𝐴𝑄𝐴) ∧ 𝐶𝑄) → (𝐶 (𝑄 𝑇) → 𝑇 (𝑄 𝐶)))
3613, 15, 29, 18, 34, 35syl131anc 1382 . . . . . . . . . . . . 13 (𝜑 → (𝐶 (𝑄 𝑇) → 𝑇 (𝑄 𝐶)))
3728, 36mpd 15 . . . . . . . . . . . 12 (𝜑𝑇 (𝑄 𝐶))
389, 3hlatjcom 37686 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑄𝐴) → (𝐶 𝑄) = (𝑄 𝐶))
3913, 15, 18, 38syl3anc 1370 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑄) = (𝑄 𝐶))
4037, 39breqtrrd 5121 . . . . . . . . . . 11 (𝜑𝑇 (𝐶 𝑄))
411, 3dalemseb 37961 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (Base‘𝐾))
427, 9, 3hlatjcl 37685 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑃𝐴) → (𝐶 𝑃) ∈ (Base‘𝐾))
4313, 15, 17, 42syl3anc 1370 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑃) ∈ (Base‘𝐾))
441, 3dalemteb 37962 . . . . . . . . . . . 12 (𝜑𝑇 ∈ (Base‘𝐾))
457, 9, 3hlatjcl 37685 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑄𝐴) → (𝐶 𝑄) ∈ (Base‘𝐾))
4613, 15, 18, 45syl3anc 1370 . . . . . . . . . . . 12 (𝜑 → (𝐶 𝑄) ∈ (Base‘𝐾))
477, 8, 9latjlej12 18271 . . . . . . . . . . . 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 1378 . . . . . . . . . . 11 (𝜑 → ((𝑆 (𝐶 𝑃) ∧ 𝑇 (𝐶 𝑄)) → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄))))
4927, 40, 48mp2and 696 . . . . . . . . . 10 (𝜑 → (𝑆 𝑇) ((𝐶 𝑃) (𝐶 𝑄)))
501, 3dalempeb 37958 . . . . . . . . . . 11 (𝜑𝑃 ∈ (Base‘𝐾))
511, 3dalemqeb 37959 . . . . . . . . . . 11 (𝜑𝑄 ∈ (Base‘𝐾))
527, 9latjjdi 18307 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (𝐶 (𝑃 𝑄)) = ((𝐶 𝑃) (𝐶 𝑄)))
532, 4, 50, 51, 52syl13anc 1371 . . . . . . . . . 10 (𝜑 → (𝐶 (𝑃 𝑄)) = ((𝐶 𝑃) (𝐶 𝑄)))
5449, 53breqtrrd 5121 . . . . . . . . 9 (𝜑 → (𝑆 𝑇) (𝐶 (𝑃 𝑄)))
551dalemclrju 37955 . . . . . . . . . . 11 (𝜑𝐶 (𝑅 𝑈))
561dalemuea 37950 . . . . . . . . . . . 12 (𝜑𝑈𝐴)
57 simp313 1321 . . . . . . . . . . . . . 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 37698 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑅𝐴𝑃𝐴) ∧ ¬ 𝐶 (𝑅 𝑃)) → 𝐶𝑅)
6013, 15, 30, 17, 58, 59syl131anc 1382 . . . . . . . . . . . 12 (𝜑𝐶𝑅)
618, 9, 3hlatexch1 37714 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑈𝐴𝑅𝐴) ∧ 𝐶𝑅) → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
6213, 15, 56, 30, 60, 61syl131anc 1382 . . . . . . . . . . 11 (𝜑 → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
6355, 62mpd 15 . . . . . . . . . 10 (𝜑𝑈 (𝑅 𝐶))
649, 3hlatjcom 37686 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑅𝐴) → (𝐶 𝑅) = (𝑅 𝐶))
6513, 15, 30, 64syl3anc 1370 . . . . . . . . . 10 (𝜑 → (𝐶 𝑅) = (𝑅 𝐶))
6663, 65breqtrrd 5121 . . . . . . . . 9 (𝜑𝑈 (𝐶 𝑅))
671, 9, 3dalemsjteb 37965 . . . . . . . . . 10 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
681, 9, 3dalempjqeb 37964 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
697, 9latjcl 18255 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾))
702, 4, 68, 69syl3anc 1370 . . . . . . . . . 10 (𝜑 → (𝐶 (𝑃 𝑄)) ∈ (Base‘𝐾))
711, 3dalemueb 37963 . . . . . . . . . 10 (𝜑𝑈 ∈ (Base‘𝐾))
727, 9, 3hlatjcl 37685 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝐶𝐴𝑅𝐴) → (𝐶 𝑅) ∈ (Base‘𝐾))
7313, 15, 30, 72syl3anc 1370 . . . . . . . . . 10 (𝜑 → (𝐶 𝑅) ∈ (Base‘𝐾))
747, 8, 9latjlej12 18271 . . . . . . . . . 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 1378 . . . . . . . . 9 (𝜑 → (((𝑆 𝑇) (𝐶 (𝑃 𝑄)) ∧ 𝑈 (𝐶 𝑅)) → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅))))
7654, 66, 75mp2and 696 . . . . . . . 8 (𝜑 → ((𝑆 𝑇) 𝑈) ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
771, 3dalemreb 37960 . . . . . . . . 9 (𝜑𝑅 ∈ (Base‘𝐾))
787, 9latjjdi 18307 . . . . . . . . 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 1371 . . . . . . . 8 (𝜑 → (𝐶 ((𝑃 𝑄) 𝑅)) = ((𝐶 (𝑃 𝑄)) (𝐶 𝑅)))
8076, 79breqtrrd 5121 . . . . . . 7 (𝜑 → ((𝑆 𝑇) 𝑈) (𝐶 ((𝑃 𝑄) 𝑅)))
81 dalem-cly.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
8214oveq2i 7349 . . . . . . 7 (𝐶 𝑌) = (𝐶 ((𝑃 𝑄) 𝑅))
8380, 81, 823brtr4g 5127 . . . . . 6 (𝜑𝑍 (𝐶 𝑌))
84 breq2 5097 . . . . . 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 37952 . . . . . 6 (𝜑𝑍𝑂)
881dalemyeo 37951 . . . . . 6 (𝜑𝑌𝑂)
898, 5lplncmp 37881 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑍𝑂𝑌𝑂) → (𝑍 𝑌𝑍 = 𝑌))
9013, 87, 88, 89syl3anc 1370 . . . . 5 (𝜑 → (𝑍 𝑌𝑍 = 𝑌))
91 eqcom 2743 . . . . 5 (𝑍 = 𝑌𝑌 = 𝑍)
9290, 91bitrdi 286 . . . 4 (𝜑 → (𝑍 𝑌𝑌 = 𝑍))
9386, 92sylibd 238 . . 3 (𝜑 → (𝐶 𝑌𝑌 = 𝑍))
9493necon3ad 2953 . 2 (𝜑 → (𝑌𝑍 → ¬ 𝐶 𝑌))
9594imp 407 1 ((𝜑𝑌𝑍) → ¬ 𝐶 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wne 2940   class class class wbr 5093  cfv 6480  (class class class)co 7338  Basecbs 17010  lecple 17067  joincjn 18127  Latclat 18247  Atomscatm 37581  HLchlt 37668  LPlanesclpl 37811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-riota 7294  df-ov 7341  df-oprab 7342  df-proset 18111  df-poset 18129  df-plt 18146  df-lub 18162  df-glb 18163  df-join 18164  df-meet 18165  df-p0 18241  df-lat 18248  df-clat 18315  df-oposet 37494  df-ol 37496  df-oml 37497  df-covers 37584  df-ats 37585  df-atl 37616  df-cvlat 37640  df-hlat 37669  df-llines 37817  df-lplanes 37818
This theorem is referenced by:  dalem9  37991
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