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Theorem dalem44 35320
Description: Lemma for dath 35340. Dummy center of perspectivity 𝑐 lies outside of plane 𝐺𝐻𝐼. (Contributed by NM, 16-Aug-2012.)
Hypotheses
Ref Expression
dalem.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)))))
dalem.l = (le‘𝐾)
dalem.j = (join‘𝐾)
dalem.a 𝐴 = (Atoms‘𝐾)
dalem.ps (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
dalem44.m = (meet‘𝐾)
dalem44.o 𝑂 = (LPlanes‘𝐾)
dalem44.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem44.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem44.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem44.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem44.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
Assertion
Ref Expression
dalem44 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 ((𝐺 𝐻) 𝐼))

Proof of Theorem dalem44
StepHypRef Expression
1 dalem.ph . . . 4 (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)))))
2 dalem.l . . . 4 = (le‘𝐾)
3 dalem.j . . . 4 = (join‘𝐾)
4 dalem.a . . . 4 𝐴 = (Atoms‘𝐾)
5 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
6 dalem44.m . . . 4 = (meet‘𝐾)
7 dalem44.o . . . 4 𝑂 = (LPlanes‘𝐾)
8 dalem44.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
9 dalem44.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
10 dalem44.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
11 dalem44.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
12 dalem44.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem43 35319 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)
1413necomd 2878 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ≠ ((𝐺 𝐻) 𝐼))
151dalemkelat 35228 . . . . . . 7 (𝜑𝐾 ∈ Lat)
16153ad2ant1 1102 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
175, 4dalemcceb 35293 . . . . . . 7 (𝜓𝑐 ∈ (Base‘𝐾))
18173ad2ant3 1104 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem42 35318 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
20 eqid 2651 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2120, 7lplnbase 35138 . . . . . . 7 (((𝐺 𝐻) 𝐼) ∈ 𝑂 → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
2219, 21syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
2320, 2, 3latleeqj1 17110 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾)) → (𝑐 ((𝐺 𝐻) 𝐼) ↔ (𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼)))
2416, 18, 22, 23syl3anc 1366 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼) ↔ (𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼)))
251, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem28 35304 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 (𝐺 𝑐))
261dalemkehl 35227 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ HL)
27263ad2ant1 1102 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
285dalemccea 35287 . . . . . . . . . . . . . 14 (𝜓𝑐𝐴)
29283ad2ant3 1104 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
301, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 35300 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
313, 4hlatjcom 34972 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐺𝐴) → (𝑐 𝐺) = (𝐺 𝑐))
3227, 29, 30, 31syl3anc 1366 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐺) = (𝐺 𝑐))
3325, 32breqtrrd 4713 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 (𝑐 𝐺))
341, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem33 35309 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝐻 𝑐))
351, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 35305 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
363, 4hlatjcom 34972 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐻𝐴) → (𝑐 𝐻) = (𝐻 𝑐))
3727, 29, 35, 36syl3anc 1366 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐻) = (𝐻 𝑐))
3834, 37breqtrrd 4713 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝑐 𝐻))
391, 4dalempeb 35243 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ (Base‘𝐾))
40393ad2ant1 1102 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 ∈ (Base‘𝐾))
4120, 3, 4hlatjcl 34971 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐺𝐴) → (𝑐 𝐺) ∈ (Base‘𝐾))
4227, 29, 30, 41syl3anc 1366 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐺) ∈ (Base‘𝐾))
431, 4dalemqeb 35244 . . . . . . . . . . . . 13 (𝜑𝑄 ∈ (Base‘𝐾))
44433ad2ant1 1102 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 ∈ (Base‘𝐾))
4520, 3, 4hlatjcl 34971 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐻𝐴) → (𝑐 𝐻) ∈ (Base‘𝐾))
4627, 29, 35, 45syl3anc 1366 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐻) ∈ (Base‘𝐾))
4720, 2, 3latjlej12 17114 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝑐 𝐺) ∈ (Base‘𝐾)) ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑐 𝐻) ∈ (Base‘𝐾))) → ((𝑃 (𝑐 𝐺) ∧ 𝑄 (𝑐 𝐻)) → (𝑃 𝑄) ((𝑐 𝐺) (𝑐 𝐻))))
4816, 40, 42, 44, 46, 47syl122anc 1375 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 (𝑐 𝐺) ∧ 𝑄 (𝑐 𝐻)) → (𝑃 𝑄) ((𝑐 𝐺) (𝑐 𝐻))))
4933, 38, 48mp2and 715 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ((𝑐 𝐺) (𝑐 𝐻)))
5020, 4atbase 34894 . . . . . . . . . . . 12 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
5130, 50syl 17 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
5220, 4atbase 34894 . . . . . . . . . . . 12 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
5335, 52syl 17 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
5420, 3latjjdi 17150 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ 𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾))) → (𝑐 (𝐺 𝐻)) = ((𝑐 𝐺) (𝑐 𝐻)))
5516, 18, 51, 53, 54syl13anc 1368 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝐻)) = ((𝑐 𝐺) (𝑐 𝐻)))
5649, 55breqtrrd 4713 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) (𝑐 (𝐺 𝐻)))
571, 2, 3, 4, 5, 6, 7, 8, 9, 12dalem37 35313 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝐼 𝑐))
581, 2, 3, 4, 5, 6, 7, 8, 9, 12dalem34 35310 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
593, 4hlatjcom 34972 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐼𝐴) → (𝑐 𝐼) = (𝐼 𝑐))
6027, 29, 58, 59syl3anc 1366 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐼) = (𝐼 𝑐))
6157, 60breqtrrd 4713 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝑐 𝐼))
621, 3, 4dalempjqeb 35249 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
63623ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
6420, 3, 4hlatjcl 34971 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
6527, 30, 35, 64syl3anc 1366 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
6620, 3latjcl 17098 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ (𝐺 𝐻) ∈ (Base‘𝐾)) → (𝑐 (𝐺 𝐻)) ∈ (Base‘𝐾))
6716, 18, 65, 66syl3anc 1366 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝐻)) ∈ (Base‘𝐾))
681, 4dalemreb 35245 . . . . . . . . . . 11 (𝜑𝑅 ∈ (Base‘𝐾))
69683ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 ∈ (Base‘𝐾))
7020, 3, 4hlatjcl 34971 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐼𝐴) → (𝑐 𝐼) ∈ (Base‘𝐾))
7127, 29, 58, 70syl3anc 1366 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐼) ∈ (Base‘𝐾))
7220, 2, 3latjlej12 17114 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑐 (𝐺 𝐻)) ∈ (Base‘𝐾)) ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝑐 𝐼) ∈ (Base‘𝐾))) → (((𝑃 𝑄) (𝑐 (𝐺 𝐻)) ∧ 𝑅 (𝑐 𝐼)) → ((𝑃 𝑄) 𝑅) ((𝑐 (𝐺 𝐻)) (𝑐 𝐼))))
7316, 63, 67, 69, 71, 72syl122anc 1375 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → (((𝑃 𝑄) (𝑐 (𝐺 𝐻)) ∧ 𝑅 (𝑐 𝐼)) → ((𝑃 𝑄) 𝑅) ((𝑐 (𝐺 𝐻)) (𝑐 𝐼))))
7456, 61, 73mp2and 715 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) ((𝑐 (𝐺 𝐻)) (𝑐 𝐼)))
7520, 4atbase 34894 . . . . . . . . . 10 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
7658, 75syl 17 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
7720, 3latjjdi 17150 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾))) → (𝑐 ((𝐺 𝐻) 𝐼)) = ((𝑐 (𝐺 𝐻)) (𝑐 𝐼)))
7816, 18, 65, 76, 77syl13anc 1368 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼)) = ((𝑐 (𝐺 𝐻)) (𝑐 𝐼)))
7974, 78breqtrrd 4713 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (𝑐 ((𝐺 𝐻) 𝐼)))
808, 79syl5eqbr 4720 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (𝑐 ((𝐺 𝐻) 𝐼)))
81 breq2 4689 . . . . . 6 ((𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼) → (𝑌 (𝑐 ((𝐺 𝐻) 𝐼)) ↔ 𝑌 ((𝐺 𝐻) 𝐼)))
8280, 81syl5ibcom 235 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼) → 𝑌 ((𝐺 𝐻) 𝐼)))
8324, 82sylbid 230 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼) → 𝑌 ((𝐺 𝐻) 𝐼)))
841dalemyeo 35236 . . . . . 6 (𝜑𝑌𝑂)
85843ad2ant1 1102 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
862, 7lplncmp 35166 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝑂 ∧ ((𝐺 𝐻) 𝐼) ∈ 𝑂) → (𝑌 ((𝐺 𝐻) 𝐼) ↔ 𝑌 = ((𝐺 𝐻) 𝐼)))
8727, 85, 19, 86syl3anc 1366 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 ((𝐺 𝐻) 𝐼) ↔ 𝑌 = ((𝐺 𝐻) 𝐼)))
8883, 87sylibd 229 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼) → 𝑌 = ((𝐺 𝐻) 𝐼)))
8988necon3ad 2836 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 ≠ ((𝐺 𝐻) 𝐼) → ¬ 𝑐 ((𝐺 𝐻) 𝐼)))
9014, 89mpd 15 1 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 ((𝐺 𝐻) 𝐼))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Latclat 17092  Atomscatm 34868  HLchlt 34955  LPlanesclpl 35096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104
This theorem is referenced by:  dalem45  35321
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