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Theorem dalem44 37005
Description: Lemma for dath 37025. 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 37004 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)
1413necomd 3045 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ≠ ((𝐺 𝐻) 𝐼))
151dalemkelat 36913 . . . . . . 7 (𝜑𝐾 ∈ Lat)
16153ad2ant1 1130 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
175, 4dalemcceb 36978 . . . . . . 7 (𝜓𝑐 ∈ (Base‘𝐾))
18173ad2ant3 1132 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem42 37003 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
20 eqid 2801 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2120, 7lplnbase 36823 . . . . . . 7 (((𝐺 𝐻) 𝐼) ∈ 𝑂 → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
2219, 21syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
2320, 2, 3latleeqj1 17668 . . . . . 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 1368 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼) ↔ (𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼)))
251, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem28 36989 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 (𝐺 𝑐))
261dalemkehl 36912 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ HL)
27263ad2ant1 1130 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
285dalemccea 36972 . . . . . . . . . . . . . 14 (𝜓𝑐𝐴)
29283ad2ant3 1132 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
301, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 36985 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
313, 4hlatjcom 36657 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐺𝐴) → (𝑐 𝐺) = (𝐺 𝑐))
3227, 29, 30, 31syl3anc 1368 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐺) = (𝐺 𝑐))
3325, 32breqtrrd 5061 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 (𝑐 𝐺))
341, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem33 36994 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝐻 𝑐))
351, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 36990 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
363, 4hlatjcom 36657 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐻𝐴) → (𝑐 𝐻) = (𝐻 𝑐))
3727, 29, 35, 36syl3anc 1368 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐻) = (𝐻 𝑐))
3834, 37breqtrrd 5061 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝑐 𝐻))
391, 4dalempeb 36928 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ (Base‘𝐾))
40393ad2ant1 1130 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 ∈ (Base‘𝐾))
4120, 3, 4hlatjcl 36656 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐺𝐴) → (𝑐 𝐺) ∈ (Base‘𝐾))
4227, 29, 30, 41syl3anc 1368 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐺) ∈ (Base‘𝐾))
431, 4dalemqeb 36929 . . . . . . . . . . . . 13 (𝜑𝑄 ∈ (Base‘𝐾))
44433ad2ant1 1130 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 ∈ (Base‘𝐾))
4520, 3, 4hlatjcl 36656 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐻𝐴) → (𝑐 𝐻) ∈ (Base‘𝐾))
4627, 29, 35, 45syl3anc 1368 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐻) ∈ (Base‘𝐾))
4720, 2, 3latjlej12 17672 . . . . . . . . . . . 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 1376 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 (𝑐 𝐺) ∧ 𝑄 (𝑐 𝐻)) → (𝑃 𝑄) ((𝑐 𝐺) (𝑐 𝐻))))
4933, 38, 48mp2and 698 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ((𝑐 𝐺) (𝑐 𝐻)))
5020, 4atbase 36578 . . . . . . . . . . . 12 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
5130, 50syl 17 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
5220, 4atbase 36578 . . . . . . . . . . . 12 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
5335, 52syl 17 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
5420, 3latjjdi 17708 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ 𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾))) → (𝑐 (𝐺 𝐻)) = ((𝑐 𝐺) (𝑐 𝐻)))
5516, 18, 51, 53, 54syl13anc 1369 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝐻)) = ((𝑐 𝐺) (𝑐 𝐻)))
5649, 55breqtrrd 5061 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) (𝑐 (𝐺 𝐻)))
571, 2, 3, 4, 5, 6, 7, 8, 9, 12dalem37 36998 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝐼 𝑐))
581, 2, 3, 4, 5, 6, 7, 8, 9, 12dalem34 36995 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
593, 4hlatjcom 36657 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐼𝐴) → (𝑐 𝐼) = (𝐼 𝑐))
6027, 29, 58, 59syl3anc 1368 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐼) = (𝐼 𝑐))
6157, 60breqtrrd 5061 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝑐 𝐼))
621, 3, 4dalempjqeb 36934 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
63623ad2ant1 1130 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
6420, 3, 4hlatjcl 36656 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
6527, 30, 35, 64syl3anc 1368 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
6620, 3latjcl 17656 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ (𝐺 𝐻) ∈ (Base‘𝐾)) → (𝑐 (𝐺 𝐻)) ∈ (Base‘𝐾))
6716, 18, 65, 66syl3anc 1368 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝐻)) ∈ (Base‘𝐾))
681, 4dalemreb 36930 . . . . . . . . . . 11 (𝜑𝑅 ∈ (Base‘𝐾))
69683ad2ant1 1130 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 ∈ (Base‘𝐾))
7020, 3, 4hlatjcl 36656 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐼𝐴) → (𝑐 𝐼) ∈ (Base‘𝐾))
7127, 29, 58, 70syl3anc 1368 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐼) ∈ (Base‘𝐾))
7220, 2, 3latjlej12 17672 . . . . . . . . . 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 1376 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → (((𝑃 𝑄) (𝑐 (𝐺 𝐻)) ∧ 𝑅 (𝑐 𝐼)) → ((𝑃 𝑄) 𝑅) ((𝑐 (𝐺 𝐻)) (𝑐 𝐼))))
7456, 61, 73mp2and 698 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) ((𝑐 (𝐺 𝐻)) (𝑐 𝐼)))
7520, 4atbase 36578 . . . . . . . . . 10 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
7658, 75syl 17 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
7720, 3latjjdi 17708 . . . . . . . . 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 1369 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼)) = ((𝑐 (𝐺 𝐻)) (𝑐 𝐼)))
7974, 78breqtrrd 5061 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (𝑐 ((𝐺 𝐻) 𝐼)))
808, 79eqbrtrid 5068 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (𝑐 ((𝐺 𝐻) 𝐼)))
81 breq2 5037 . . . . . 6 ((𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼) → (𝑌 (𝑐 ((𝐺 𝐻) 𝐼)) ↔ 𝑌 ((𝐺 𝐻) 𝐼)))
8280, 81syl5ibcom 248 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼) → 𝑌 ((𝐺 𝐻) 𝐼)))
8324, 82sylbid 243 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼) → 𝑌 ((𝐺 𝐻) 𝐼)))
841dalemyeo 36921 . . . . . 6 (𝜑𝑌𝑂)
85843ad2ant1 1130 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
862, 7lplncmp 36851 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝑂 ∧ ((𝐺 𝐻) 𝐼) ∈ 𝑂) → (𝑌 ((𝐺 𝐻) 𝐼) ↔ 𝑌 = ((𝐺 𝐻) 𝐼)))
8727, 85, 19, 86syl3anc 1368 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 ((𝐺 𝐻) 𝐼) ↔ 𝑌 = ((𝐺 𝐻) 𝐼)))
8883, 87sylibd 242 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼) → 𝑌 = ((𝐺 𝐻) 𝐼)))
8988necon3ad 3003 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16478  lecple 16567  joincjn 17549  meetcmee 17550  Latclat 17650  Atomscatm 36552  HLchlt 36639  LPlanesclpl 36781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-lat 17651  df-clat 17713  df-oposet 36465  df-ol 36467  df-oml 36468  df-covers 36555  df-ats 36556  df-atl 36587  df-cvlat 36611  df-hlat 36640  df-llines 36787  df-lplanes 36788  df-lvols 36789
This theorem is referenced by:  dalem45  37006
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