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Theorem dalem44 34482
Description: Lemma for dath 34502. 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 34481 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)
1413necomd 2845 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ≠ ((𝐺 𝐻) 𝐼))
151dalemkelat 34390 . . . . . . 7 (𝜑𝐾 ∈ Lat)
16153ad2ant1 1080 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
175, 4dalemcceb 34455 . . . . . . 7 (𝜓𝑐 ∈ (Base‘𝐾))
18173ad2ant3 1082 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem42 34480 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
20 eqid 2621 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2120, 7lplnbase 34300 . . . . . . 7 (((𝐺 𝐻) 𝐼) ∈ 𝑂 → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
2219, 21syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
2320, 2, 3latleeqj1 16984 . . . . . 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 1323 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼) ↔ (𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼)))
251, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem28 34466 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 (𝐺 𝑐))
261dalemkehl 34389 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ HL)
27263ad2ant1 1080 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
285dalemccea 34449 . . . . . . . . . . . . . 14 (𝜓𝑐𝐴)
29283ad2ant3 1082 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
301, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 34462 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
313, 4hlatjcom 34134 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐺𝐴) → (𝑐 𝐺) = (𝐺 𝑐))
3227, 29, 30, 31syl3anc 1323 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐺) = (𝐺 𝑐))
3325, 32breqtrrd 4641 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 (𝑐 𝐺))
341, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem33 34471 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝐻 𝑐))
351, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 34467 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
363, 4hlatjcom 34134 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐻𝐴) → (𝑐 𝐻) = (𝐻 𝑐))
3727, 29, 35, 36syl3anc 1323 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐻) = (𝐻 𝑐))
3834, 37breqtrrd 4641 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝑐 𝐻))
391, 4dalempeb 34405 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ (Base‘𝐾))
40393ad2ant1 1080 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 ∈ (Base‘𝐾))
4120, 3, 4hlatjcl 34133 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐺𝐴) → (𝑐 𝐺) ∈ (Base‘𝐾))
4227, 29, 30, 41syl3anc 1323 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐺) ∈ (Base‘𝐾))
431, 4dalemqeb 34406 . . . . . . . . . . . . 13 (𝜑𝑄 ∈ (Base‘𝐾))
44433ad2ant1 1080 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 ∈ (Base‘𝐾))
4520, 3, 4hlatjcl 34133 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐻𝐴) → (𝑐 𝐻) ∈ (Base‘𝐾))
4627, 29, 35, 45syl3anc 1323 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐻) ∈ (Base‘𝐾))
4720, 2, 3latjlej12 16988 . . . . . . . . . . . 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 1332 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 (𝑐 𝐺) ∧ 𝑄 (𝑐 𝐻)) → (𝑃 𝑄) ((𝑐 𝐺) (𝑐 𝐻))))
4933, 38, 48mp2and 714 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ((𝑐 𝐺) (𝑐 𝐻)))
5020, 4atbase 34056 . . . . . . . . . . . 12 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
5130, 50syl 17 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
5220, 4atbase 34056 . . . . . . . . . . . 12 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
5335, 52syl 17 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
5420, 3latjjdi 17024 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ 𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾))) → (𝑐 (𝐺 𝐻)) = ((𝑐 𝐺) (𝑐 𝐻)))
5516, 18, 51, 53, 54syl13anc 1325 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝐻)) = ((𝑐 𝐺) (𝑐 𝐻)))
5649, 55breqtrrd 4641 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) (𝑐 (𝐺 𝐻)))
571, 2, 3, 4, 5, 6, 7, 8, 9, 12dalem37 34475 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝐼 𝑐))
581, 2, 3, 4, 5, 6, 7, 8, 9, 12dalem34 34472 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
593, 4hlatjcom 34134 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐼𝐴) → (𝑐 𝐼) = (𝐼 𝑐))
6027, 29, 58, 59syl3anc 1323 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐼) = (𝐼 𝑐))
6157, 60breqtrrd 4641 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝑐 𝐼))
621, 3, 4dalempjqeb 34411 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
63623ad2ant1 1080 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
6420, 3, 4hlatjcl 34133 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
6527, 30, 35, 64syl3anc 1323 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
6620, 3latjcl 16972 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ (𝐺 𝐻) ∈ (Base‘𝐾)) → (𝑐 (𝐺 𝐻)) ∈ (Base‘𝐾))
6716, 18, 65, 66syl3anc 1323 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝐻)) ∈ (Base‘𝐾))
681, 4dalemreb 34407 . . . . . . . . . . 11 (𝜑𝑅 ∈ (Base‘𝐾))
69683ad2ant1 1080 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 ∈ (Base‘𝐾))
7020, 3, 4hlatjcl 34133 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐼𝐴) → (𝑐 𝐼) ∈ (Base‘𝐾))
7127, 29, 58, 70syl3anc 1323 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐼) ∈ (Base‘𝐾))
7220, 2, 3latjlej12 16988 . . . . . . . . . 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 1332 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → (((𝑃 𝑄) (𝑐 (𝐺 𝐻)) ∧ 𝑅 (𝑐 𝐼)) → ((𝑃 𝑄) 𝑅) ((𝑐 (𝐺 𝐻)) (𝑐 𝐼))))
7456, 61, 73mp2and 714 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) ((𝑐 (𝐺 𝐻)) (𝑐 𝐼)))
7520, 4atbase 34056 . . . . . . . . . 10 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
7658, 75syl 17 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
7720, 3latjjdi 17024 . . . . . . . . 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 1325 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼)) = ((𝑐 (𝐺 𝐻)) (𝑐 𝐼)))
7974, 78breqtrrd 4641 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (𝑐 ((𝐺 𝐻) 𝐼)))
808, 79syl5eqbr 4648 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (𝑐 ((𝐺 𝐻) 𝐼)))
81 breq2 4617 . . . . . 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 34398 . . . . . 6 (𝜑𝑌𝑂)
85843ad2ant1 1080 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
862, 7lplncmp 34328 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝑂 ∧ ((𝐺 𝐻) 𝐼) ∈ 𝑂) → (𝑌 ((𝐺 𝐻) 𝐼) ↔ 𝑌 = ((𝐺 𝐻) 𝐼)))
8727, 85, 19, 86syl3anc 1323 . . . 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 2803 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  meetcmee 16866  Latclat 16966  Atomscatm 34030  HLchlt 34117  LPlanesclpl 34258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-clat 17029  df-oposet 33943  df-ol 33945  df-oml 33946  df-covers 34033  df-ats 34034  df-atl 34065  df-cvlat 34089  df-hlat 34118  df-llines 34264  df-lplanes 34265  df-lvols 34266
This theorem is referenced by:  dalem45  34483
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