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Theorem dalem44 39718
Description: Lemma for dath 39738. 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 39717 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ≠ 𝑌)
1413necomd 2996 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ≠ ((𝐺 𝐻) 𝐼))
151dalemkelat 39626 . . . . . . 7 (𝜑𝐾 ∈ Lat)
16153ad2ant1 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
175, 4dalemcceb 39691 . . . . . . 7 (𝜓𝑐 ∈ (Base‘𝐾))
18173ad2ant3 1136 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem42 39716 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
20 eqid 2737 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2120, 7lplnbase 39536 . . . . . . 7 (((𝐺 𝐻) 𝐼) ∈ 𝑂 → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
2219, 21syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
2320, 2, 3latleeqj1 18496 . . . . . 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 1373 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼) ↔ (𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼)))
251, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem28 39702 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 (𝐺 𝑐))
261dalemkehl 39625 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ HL)
27263ad2ant1 1134 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
285dalemccea 39685 . . . . . . . . . . . . . 14 (𝜓𝑐𝐴)
29283ad2ant3 1136 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
301, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 39698 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
313, 4hlatjcom 39369 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐺𝐴) → (𝑐 𝐺) = (𝐺 𝑐))
3227, 29, 30, 31syl3anc 1373 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐺) = (𝐺 𝑐))
3325, 32breqtrrd 5171 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 (𝑐 𝐺))
341, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem33 39707 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝐻 𝑐))
351, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 39703 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
363, 4hlatjcom 39369 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐻𝐴) → (𝑐 𝐻) = (𝐻 𝑐))
3727, 29, 35, 36syl3anc 1373 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐻) = (𝐻 𝑐))
3834, 37breqtrrd 5171 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝑐 𝐻))
391, 4dalempeb 39641 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ (Base‘𝐾))
40393ad2ant1 1134 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 ∈ (Base‘𝐾))
4120, 3, 4hlatjcl 39368 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐺𝐴) → (𝑐 𝐺) ∈ (Base‘𝐾))
4227, 29, 30, 41syl3anc 1373 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐺) ∈ (Base‘𝐾))
431, 4dalemqeb 39642 . . . . . . . . . . . . 13 (𝜑𝑄 ∈ (Base‘𝐾))
44433ad2ant1 1134 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 ∈ (Base‘𝐾))
4520, 3, 4hlatjcl 39368 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐻𝐴) → (𝑐 𝐻) ∈ (Base‘𝐾))
4627, 29, 35, 45syl3anc 1373 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐻) ∈ (Base‘𝐾))
4720, 2, 3latjlej12 18500 . . . . . . . . . . . 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 1381 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 (𝑐 𝐺) ∧ 𝑄 (𝑐 𝐻)) → (𝑃 𝑄) ((𝑐 𝐺) (𝑐 𝐻))))
4933, 38, 48mp2and 699 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ((𝑐 𝐺) (𝑐 𝐻)))
5020, 4atbase 39290 . . . . . . . . . . . 12 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
5130, 50syl 17 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
5220, 4atbase 39290 . . . . . . . . . . . 12 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
5335, 52syl 17 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
5420, 3latjjdi 18536 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ 𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾))) → (𝑐 (𝐺 𝐻)) = ((𝑐 𝐺) (𝑐 𝐻)))
5516, 18, 51, 53, 54syl13anc 1374 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝐻)) = ((𝑐 𝐺) (𝑐 𝐻)))
5649, 55breqtrrd 5171 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) (𝑐 (𝐺 𝐻)))
571, 2, 3, 4, 5, 6, 7, 8, 9, 12dalem37 39711 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝐼 𝑐))
581, 2, 3, 4, 5, 6, 7, 8, 9, 12dalem34 39708 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
593, 4hlatjcom 39369 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐼𝐴) → (𝑐 𝐼) = (𝐼 𝑐))
6027, 29, 58, 59syl3anc 1373 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐼) = (𝐼 𝑐))
6157, 60breqtrrd 5171 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝑐 𝐼))
621, 3, 4dalempjqeb 39647 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
63623ad2ant1 1134 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
6420, 3, 4hlatjcl 39368 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
6527, 30, 35, 64syl3anc 1373 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
6620, 3latjcl 18484 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ (𝐺 𝐻) ∈ (Base‘𝐾)) → (𝑐 (𝐺 𝐻)) ∈ (Base‘𝐾))
6716, 18, 65, 66syl3anc 1373 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 (𝐺 𝐻)) ∈ (Base‘𝐾))
681, 4dalemreb 39643 . . . . . . . . . . 11 (𝜑𝑅 ∈ (Base‘𝐾))
69683ad2ant1 1134 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 ∈ (Base‘𝐾))
7020, 3, 4hlatjcl 39368 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝐼𝐴) → (𝑐 𝐼) ∈ (Base‘𝐾))
7127, 29, 58, 70syl3anc 1373 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝐼) ∈ (Base‘𝐾))
7220, 2, 3latjlej12 18500 . . . . . . . . . 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 1381 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → (((𝑃 𝑄) (𝑐 (𝐺 𝐻)) ∧ 𝑅 (𝑐 𝐼)) → ((𝑃 𝑄) 𝑅) ((𝑐 (𝐺 𝐻)) (𝑐 𝐼))))
7456, 61, 73mp2and 699 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) ((𝑐 (𝐺 𝐻)) (𝑐 𝐼)))
7520, 4atbase 39290 . . . . . . . . . 10 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
7658, 75syl 17 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
7720, 3latjjdi 18536 . . . . . . . . 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 1374 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼)) = ((𝑐 (𝐺 𝐻)) (𝑐 𝐼)))
7974, 78breqtrrd 5171 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (𝑐 ((𝐺 𝐻) 𝐼)))
808, 79eqbrtrid 5178 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (𝑐 ((𝐺 𝐻) 𝐼)))
81 breq2 5147 . . . . . 6 ((𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼) → (𝑌 (𝑐 ((𝐺 𝐻) 𝐼)) ↔ 𝑌 ((𝐺 𝐻) 𝐼)))
8280, 81syl5ibcom 245 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 ((𝐺 𝐻) 𝐼)) = ((𝐺 𝐻) 𝐼) → 𝑌 ((𝐺 𝐻) 𝐼)))
8324, 82sylbid 240 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼) → 𝑌 ((𝐺 𝐻) 𝐼)))
841dalemyeo 39634 . . . . . 6 (𝜑𝑌𝑂)
85843ad2ant1 1134 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
862, 7lplncmp 39564 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝑂 ∧ ((𝐺 𝐻) 𝐼) ∈ 𝑂) → (𝑌 ((𝐺 𝐻) 𝐼) ↔ 𝑌 = ((𝐺 𝐻) 𝐼)))
8727, 85, 19, 86syl3anc 1373 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 ((𝐺 𝐻) 𝐼) ↔ 𝑌 = ((𝐺 𝐻) 𝐼)))
8883, 87sylibd 239 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 ((𝐺 𝐻) 𝐼) → 𝑌 = ((𝐺 𝐻) 𝐼)))
8988necon3ad 2953 . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  Latclat 18476  Atomscatm 39264  HLchlt 39351  LPlanesclpl 39494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502
This theorem is referenced by:  dalem45  39719
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