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Theorem dalem38 39712
Description: Lemma for dath 39738. Plane 𝑌 belongs to the 3-dimensional volume 𝐺𝐻𝐼𝑐. (Contributed by NM, 5-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))))
dalem38.m = (meet‘𝐾)
dalem38.o 𝑂 = (LPlanes‘𝐾)
dalem38.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem38.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem38.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem38.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem38.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
Assertion
Ref Expression
dalem38 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))

Proof of Theorem dalem38
StepHypRef Expression
1 dalem38.y . 2 𝑌 = ((𝑃 𝑄) 𝑅)
2 dalem.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)))))
3 dalem.l . . . . . . 7 = (le‘𝐾)
4 dalem.j . . . . . . 7 = (join‘𝐾)
5 dalem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
6 dalem.ps . . . . . . 7 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
7 dalem38.m . . . . . . 7 = (meet‘𝐾)
8 dalem38.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
9 dalem38.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
10 dalem38.g . . . . . . 7 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
112, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem28 39702 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 (𝐺 𝑐))
12 dalem38.h . . . . . . 7 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
132, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem33 39707 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝐻 𝑐))
142dalemkelat 39626 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
15143ad2ant1 1134 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
162, 5dalempeb 39641 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
17163ad2ant1 1134 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 ∈ (Base‘𝐾))
182dalemkehl 39625 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
19183ad2ant1 1134 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
202, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem23 39698 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
216dalemccea 39685 . . . . . . . . 9 (𝜓𝑐𝐴)
22213ad2ant3 1136 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
23 eqid 2737 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2423, 4, 5hlatjcl 39368 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐺𝐴𝑐𝐴) → (𝐺 𝑐) ∈ (Base‘𝐾))
2519, 20, 22, 24syl3anc 1373 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑐) ∈ (Base‘𝐾))
262, 5dalemqeb 39642 . . . . . . . 8 (𝜑𝑄 ∈ (Base‘𝐾))
27263ad2ant1 1134 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 ∈ (Base‘𝐾))
282, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem29 39703 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
2923, 4, 5hlatjcl 39368 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐻𝐴𝑐𝐴) → (𝐻 𝑐) ∈ (Base‘𝐾))
3019, 28, 22, 29syl3anc 1373 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐻 𝑐) ∈ (Base‘𝐾))
3123, 3, 4latjlej12 18500 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝐺 𝑐) ∈ (Base‘𝐾)) ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝐻 𝑐) ∈ (Base‘𝐾))) → ((𝑃 (𝐺 𝑐) ∧ 𝑄 (𝐻 𝑐)) → (𝑃 𝑄) ((𝐺 𝑐) (𝐻 𝑐))))
3215, 17, 25, 27, 30, 31syl122anc 1381 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 (𝐺 𝑐) ∧ 𝑄 (𝐻 𝑐)) → (𝑃 𝑄) ((𝐺 𝑐) (𝐻 𝑐))))
3311, 13, 32mp2and 699 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ((𝐺 𝑐) (𝐻 𝑐)))
3423, 5atbase 39290 . . . . . . 7 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
3520, 34syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
3623, 5atbase 39290 . . . . . . 7 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
3728, 36syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
386, 5dalemcceb 39691 . . . . . . 7 (𝜓𝑐 ∈ (Base‘𝐾))
39383ad2ant3 1136 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4023, 4latjjdir 18537 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → ((𝐺 𝐻) 𝑐) = ((𝐺 𝑐) (𝐻 𝑐)))
4115, 35, 37, 39, 40syl13anc 1374 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑐) = ((𝐺 𝑐) (𝐻 𝑐)))
4233, 41breqtrrd 5171 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ((𝐺 𝐻) 𝑐))
43 dalem38.i . . . . 5 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
442, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem37 39711 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝐼 𝑐))
452, 4, 5dalempjqeb 39647 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
46453ad2ant1 1134 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
4723, 4, 5hlatjcl 39368 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
4819, 20, 28, 47syl3anc 1373 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
4923, 4latjcl 18484 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝑐) ∈ (Base‘𝐾))
5015, 48, 39, 49syl3anc 1373 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑐) ∈ (Base‘𝐾))
512, 5dalemreb 39643 . . . . . 6 (𝜑𝑅 ∈ (Base‘𝐾))
52513ad2ant1 1134 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 ∈ (Base‘𝐾))
532, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem34 39708 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
5423, 4, 5hlatjcl 39368 . . . . . 6 ((𝐾 ∈ HL ∧ 𝐼𝐴𝑐𝐴) → (𝐼 𝑐) ∈ (Base‘𝐾))
5519, 53, 22, 54syl3anc 1373 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐼 𝑐) ∈ (Base‘𝐾))
5623, 3, 4latjlej12 18500 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ ((𝐺 𝐻) 𝑐) ∈ (Base‘𝐾)) ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝐼 𝑐) ∈ (Base‘𝐾))) → (((𝑃 𝑄) ((𝐺 𝐻) 𝑐) ∧ 𝑅 (𝐼 𝑐)) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝑐) (𝐼 𝑐))))
5715, 46, 50, 52, 55, 56syl122anc 1381 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝑃 𝑄) ((𝐺 𝐻) 𝑐) ∧ 𝑅 (𝐼 𝑐)) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝑐) (𝐼 𝑐))))
5842, 44, 57mp2and 699 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝑐) (𝐼 𝑐)))
5923, 5atbase 39290 . . . . 5 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
6053, 59syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
6123, 4latjjdir 18537 . . . 4 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐺 𝐻) 𝑐) (𝐼 𝑐)))
6215, 48, 60, 39, 61syl13anc 1374 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐺 𝐻) 𝑐) (𝐼 𝑐)))
6358, 62breqtrrd 5171 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝐼) 𝑐))
641, 63eqbrtrid 5178 1 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))
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
This theorem is referenced by:  dalem39  39713
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