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Theorem dalem39 39713
Description: Lemma for dath 39738. Auxiliary atoms 𝐺, 𝐻, and 𝐼 are not colinear. (Contributed by NM, 4-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
dalem39 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐻 (𝐼 𝐺))

Proof of Theorem dalem39
StepHypRef Expression
1 dalem.ph . . . . 5 (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)))))
21dalemkehl 39625 . . . 4 (𝜑𝐾 ∈ HL)
323ad2ant1 1134 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
41dalemyeo 39634 . . . . 5 (𝜑𝑌𝑂)
543ad2ant1 1134 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
6 dalem.ps . . . . . 6 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
76dalemccea 39685 . . . . 5 (𝜓𝑐𝐴)
873ad2ant3 1136 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
96dalem-ccly 39687 . . . . 5 (𝜓 → ¬ 𝑐 𝑌)
1093ad2ant3 1136 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑌)
11 dalem.l . . . . 5 = (le‘𝐾)
12 dalem.j . . . . 5 = (join‘𝐾)
13 dalem.a . . . . 5 𝐴 = (Atoms‘𝐾)
14 dalem38.o . . . . 5 𝑂 = (LPlanes‘𝐾)
15 eqid 2737 . . . . 5 (LVols‘𝐾) = (LVols‘𝐾)
1611, 12, 13, 14, 15lvoli3 39579 . . . 4 (((𝐾 ∈ HL ∧ 𝑌𝑂𝑐𝐴) ∧ ¬ 𝑐 𝑌) → (𝑌 𝑐) ∈ (LVols‘𝐾))
173, 5, 8, 10, 16syl31anc 1375 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 𝑐) ∈ (LVols‘𝐾))
18 dalem38.m . . . 4 = (meet‘𝐾)
19 dalem38.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
20 dalem38.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
21 dalem38.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
221, 11, 12, 13, 6, 18, 14, 19, 20, 21dalem34 39708 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
23 dalem38.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
241, 11, 12, 13, 6, 18, 14, 19, 20, 23dalem23 39698 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
2511, 12, 13, 15lvolnle3at 39584 . . 3 (((𝐾 ∈ HL ∧ (𝑌 𝑐) ∈ (LVols‘𝐾)) ∧ (𝐼𝐴𝐺𝐴𝑐𝐴)) → ¬ (𝑌 𝑐) ((𝐼 𝐺) 𝑐))
263, 17, 22, 24, 8, 25syl23anc 1379 . 2 ((𝜑𝑌 = 𝑍𝜓) → ¬ (𝑌 𝑐) ((𝐼 𝐺) 𝑐))
27 dalem38.h . . . . . . 7 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
281, 11, 12, 13, 6, 18, 14, 19, 20, 23, 27, 21dalem38 39712 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))
291dalemkelat 39626 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
30293ad2ant1 1134 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
311, 11, 12, 13, 6, 18, 14, 19, 20, 27dalem29 39703 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
32 eqid 2737 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
3332, 12, 13hlatjcl 39368 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
343, 24, 31, 33syl3anc 1373 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
3532, 13atbase 39290 . . . . . . . . 9 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
3622, 35syl 17 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
3732, 12latjcl 18484 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3830, 34, 36, 37syl3anc 1373 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
396, 13dalemcceb 39691 . . . . . . . 8 (𝜓𝑐 ∈ (Base‘𝐾))
40393ad2ant3 1136 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4132, 11, 12latlej2 18494 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → 𝑐 (((𝐺 𝐻) 𝐼) 𝑐))
4230, 38, 40, 41syl3anc 1373 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (((𝐺 𝐻) 𝐼) 𝑐))
431, 14dalemyeb 39651 . . . . . . . 8 (𝜑𝑌 ∈ (Base‘𝐾))
44433ad2ant1 1134 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
4532, 12latjcl 18484 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → (((𝐺 𝐻) 𝐼) 𝑐) ∈ (Base‘𝐾))
4630, 38, 40, 45syl3anc 1373 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) ∈ (Base‘𝐾))
4732, 11, 12latjle12 18495 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑌 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾) ∧ (((𝐺 𝐻) 𝐼) 𝑐) ∈ (Base‘𝐾))) → ((𝑌 (((𝐺 𝐻) 𝐼) 𝑐) ∧ 𝑐 (((𝐺 𝐻) 𝐼) 𝑐)) ↔ (𝑌 𝑐) (((𝐺 𝐻) 𝐼) 𝑐)))
4830, 44, 40, 46, 47syl13anc 1374 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝑌 (((𝐺 𝐻) 𝐼) 𝑐) ∧ 𝑐 (((𝐺 𝐻) 𝐼) 𝑐)) ↔ (𝑌 𝑐) (((𝐺 𝐻) 𝐼) 𝑐)))
4928, 42, 48mpbi2and 712 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 𝑐) (((𝐺 𝐻) 𝐼) 𝑐))
5012, 13hlatjrot 39374 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝐺𝐴𝐻𝐴𝐼𝐴)) → ((𝐺 𝐻) 𝐼) = ((𝐼 𝐺) 𝐻))
513, 24, 31, 22, 50syl13anc 1374 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) = ((𝐼 𝐺) 𝐻))
5251oveq1d 7446 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐼 𝐺) 𝐻) 𝑐))
5349, 52breqtrd 5169 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 𝑐) (((𝐼 𝐺) 𝐻) 𝑐))
5453adantr 480 . . 3 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (𝑌 𝑐) (((𝐼 𝐺) 𝐻) 𝑐))
5532, 13atbase 39290 . . . . . . 7 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
5631, 55syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
5732, 12, 13hlatjcl 39368 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝐼𝐴𝐺𝐴) → (𝐼 𝐺) ∈ (Base‘𝐾))
583, 22, 24, 57syl3anc 1373 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐼 𝐺) ∈ (Base‘𝐾))
5932, 11, 12latleeqj2 18497 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐻 ∈ (Base‘𝐾) ∧ (𝐼 𝐺) ∈ (Base‘𝐾)) → (𝐻 (𝐼 𝐺) ↔ ((𝐼 𝐺) 𝐻) = (𝐼 𝐺)))
6030, 56, 58, 59syl3anc 1373 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐻 (𝐼 𝐺) ↔ ((𝐼 𝐺) 𝐻) = (𝐼 𝐺)))
6160biimpa 476 . . . 4 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → ((𝐼 𝐺) 𝐻) = (𝐼 𝐺))
6261oveq1d 7446 . . 3 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (((𝐼 𝐺) 𝐻) 𝑐) = ((𝐼 𝐺) 𝑐))
6354, 62breqtrd 5169 . 2 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (𝑌 𝑐) ((𝐼 𝐺) 𝑐))
6426, 63mtand 816 1 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐻 (𝐼 𝐺))
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  LVolsclvol 39495
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:  dalem40  39714  dalem41  39715
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