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Theorem dalem39 37462
Description: Lemma for dath 37487. 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 37374 . . . 4 (𝜑𝐾 ∈ HL)
323ad2ant1 1135 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
41dalemyeo 37383 . . . . 5 (𝜑𝑌𝑂)
543ad2ant1 1135 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
6 dalem.ps . . . . . 6 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
76dalemccea 37434 . . . . 5 (𝜓𝑐𝐴)
873ad2ant3 1137 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
96dalem-ccly 37436 . . . . 5 (𝜓 → ¬ 𝑐 𝑌)
1093ad2ant3 1137 . . . 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 37328 . . . 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 37457 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
23 dalem38.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
241, 11, 12, 13, 6, 18, 14, 19, 20, 23dalem23 37447 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
2511, 12, 13, 15lvolnle3at 37333 . . 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 37461 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))
291dalemkelat 37375 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
30293ad2ant1 1135 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
311, 11, 12, 13, 6, 18, 14, 19, 20, 27dalem29 37452 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
32 eqid 2737 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
3332, 12, 13hlatjcl 37118 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
343, 24, 31, 33syl3anc 1373 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
3532, 13atbase 37040 . . . . . . . . 9 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
3622, 35syl 17 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
3732, 12latjcl 17945 . . . . . . . 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 37440 . . . . . . . 8 (𝜓𝑐 ∈ (Base‘𝐾))
40393ad2ant3 1137 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4132, 11, 12latlej2 17955 . . . . . . 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 37400 . . . . . . . 8 (𝜑𝑌 ∈ (Base‘𝐾))
44433ad2ant1 1135 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
4532, 12latjcl 17945 . . . . . . . 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 17956 . . . . . . 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 37124 . . . . . . 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 7228 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐼 𝐺) 𝐻) 𝑐))
5349, 52breqtrd 5079 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 𝑐) (((𝐼 𝐺) 𝐻) 𝑐))
5453adantr 484 . . 3 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (𝑌 𝑐) (((𝐼 𝐺) 𝐻) 𝑐))
5532, 13atbase 37040 . . . . . . 7 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
5631, 55syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
5732, 12, 13hlatjcl 37118 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝐼𝐴𝐺𝐴) → (𝐼 𝐺) ∈ (Base‘𝐾))
583, 22, 24, 57syl3anc 1373 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐼 𝐺) ∈ (Base‘𝐾))
5932, 11, 12latleeqj2 17958 . . . . . 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 480 . . . 4 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → ((𝐼 𝐺) 𝐻) = (𝐼 𝐺))
6261oveq1d 7228 . . 3 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (((𝐼 𝐺) 𝐻) 𝑐) = ((𝐼 𝐺) 𝑐))
6354, 62breqtrd 5079 . 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940   class class class wbr 5053  cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  joincjn 17818  meetcmee 17819  Latclat 17937  Atomscatm 37014  HLchlt 37101  LPlanesclpl 37243  LVolsclvol 37244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-lat 17938  df-clat 18005  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-llines 37249  df-lplanes 37250  df-lvols 37251
This theorem is referenced by:  dalem40  37463  dalem41  37464
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