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Theorem dalem39 34477
Description: Lemma for dath 34502. 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 34389 . . . 4 (𝜑𝐾 ∈ HL)
323ad2ant1 1080 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
41dalemyeo 34398 . . . . 5 (𝜑𝑌𝑂)
543ad2ant1 1080 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
6 dalem.ps . . . . . 6 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
76dalemccea 34449 . . . . 5 (𝜓𝑐𝐴)
873ad2ant3 1082 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
96dalem-ccly 34451 . . . . 5 (𝜓 → ¬ 𝑐 𝑌)
1093ad2ant3 1082 . . . 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 2621 . . . . 5 (LVols‘𝐾) = (LVols‘𝐾)
1611, 12, 13, 14, 15lvoli3 34343 . . . 4 (((𝐾 ∈ HL ∧ 𝑌𝑂𝑐𝐴) ∧ ¬ 𝑐 𝑌) → (𝑌 𝑐) ∈ (LVols‘𝐾))
173, 5, 8, 10, 16syl31anc 1326 . . 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 34472 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
23 dalem38.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
241, 11, 12, 13, 6, 18, 14, 19, 20, 23dalem23 34462 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
2511, 12, 13, 15lvolnle3at 34348 . . 3 (((𝐾 ∈ HL ∧ (𝑌 𝑐) ∈ (LVols‘𝐾)) ∧ (𝐼𝐴𝐺𝐴𝑐𝐴)) → ¬ (𝑌 𝑐) ((𝐼 𝐺) 𝑐))
263, 17, 22, 24, 8, 25syl23anc 1330 . 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 34476 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))
291dalemkelat 34390 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
30293ad2ant1 1080 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
311, 11, 12, 13, 6, 18, 14, 19, 20, 27dalem29 34467 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
32 eqid 2621 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
3332, 12, 13hlatjcl 34133 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
343, 24, 31, 33syl3anc 1323 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
3532, 13atbase 34056 . . . . . . . . 9 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
3622, 35syl 17 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
3732, 12latjcl 16972 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3830, 34, 36, 37syl3anc 1323 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
396, 13dalemcceb 34455 . . . . . . . 8 (𝜓𝑐 ∈ (Base‘𝐾))
40393ad2ant3 1082 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4132, 11, 12latlej2 16982 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → 𝑐 (((𝐺 𝐻) 𝐼) 𝑐))
4230, 38, 40, 41syl3anc 1323 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (((𝐺 𝐻) 𝐼) 𝑐))
431, 14dalemyeb 34415 . . . . . . . 8 (𝜑𝑌 ∈ (Base‘𝐾))
44433ad2ant1 1080 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
4532, 12latjcl 16972 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → (((𝐺 𝐻) 𝐼) 𝑐) ∈ (Base‘𝐾))
4630, 38, 40, 45syl3anc 1323 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) ∈ (Base‘𝐾))
4732, 11, 12latjle12 16983 . . . . . . 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 1325 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝑌 (((𝐺 𝐻) 𝐼) 𝑐) ∧ 𝑐 (((𝐺 𝐻) 𝐼) 𝑐)) ↔ (𝑌 𝑐) (((𝐺 𝐻) 𝐼) 𝑐)))
4928, 42, 48mpbi2and 955 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 𝑐) (((𝐺 𝐻) 𝐼) 𝑐))
5012, 13hlatjrot 34139 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝐺𝐴𝐻𝐴𝐼𝐴)) → ((𝐺 𝐻) 𝐼) = ((𝐼 𝐺) 𝐻))
513, 24, 31, 22, 50syl13anc 1325 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) = ((𝐼 𝐺) 𝐻))
5251oveq1d 6619 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐼 𝐺) 𝐻) 𝑐))
5349, 52breqtrd 4639 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 𝑐) (((𝐼 𝐺) 𝐻) 𝑐))
5453adantr 481 . . 3 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (𝑌 𝑐) (((𝐼 𝐺) 𝐻) 𝑐))
5532, 13atbase 34056 . . . . . . 7 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
5631, 55syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
5732, 12, 13hlatjcl 34133 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝐼𝐴𝐺𝐴) → (𝐼 𝐺) ∈ (Base‘𝐾))
583, 22, 24, 57syl3anc 1323 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐼 𝐺) ∈ (Base‘𝐾))
5932, 11, 12latleeqj2 16985 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐻 ∈ (Base‘𝐾) ∧ (𝐼 𝐺) ∈ (Base‘𝐾)) → (𝐻 (𝐼 𝐺) ↔ ((𝐼 𝐺) 𝐻) = (𝐼 𝐺)))
6030, 56, 58, 59syl3anc 1323 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐻 (𝐼 𝐺) ↔ ((𝐼 𝐺) 𝐻) = (𝐼 𝐺)))
6160biimpa 501 . . . 4 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → ((𝐼 𝐺) 𝐻) = (𝐼 𝐺))
6261oveq1d 6619 . . 3 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (((𝐼 𝐺) 𝐻) 𝑐) = ((𝐼 𝐺) 𝑐))
6354, 62breqtrd 4639 . 2 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (𝑌 𝑐) ((𝐼 𝐺) 𝑐))
6426, 63mtand 690 1 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐻 (𝐼 𝐺))
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  LVolsclvol 34259
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:  dalem40  34478  dalem41  34479
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