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Theorem dalem39 35519
Description: Lemma for dath 35544. 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 35431 . . . 4 (𝜑𝐾 ∈ HL)
323ad2ant1 1127 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
41dalemyeo 35440 . . . . 5 (𝜑𝑌𝑂)
543ad2ant1 1127 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
6 dalem.ps . . . . . 6 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
76dalemccea 35491 . . . . 5 (𝜓𝑐𝐴)
873ad2ant3 1129 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
96dalem-ccly 35493 . . . . 5 (𝜓 → ¬ 𝑐 𝑌)
1093ad2ant3 1129 . . . 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 2771 . . . . 5 (LVols‘𝐾) = (LVols‘𝐾)
1611, 12, 13, 14, 15lvoli3 35385 . . . 4 (((𝐾 ∈ HL ∧ 𝑌𝑂𝑐𝐴) ∧ ¬ 𝑐 𝑌) → (𝑌 𝑐) ∈ (LVols‘𝐾))
173, 5, 8, 10, 16syl31anc 1479 . . 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 35514 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
23 dalem38.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
241, 11, 12, 13, 6, 18, 14, 19, 20, 23dalem23 35504 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
2511, 12, 13, 15lvolnle3at 35390 . . 3 (((𝐾 ∈ HL ∧ (𝑌 𝑐) ∈ (LVols‘𝐾)) ∧ (𝐼𝐴𝐺𝐴𝑐𝐴)) → ¬ (𝑌 𝑐) ((𝐼 𝐺) 𝑐))
263, 17, 22, 24, 8, 25syl23anc 1483 . 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 35518 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))
291dalemkelat 35432 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
30293ad2ant1 1127 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
311, 11, 12, 13, 6, 18, 14, 19, 20, 27dalem29 35509 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
32 eqid 2771 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
3332, 12, 13hlatjcl 35175 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
343, 24, 31, 33syl3anc 1476 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
3532, 13atbase 35098 . . . . . . . . 9 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
3622, 35syl 17 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
3732, 12latjcl 17259 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3830, 34, 36, 37syl3anc 1476 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
396, 13dalemcceb 35497 . . . . . . . 8 (𝜓𝑐 ∈ (Base‘𝐾))
40393ad2ant3 1129 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4132, 11, 12latlej2 17269 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → 𝑐 (((𝐺 𝐻) 𝐼) 𝑐))
4230, 38, 40, 41syl3anc 1476 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (((𝐺 𝐻) 𝐼) 𝑐))
431, 14dalemyeb 35457 . . . . . . . 8 (𝜑𝑌 ∈ (Base‘𝐾))
44433ad2ant1 1127 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
4532, 12latjcl 17259 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → (((𝐺 𝐻) 𝐼) 𝑐) ∈ (Base‘𝐾))
4630, 38, 40, 45syl3anc 1476 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) ∈ (Base‘𝐾))
4732, 11, 12latjle12 17270 . . . . . . 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 1478 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝑌 (((𝐺 𝐻) 𝐼) 𝑐) ∧ 𝑐 (((𝐺 𝐻) 𝐼) 𝑐)) ↔ (𝑌 𝑐) (((𝐺 𝐻) 𝐼) 𝑐)))
4928, 42, 48mpbi2and 691 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 𝑐) (((𝐺 𝐻) 𝐼) 𝑐))
5012, 13hlatjrot 35181 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝐺𝐴𝐻𝐴𝐼𝐴)) → ((𝐺 𝐻) 𝐼) = ((𝐼 𝐺) 𝐻))
513, 24, 31, 22, 50syl13anc 1478 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) = ((𝐼 𝐺) 𝐻))
5251oveq1d 6808 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐼 𝐺) 𝐻) 𝑐))
5349, 52breqtrd 4812 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 𝑐) (((𝐼 𝐺) 𝐻) 𝑐))
5453adantr 466 . . 3 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (𝑌 𝑐) (((𝐼 𝐺) 𝐻) 𝑐))
5532, 13atbase 35098 . . . . . . 7 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
5631, 55syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
5732, 12, 13hlatjcl 35175 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝐼𝐴𝐺𝐴) → (𝐼 𝐺) ∈ (Base‘𝐾))
583, 22, 24, 57syl3anc 1476 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐼 𝐺) ∈ (Base‘𝐾))
5932, 11, 12latleeqj2 17272 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐻 ∈ (Base‘𝐾) ∧ (𝐼 𝐺) ∈ (Base‘𝐾)) → (𝐻 (𝐼 𝐺) ↔ ((𝐼 𝐺) 𝐻) = (𝐼 𝐺)))
6030, 56, 58, 59syl3anc 1476 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐻 (𝐼 𝐺) ↔ ((𝐼 𝐺) 𝐻) = (𝐼 𝐺)))
6160biimpa 462 . . . 4 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → ((𝐼 𝐺) 𝐻) = (𝐼 𝐺))
6261oveq1d 6808 . . 3 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (((𝐼 𝐺) 𝐻) 𝑐) = ((𝐼 𝐺) 𝑐))
6354, 62breqtrd 4812 . 2 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (𝑌 𝑐) ((𝐼 𝐺) 𝑐))
6426, 63mtand 817 1 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐻 (𝐼 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Latclat 17253  Atomscatm 35072  HLchlt 35159  LPlanesclpl 35300  LVolsclvol 35301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307  df-lvols 35308
This theorem is referenced by:  dalem40  35520  dalem41  35521
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