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Theorem dalem39 39410
Description: Lemma for dath 39435. 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 39322 . . . 4 (𝜑𝐾 ∈ HL)
323ad2ant1 1130 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
41dalemyeo 39331 . . . . 5 (𝜑𝑌𝑂)
543ad2ant1 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑌𝑂)
6 dalem.ps . . . . . 6 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
76dalemccea 39382 . . . . 5 (𝜓𝑐𝐴)
873ad2ant3 1132 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
96dalem-ccly 39384 . . . . 5 (𝜓 → ¬ 𝑐 𝑌)
1093ad2ant3 1132 . . . 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 2726 . . . . 5 (LVols‘𝐾) = (LVols‘𝐾)
1611, 12, 13, 14, 15lvoli3 39276 . . . 4 (((𝐾 ∈ HL ∧ 𝑌𝑂𝑐𝐴) ∧ ¬ 𝑐 𝑌) → (𝑌 𝑐) ∈ (LVols‘𝐾))
173, 5, 8, 10, 16syl31anc 1370 . . 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 39405 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
23 dalem38.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
241, 11, 12, 13, 6, 18, 14, 19, 20, 23dalem23 39395 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
2511, 12, 13, 15lvolnle3at 39281 . . 3 (((𝐾 ∈ HL ∧ (𝑌 𝑐) ∈ (LVols‘𝐾)) ∧ (𝐼𝐴𝐺𝐴𝑐𝐴)) → ¬ (𝑌 𝑐) ((𝐼 𝐺) 𝑐))
263, 17, 22, 24, 8, 25syl23anc 1374 . 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 39409 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))
291dalemkelat 39323 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
30293ad2ant1 1130 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
311, 11, 12, 13, 6, 18, 14, 19, 20, 27dalem29 39400 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
32 eqid 2726 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
3332, 12, 13hlatjcl 39065 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
343, 24, 31, 33syl3anc 1368 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
3532, 13atbase 38987 . . . . . . . . 9 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
3622, 35syl 17 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
3732, 12latjcl 18464 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3830, 34, 36, 37syl3anc 1368 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
396, 13dalemcceb 39388 . . . . . . . 8 (𝜓𝑐 ∈ (Base‘𝐾))
40393ad2ant3 1132 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4132, 11, 12latlej2 18474 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → 𝑐 (((𝐺 𝐻) 𝐼) 𝑐))
4230, 38, 40, 41syl3anc 1368 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (((𝐺 𝐻) 𝐼) 𝑐))
431, 14dalemyeb 39348 . . . . . . . 8 (𝜑𝑌 ∈ (Base‘𝐾))
44433ad2ant1 1130 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
4532, 12latjcl 18464 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → (((𝐺 𝐻) 𝐼) 𝑐) ∈ (Base‘𝐾))
4630, 38, 40, 45syl3anc 1368 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) ∈ (Base‘𝐾))
4732, 11, 12latjle12 18475 . . . . . . 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 1369 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝑌 (((𝐺 𝐻) 𝐼) 𝑐) ∧ 𝑐 (((𝐺 𝐻) 𝐼) 𝑐)) ↔ (𝑌 𝑐) (((𝐺 𝐻) 𝐼) 𝑐)))
4928, 42, 48mpbi2and 710 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 𝑐) (((𝐺 𝐻) 𝐼) 𝑐))
5012, 13hlatjrot 39071 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝐺𝐴𝐻𝐴𝐼𝐴)) → ((𝐺 𝐻) 𝐼) = ((𝐼 𝐺) 𝐻))
513, 24, 31, 22, 50syl13anc 1369 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) = ((𝐼 𝐺) 𝐻))
5251oveq1d 7439 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐼 𝐺) 𝐻) 𝑐))
5349, 52breqtrd 5179 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑌 𝑐) (((𝐼 𝐺) 𝐻) 𝑐))
5453adantr 479 . . 3 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (𝑌 𝑐) (((𝐼 𝐺) 𝐻) 𝑐))
5532, 13atbase 38987 . . . . . . 7 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
5631, 55syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
5732, 12, 13hlatjcl 39065 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝐼𝐴𝐺𝐴) → (𝐼 𝐺) ∈ (Base‘𝐾))
583, 22, 24, 57syl3anc 1368 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐼 𝐺) ∈ (Base‘𝐾))
5932, 11, 12latleeqj2 18477 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐻 ∈ (Base‘𝐾) ∧ (𝐼 𝐺) ∈ (Base‘𝐾)) → (𝐻 (𝐼 𝐺) ↔ ((𝐼 𝐺) 𝐻) = (𝐼 𝐺)))
6030, 56, 58, 59syl3anc 1368 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐻 (𝐼 𝐺) ↔ ((𝐼 𝐺) 𝐻) = (𝐼 𝐺)))
6160biimpa 475 . . . 4 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → ((𝐼 𝐺) 𝐻) = (𝐼 𝐺))
6261oveq1d 7439 . . 3 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (((𝐼 𝐺) 𝐻) 𝑐) = ((𝐼 𝐺) 𝑐))
6354, 62breqtrd 5179 . 2 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝐻 (𝐼 𝐺)) → (𝑌 𝑐) ((𝐼 𝐺) 𝑐))
6426, 63mtand 814 1 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐻 (𝐼 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930   class class class wbr 5153  cfv 6554  (class class class)co 7424  Basecbs 17213  lecple 17273  joincjn 18336  meetcmee 18337  Latclat 18456  Atomscatm 38961  HLchlt 39048  LPlanesclpl 39191  LVolsclvol 39192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-lat 18457  df-clat 18524  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-llines 39197  df-lplanes 39198  df-lvols 39199
This theorem is referenced by:  dalem40  39411  dalem41  39412
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