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Theorem dalem38 36840
Description: Lemma for dath 36866. Plane 𝑌 belongs to the 3-dimensional volume 𝐺𝐻𝐼𝑐. (Contributed by NM, 5-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
dalem38 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))

Proof of Theorem dalem38
StepHypRef Expression
1 dalem38.y . 2 𝑌 = ((𝑃 𝑄) 𝑅)
2 dalem.ph . . . . . . 7 (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)))))
3 dalem.l . . . . . . 7 = (le‘𝐾)
4 dalem.j . . . . . . 7 = (join‘𝐾)
5 dalem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
6 dalem.ps . . . . . . 7 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
7 dalem38.m . . . . . . 7 = (meet‘𝐾)
8 dalem38.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
9 dalem38.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
10 dalem38.g . . . . . . 7 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
112, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem28 36830 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 (𝐺 𝑐))
12 dalem38.h . . . . . . 7 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
132, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem33 36835 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝐻 𝑐))
142dalemkelat 36754 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
15143ad2ant1 1129 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
162, 5dalempeb 36769 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
17163ad2ant1 1129 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 ∈ (Base‘𝐾))
182dalemkehl 36753 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
19183ad2ant1 1129 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
202, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem23 36826 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
216dalemccea 36813 . . . . . . . . 9 (𝜓𝑐𝐴)
22213ad2ant3 1131 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
23 eqid 2821 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2423, 4, 5hlatjcl 36497 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐺𝐴𝑐𝐴) → (𝐺 𝑐) ∈ (Base‘𝐾))
2519, 20, 22, 24syl3anc 1367 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑐) ∈ (Base‘𝐾))
262, 5dalemqeb 36770 . . . . . . . 8 (𝜑𝑄 ∈ (Base‘𝐾))
27263ad2ant1 1129 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 ∈ (Base‘𝐾))
282, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem29 36831 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
2923, 4, 5hlatjcl 36497 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐻𝐴𝑐𝐴) → (𝐻 𝑐) ∈ (Base‘𝐾))
3019, 28, 22, 29syl3anc 1367 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐻 𝑐) ∈ (Base‘𝐾))
3123, 3, 4latjlej12 17671 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝐺 𝑐) ∈ (Base‘𝐾)) ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝐻 𝑐) ∈ (Base‘𝐾))) → ((𝑃 (𝐺 𝑐) ∧ 𝑄 (𝐻 𝑐)) → (𝑃 𝑄) ((𝐺 𝑐) (𝐻 𝑐))))
3215, 17, 25, 27, 30, 31syl122anc 1375 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 (𝐺 𝑐) ∧ 𝑄 (𝐻 𝑐)) → (𝑃 𝑄) ((𝐺 𝑐) (𝐻 𝑐))))
3311, 13, 32mp2and 697 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ((𝐺 𝑐) (𝐻 𝑐)))
3423, 5atbase 36419 . . . . . . 7 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
3520, 34syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
3623, 5atbase 36419 . . . . . . 7 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
3728, 36syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
386, 5dalemcceb 36819 . . . . . . 7 (𝜓𝑐 ∈ (Base‘𝐾))
39383ad2ant3 1131 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4023, 4latjjdir 17708 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → ((𝐺 𝐻) 𝑐) = ((𝐺 𝑐) (𝐻 𝑐)))
4115, 35, 37, 39, 40syl13anc 1368 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑐) = ((𝐺 𝑐) (𝐻 𝑐)))
4233, 41breqtrrd 5087 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ((𝐺 𝐻) 𝑐))
43 dalem38.i . . . . 5 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
442, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem37 36839 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝐼 𝑐))
452, 4, 5dalempjqeb 36775 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
46453ad2ant1 1129 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
4723, 4, 5hlatjcl 36497 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
4819, 20, 28, 47syl3anc 1367 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
4923, 4latjcl 17655 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝑐) ∈ (Base‘𝐾))
5015, 48, 39, 49syl3anc 1367 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑐) ∈ (Base‘𝐾))
512, 5dalemreb 36771 . . . . . 6 (𝜑𝑅 ∈ (Base‘𝐾))
52513ad2ant1 1129 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 ∈ (Base‘𝐾))
532, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem34 36836 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
5423, 4, 5hlatjcl 36497 . . . . . 6 ((𝐾 ∈ HL ∧ 𝐼𝐴𝑐𝐴) → (𝐼 𝑐) ∈ (Base‘𝐾))
5519, 53, 22, 54syl3anc 1367 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐼 𝑐) ∈ (Base‘𝐾))
5623, 3, 4latjlej12 17671 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ ((𝐺 𝐻) 𝑐) ∈ (Base‘𝐾)) ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝐼 𝑐) ∈ (Base‘𝐾))) → (((𝑃 𝑄) ((𝐺 𝐻) 𝑐) ∧ 𝑅 (𝐼 𝑐)) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝑐) (𝐼 𝑐))))
5715, 46, 50, 52, 55, 56syl122anc 1375 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝑃 𝑄) ((𝐺 𝐻) 𝑐) ∧ 𝑅 (𝐼 𝑐)) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝑐) (𝐼 𝑐))))
5842, 44, 57mp2and 697 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝑐) (𝐼 𝑐)))
5923, 5atbase 36419 . . . . 5 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
6053, 59syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
6123, 4latjjdir 17708 . . . 4 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐺 𝐻) 𝑐) (𝐼 𝑐)))
6215, 48, 60, 39, 61syl13anc 1368 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐺 𝐻) 𝑐) (𝐼 𝑐)))
6358, 62breqtrrd 5087 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝐼) 𝑐))
641, 63eqbrtrid 5094 1 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wne 3016   class class class wbr 5059  cfv 6350  (class class class)co 7150  Basecbs 16477  lecple 16566  joincjn 17548  meetcmee 17549  Latclat 17649  Atomscatm 36393  HLchlt 36480  LPlanesclpl 36622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17532  df-poset 17550  df-plt 17562  df-lub 17578  df-glb 17579  df-join 17580  df-meet 17581  df-p0 17643  df-lat 17650  df-clat 17712  df-oposet 36306  df-ol 36308  df-oml 36309  df-covers 36396  df-ats 36397  df-atl 36428  df-cvlat 36452  df-hlat 36481  df-llines 36628  df-lplanes 36629
This theorem is referenced by:  dalem39  36841
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