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Theorem dalem38 39692
Description: Lemma for dath 39718. 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 39682 . . . . . 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 39687 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝐻 𝑐))
142dalemkelat 39606 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
15143ad2ant1 1132 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
162, 5dalempeb 39621 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
17163ad2ant1 1132 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 ∈ (Base‘𝐾))
182dalemkehl 39605 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
19183ad2ant1 1132 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
202, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem23 39678 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
216dalemccea 39665 . . . . . . . . 9 (𝜓𝑐𝐴)
22213ad2ant3 1134 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
23 eqid 2734 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2423, 4, 5hlatjcl 39348 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐺𝐴𝑐𝐴) → (𝐺 𝑐) ∈ (Base‘𝐾))
2519, 20, 22, 24syl3anc 1370 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑐) ∈ (Base‘𝐾))
262, 5dalemqeb 39622 . . . . . . . 8 (𝜑𝑄 ∈ (Base‘𝐾))
27263ad2ant1 1132 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 ∈ (Base‘𝐾))
282, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem29 39683 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
2923, 4, 5hlatjcl 39348 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐻𝐴𝑐𝐴) → (𝐻 𝑐) ∈ (Base‘𝐾))
3019, 28, 22, 29syl3anc 1370 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐻 𝑐) ∈ (Base‘𝐾))
3123, 3, 4latjlej12 18512 . . . . . . 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 1378 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 (𝐺 𝑐) ∧ 𝑄 (𝐻 𝑐)) → (𝑃 𝑄) ((𝐺 𝑐) (𝐻 𝑐))))
3311, 13, 32mp2and 699 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ((𝐺 𝑐) (𝐻 𝑐)))
3423, 5atbase 39270 . . . . . . 7 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
3520, 34syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
3623, 5atbase 39270 . . . . . . 7 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
3728, 36syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
386, 5dalemcceb 39671 . . . . . . 7 (𝜓𝑐 ∈ (Base‘𝐾))
39383ad2ant3 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4023, 4latjjdir 18549 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → ((𝐺 𝐻) 𝑐) = ((𝐺 𝑐) (𝐻 𝑐)))
4115, 35, 37, 39, 40syl13anc 1371 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑐) = ((𝐺 𝑐) (𝐻 𝑐)))
4233, 41breqtrrd 5175 . . . 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 39691 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝐼 𝑐))
452, 4, 5dalempjqeb 39627 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
46453ad2ant1 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
4723, 4, 5hlatjcl 39348 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
4819, 20, 28, 47syl3anc 1370 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
4923, 4latjcl 18496 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝑐) ∈ (Base‘𝐾))
5015, 48, 39, 49syl3anc 1370 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑐) ∈ (Base‘𝐾))
512, 5dalemreb 39623 . . . . . 6 (𝜑𝑅 ∈ (Base‘𝐾))
52513ad2ant1 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 ∈ (Base‘𝐾))
532, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem34 39688 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
5423, 4, 5hlatjcl 39348 . . . . . 6 ((𝐾 ∈ HL ∧ 𝐼𝐴𝑐𝐴) → (𝐼 𝑐) ∈ (Base‘𝐾))
5519, 53, 22, 54syl3anc 1370 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐼 𝑐) ∈ (Base‘𝐾))
5623, 3, 4latjlej12 18512 . . . . 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 1378 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝑃 𝑄) ((𝐺 𝐻) 𝑐) ∧ 𝑅 (𝐼 𝑐)) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝑐) (𝐼 𝑐))))
5842, 44, 57mp2and 699 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝑐) (𝐼 𝑐)))
5923, 5atbase 39270 . . . . 5 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
6053, 59syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
6123, 4latjjdir 18549 . . . 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 1371 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐺 𝐻) 𝑐) (𝐼 𝑐)))
6358, 62breqtrrd 5175 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝐼) 𝑐))
641, 63eqbrtrid 5182 1 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937   class class class wbr 5147  cfv 6562  (class class class)co 7430  Basecbs 17244  lecple 17304  joincjn 18368  meetcmee 18369  Latclat 18488  Atomscatm 39244  HLchlt 39331  LPlanesclpl 39474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-proset 18351  df-poset 18370  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-p0 18482  df-lat 18489  df-clat 18556  df-oposet 39157  df-ol 39159  df-oml 39160  df-covers 39247  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332  df-llines 39480  df-lplanes 39481
This theorem is referenced by:  dalem39  39693
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