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Theorem dalem38 35780
 Description: Lemma for dath 35806. 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 35770 . . . . . 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 35775 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 (𝐻 𝑐))
142dalemkelat 35694 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
15143ad2ant1 1167 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
162, 5dalempeb 35709 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
17163ad2ant1 1167 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 ∈ (Base‘𝐾))
182dalemkehl 35693 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
19183ad2ant1 1167 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
202, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem23 35766 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
216dalemccea 35753 . . . . . . . . 9 (𝜓𝑐𝐴)
22213ad2ant3 1169 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
23 eqid 2825 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2423, 4, 5hlatjcl 35437 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐺𝐴𝑐𝐴) → (𝐺 𝑐) ∈ (Base‘𝐾))
2519, 20, 22, 24syl3anc 1494 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑐) ∈ (Base‘𝐾))
262, 5dalemqeb 35710 . . . . . . . 8 (𝜑𝑄 ∈ (Base‘𝐾))
27263ad2ant1 1167 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝑄 ∈ (Base‘𝐾))
282, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem29 35771 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
2923, 4, 5hlatjcl 35437 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐻𝐴𝑐𝐴) → (𝐻 𝑐) ∈ (Base‘𝐾))
3019, 28, 22, 29syl3anc 1494 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐻 𝑐) ∈ (Base‘𝐾))
3123, 3, 4latjlej12 17427 . . . . . . 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 1502 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 (𝐺 𝑐) ∧ 𝑄 (𝐻 𝑐)) → (𝑃 𝑄) ((𝐺 𝑐) (𝐻 𝑐))))
3311, 13, 32mp2and 690 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ((𝐺 𝑐) (𝐻 𝑐)))
3423, 5atbase 35359 . . . . . . 7 (𝐺𝐴𝐺 ∈ (Base‘𝐾))
3520, 34syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 ∈ (Base‘𝐾))
3623, 5atbase 35359 . . . . . . 7 (𝐻𝐴𝐻 ∈ (Base‘𝐾))
3728, 36syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐻 ∈ (Base‘𝐾))
386, 5dalemcceb 35759 . . . . . . 7 (𝜓𝑐 ∈ (Base‘𝐾))
39383ad2ant3 1169 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4023, 4latjjdir 17464 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → ((𝐺 𝐻) 𝑐) = ((𝐺 𝑐) (𝐻 𝑐)))
4115, 35, 37, 39, 40syl13anc 1495 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑐) = ((𝐺 𝑐) (𝐻 𝑐)))
4233, 41breqtrrd 4903 . . . 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 35779 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 (𝐼 𝑐))
452, 4, 5dalempjqeb 35715 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
46453ad2ant1 1167 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
4723, 4, 5hlatjcl 35437 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
4819, 20, 28, 47syl3anc 1494 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
4923, 4latjcl 17411 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝑐) ∈ (Base‘𝐾))
5015, 48, 39, 49syl3anc 1494 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝑐) ∈ (Base‘𝐾))
512, 5dalemreb 35711 . . . . . 6 (𝜑𝑅 ∈ (Base‘𝐾))
52513ad2ant1 1167 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑅 ∈ (Base‘𝐾))
532, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem34 35776 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
5423, 4, 5hlatjcl 35437 . . . . . 6 ((𝐾 ∈ HL ∧ 𝐼𝐴𝑐𝐴) → (𝐼 𝑐) ∈ (Base‘𝐾))
5519, 53, 22, 54syl3anc 1494 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐼 𝑐) ∈ (Base‘𝐾))
5623, 3, 4latjlej12 17427 . . . . 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 1502 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝑃 𝑄) ((𝐺 𝐻) 𝑐) ∧ 𝑅 (𝐼 𝑐)) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝑐) (𝐼 𝑐))))
5842, 44, 57mp2and 690 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝑐) (𝐼 𝑐)))
5923, 5atbase 35359 . . . . 5 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
6053, 59syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
6123, 4latjjdir 17464 . . . 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 1495 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐼) 𝑐) = (((𝐺 𝐻) 𝑐) (𝐼 𝑐)))
6358, 62breqtrrd 4903 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑄) 𝑅) (((𝐺 𝐻) 𝐼) 𝑐))
641, 63syl5eqbr 4910 1 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 (((𝐺 𝐻) 𝐼) 𝑐))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999   class class class wbr 4875  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  meetcmee 17305  Latclat 17405  Atomscatm 35333  HLchlt 35420  LPlanesclpl 35562 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-lat 17406  df-clat 17468  df-oposet 35246  df-ol 35248  df-oml 35249  df-covers 35336  df-ats 35337  df-atl 35368  df-cvlat 35392  df-hlat 35421  df-llines 35568  df-lplanes 35569 This theorem is referenced by:  dalem39  35781
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