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Theorem dalem3 37111
 Description: Lemma for dalemdnee 37113. (Contributed by NM, 10-Aug-2012.)
Hypotheses
Ref Expression
dalema.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)))))
dalemc.l = (le‘𝐾)
dalemc.j = (join‘𝐾)
dalemc.a 𝐴 = (Atoms‘𝐾)
dalem3.m = (meet‘𝐾)
dalem3.o 𝑂 = (LPlanes‘𝐾)
dalem3.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem3.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem3.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
dalem3.e 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
Assertion
Ref Expression
dalem3 ((𝜑𝐷𝑄) → 𝐷𝐸)

Proof of Theorem dalem3
StepHypRef Expression
1 dalema.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 37070 . . . 4 (𝜑𝐾 ∈ HL)
31dalempea 37073 . . . 4 (𝜑𝑃𝐴)
41dalemqea 37074 . . . 4 (𝜑𝑄𝐴)
51dalemrea 37075 . . . 4 (𝜑𝑅𝐴)
61dalemyeo 37079 . . . 4 (𝜑𝑌𝑂)
7 dalemc.l . . . . 5 = (le‘𝐾)
8 dalemc.j . . . . 5 = (join‘𝐾)
9 dalemc.a . . . . 5 𝐴 = (Atoms‘𝐾)
10 dalem3.o . . . . 5 𝑂 = (LPlanes‘𝐾)
11 dalem3.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
127, 8, 9, 10, 11lplnric 36999 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑌𝑂) → ¬ 𝑅 (𝑃 𝑄))
132, 3, 4, 5, 6, 12syl131anc 1380 . . 3 (𝜑 → ¬ 𝑅 (𝑃 𝑄))
1413adantr 484 . 2 ((𝜑𝐷𝑄) → ¬ 𝑅 (𝑃 𝑄))
15 dalem3.e . . . . . . 7 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
161dalemkelat 37071 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
17 eqid 2798 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
1817, 8, 9hlatjcl 36814 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
192, 4, 5, 18syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑄 𝑅) ∈ (Base‘𝐾))
201, 8, 9dalemtjueb 37094 . . . . . . . 8 (𝜑 → (𝑇 𝑈) ∈ (Base‘𝐾))
21 dalem3.m . . . . . . . . 9 = (meet‘𝐾)
2217, 7, 21latmle1 17698 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → ((𝑄 𝑅) (𝑇 𝑈)) (𝑄 𝑅))
2316, 19, 20, 22syl3anc 1368 . . . . . . 7 (𝜑 → ((𝑄 𝑅) (𝑇 𝑈)) (𝑄 𝑅))
2415, 23eqbrtrid 5069 . . . . . 6 (𝜑𝐸 (𝑄 𝑅))
25 breq1 5037 . . . . . 6 (𝐷 = 𝐸 → (𝐷 (𝑄 𝑅) ↔ 𝐸 (𝑄 𝑅)))
2624, 25syl5ibrcom 250 . . . . 5 (𝜑 → (𝐷 = 𝐸𝐷 (𝑄 𝑅)))
2726adantr 484 . . . 4 ((𝜑𝐷𝑄) → (𝐷 = 𝐸𝐷 (𝑄 𝑅)))
282adantr 484 . . . . 5 ((𝜑𝐷𝑄) → 𝐾 ∈ HL)
29 dalem3.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
30 dalem3.d . . . . . . 7 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
311, 7, 8, 9, 21, 10, 11, 29, 30dalemdea 37109 . . . . . 6 (𝜑𝐷𝐴)
3231adantr 484 . . . . 5 ((𝜑𝐷𝑄) → 𝐷𝐴)
335adantr 484 . . . . 5 ((𝜑𝐷𝑄) → 𝑅𝐴)
344adantr 484 . . . . 5 ((𝜑𝐷𝑄) → 𝑄𝐴)
35 simpr 488 . . . . 5 ((𝜑𝐷𝑄) → 𝐷𝑄)
367, 8, 9hlatexch1 36842 . . . . 5 ((𝐾 ∈ HL ∧ (𝐷𝐴𝑅𝐴𝑄𝐴) ∧ 𝐷𝑄) → (𝐷 (𝑄 𝑅) → 𝑅 (𝑄 𝐷)))
3728, 32, 33, 34, 35, 36syl131anc 1380 . . . 4 ((𝜑𝐷𝑄) → (𝐷 (𝑄 𝑅) → 𝑅 (𝑄 𝐷)))
387, 8, 9hlatlej2 36823 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
392, 3, 4, 38syl3anc 1368 . . . . . . 7 (𝜑𝑄 (𝑃 𝑄))
401, 8, 9dalempjqeb 37092 . . . . . . . . 9 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
411, 8, 9dalemsjteb 37093 . . . . . . . . 9 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
4217, 7, 21latmle1 17698 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
4316, 40, 41, 42syl3anc 1368 . . . . . . . 8 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
4430, 43eqbrtrid 5069 . . . . . . 7 (𝜑𝐷 (𝑃 𝑄))
451, 9dalemqeb 37087 . . . . . . . 8 (𝜑𝑄 ∈ (Base‘𝐾))
4617, 9atbase 36736 . . . . . . . . 9 (𝐷𝐴𝐷 ∈ (Base‘𝐾))
4731, 46syl 17 . . . . . . . 8 (𝜑𝐷 ∈ (Base‘𝐾))
4817, 7, 8latjle12 17684 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝐷 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑄 (𝑃 𝑄) ∧ 𝐷 (𝑃 𝑄)) ↔ (𝑄 𝐷) (𝑃 𝑄)))
4916, 45, 47, 40, 48syl13anc 1369 . . . . . . 7 (𝜑 → ((𝑄 (𝑃 𝑄) ∧ 𝐷 (𝑃 𝑄)) ↔ (𝑄 𝐷) (𝑃 𝑄)))
5039, 44, 49mpbi2and 711 . . . . . 6 (𝜑 → (𝑄 𝐷) (𝑃 𝑄))
511, 9dalemreb 37088 . . . . . . 7 (𝜑𝑅 ∈ (Base‘𝐾))
5217, 8, 9hlatjcl 36814 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴𝐷𝐴) → (𝑄 𝐷) ∈ (Base‘𝐾))
532, 4, 31, 52syl3anc 1368 . . . . . . 7 (𝜑 → (𝑄 𝐷) ∈ (Base‘𝐾))
5417, 7lattr 17678 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝑄 𝐷) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑅 (𝑄 𝐷) ∧ (𝑄 𝐷) (𝑃 𝑄)) → 𝑅 (𝑃 𝑄)))
5516, 51, 53, 40, 54syl13anc 1369 . . . . . 6 (𝜑 → ((𝑅 (𝑄 𝐷) ∧ (𝑄 𝐷) (𝑃 𝑄)) → 𝑅 (𝑃 𝑄)))
5650, 55mpan2d 693 . . . . 5 (𝜑 → (𝑅 (𝑄 𝐷) → 𝑅 (𝑃 𝑄)))
5756adantr 484 . . . 4 ((𝜑𝐷𝑄) → (𝑅 (𝑄 𝐷) → 𝑅 (𝑃 𝑄)))
5827, 37, 573syld 60 . . 3 ((𝜑𝐷𝑄) → (𝐷 = 𝐸𝑅 (𝑃 𝑄)))
5958necon3bd 3001 . 2 ((𝜑𝐷𝑄) → (¬ 𝑅 (𝑃 𝑄) → 𝐷𝐸))
6014, 59mpd 15 1 ((𝜑𝐷𝑄) → 𝐷𝐸)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   class class class wbr 5034  ‘cfv 6332  (class class class)co 7145  Basecbs 16495  lecple 16584  joincjn 17566  meetcmee 17567  Latclat 17667  Atomscatm 36710  HLchlt 36797  LPlanesclpl 36939 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17550  df-poset 17568  df-plt 17580  df-lub 17596  df-glb 17597  df-join 17598  df-meet 17599  df-p0 17661  df-lat 17668  df-clat 17730  df-oposet 36623  df-ol 36625  df-oml 36626  df-covers 36713  df-ats 36714  df-atl 36745  df-cvlat 36769  df-hlat 36798  df-llines 36945  df-lplanes 36946 This theorem is referenced by:  dalem4  37112  dalemdnee  37113
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