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Theorem dalem3 35268
Description: Lemma for dalemdnee 35270. (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 35227 . . . 4 (𝜑𝐾 ∈ HL)
31dalempea 35230 . . . 4 (𝜑𝑃𝐴)
41dalemqea 35231 . . . 4 (𝜑𝑄𝐴)
51dalemrea 35232 . . . 4 (𝜑𝑅𝐴)
61dalemyeo 35236 . . . 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 35156 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑌𝑂) → ¬ 𝑅 (𝑃 𝑄))
132, 3, 4, 5, 6, 12syl131anc 1379 . . 3 (𝜑 → ¬ 𝑅 (𝑃 𝑄))
1413adantr 480 . 2 ((𝜑𝐷𝑄) → ¬ 𝑅 (𝑃 𝑄))
15 dalem3.e . . . . . . 7 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
161dalemkelat 35228 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
17 eqid 2651 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
1817, 8, 9hlatjcl 34971 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
192, 4, 5, 18syl3anc 1366 . . . . . . . 8 (𝜑 → (𝑄 𝑅) ∈ (Base‘𝐾))
201, 8, 9dalemtjueb 35251 . . . . . . . 8 (𝜑 → (𝑇 𝑈) ∈ (Base‘𝐾))
21 dalem3.m . . . . . . . . 9 = (meet‘𝐾)
2217, 7, 21latmle1 17123 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → ((𝑄 𝑅) (𝑇 𝑈)) (𝑄 𝑅))
2316, 19, 20, 22syl3anc 1366 . . . . . . 7 (𝜑 → ((𝑄 𝑅) (𝑇 𝑈)) (𝑄 𝑅))
2415, 23syl5eqbr 4720 . . . . . 6 (𝜑𝐸 (𝑄 𝑅))
25 breq1 4688 . . . . . 6 (𝐷 = 𝐸 → (𝐷 (𝑄 𝑅) ↔ 𝐸 (𝑄 𝑅)))
2624, 25syl5ibrcom 237 . . . . 5 (𝜑 → (𝐷 = 𝐸𝐷 (𝑄 𝑅)))
2726adantr 480 . . . 4 ((𝜑𝐷𝑄) → (𝐷 = 𝐸𝐷 (𝑄 𝑅)))
282adantr 480 . . . . 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 35266 . . . . . 6 (𝜑𝐷𝐴)
3231adantr 480 . . . . 5 ((𝜑𝐷𝑄) → 𝐷𝐴)
335adantr 480 . . . . 5 ((𝜑𝐷𝑄) → 𝑅𝐴)
344adantr 480 . . . . 5 ((𝜑𝐷𝑄) → 𝑄𝐴)
35 simpr 476 . . . . 5 ((𝜑𝐷𝑄) → 𝐷𝑄)
367, 8, 9hlatexch1 34999 . . . . 5 ((𝐾 ∈ HL ∧ (𝐷𝐴𝑅𝐴𝑄𝐴) ∧ 𝐷𝑄) → (𝐷 (𝑄 𝑅) → 𝑅 (𝑄 𝐷)))
3728, 32, 33, 34, 35, 36syl131anc 1379 . . . 4 ((𝜑𝐷𝑄) → (𝐷 (𝑄 𝑅) → 𝑅 (𝑄 𝐷)))
387, 8, 9hlatlej2 34980 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
392, 3, 4, 38syl3anc 1366 . . . . . . 7 (𝜑𝑄 (𝑃 𝑄))
401, 8, 9dalempjqeb 35249 . . . . . . . . 9 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
411, 8, 9dalemsjteb 35250 . . . . . . . . 9 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
4217, 7, 21latmle1 17123 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
4316, 40, 41, 42syl3anc 1366 . . . . . . . 8 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
4430, 43syl5eqbr 4720 . . . . . . 7 (𝜑𝐷 (𝑃 𝑄))
451, 9dalemqeb 35244 . . . . . . . 8 (𝜑𝑄 ∈ (Base‘𝐾))
4617, 9atbase 34894 . . . . . . . . 9 (𝐷𝐴𝐷 ∈ (Base‘𝐾))
4731, 46syl 17 . . . . . . . 8 (𝜑𝐷 ∈ (Base‘𝐾))
4817, 7, 8latjle12 17109 . . . . . . . 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 1368 . . . . . . 7 (𝜑 → ((𝑄 (𝑃 𝑄) ∧ 𝐷 (𝑃 𝑄)) ↔ (𝑄 𝐷) (𝑃 𝑄)))
5039, 44, 49mpbi2and 976 . . . . . 6 (𝜑 → (𝑄 𝐷) (𝑃 𝑄))
511, 9dalemreb 35245 . . . . . . 7 (𝜑𝑅 ∈ (Base‘𝐾))
5217, 8, 9hlatjcl 34971 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴𝐷𝐴) → (𝑄 𝐷) ∈ (Base‘𝐾))
532, 4, 31, 52syl3anc 1366 . . . . . . 7 (𝜑 → (𝑄 𝐷) ∈ (Base‘𝐾))
5417, 7lattr 17103 . . . . . . 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 1368 . . . . . 6 (𝜑 → ((𝑅 (𝑄 𝐷) ∧ (𝑄 𝐷) (𝑃 𝑄)) → 𝑅 (𝑃 𝑄)))
5650, 55mpan2d 710 . . . . 5 (𝜑 → (𝑅 (𝑄 𝐷) → 𝑅 (𝑃 𝑄)))
5756adantr 480 . . . 4 ((𝜑𝐷𝑄) → (𝑅 (𝑄 𝐷) → 𝑅 (𝑃 𝑄)))
5827, 37, 573syld 60 . . 3 ((𝜑𝐷𝑄) → (𝐷 = 𝐸𝑅 (𝑃 𝑄)))
5958necon3bd 2837 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Latclat 17092  Atomscatm 34868  HLchlt 34955  LPlanesclpl 35096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103
This theorem is referenced by:  dalem4  35269  dalemdnee  35270
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