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Theorem dalem3 35471
Description: Lemma for dalemdnee 35473. (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 35430 . . . 4 (𝜑𝐾 ∈ HL)
31dalempea 35433 . . . 4 (𝜑𝑃𝐴)
41dalemqea 35434 . . . 4 (𝜑𝑄𝐴)
51dalemrea 35435 . . . 4 (𝜑𝑅𝐴)
61dalemyeo 35439 . . . 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 35359 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑌𝑂) → ¬ 𝑅 (𝑃 𝑄))
132, 3, 4, 5, 6, 12syl131anc 1490 . . 3 (𝜑 → ¬ 𝑅 (𝑃 𝑄))
1413adantr 472 . 2 ((𝜑𝐷𝑄) → ¬ 𝑅 (𝑃 𝑄))
15 dalem3.e . . . . . . 7 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
161dalemkelat 35431 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
17 eqid 2760 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
1817, 8, 9hlatjcl 35174 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
192, 4, 5, 18syl3anc 1477 . . . . . . . 8 (𝜑 → (𝑄 𝑅) ∈ (Base‘𝐾))
201, 8, 9dalemtjueb 35454 . . . . . . . 8 (𝜑 → (𝑇 𝑈) ∈ (Base‘𝐾))
21 dalem3.m . . . . . . . . 9 = (meet‘𝐾)
2217, 7, 21latmle1 17297 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → ((𝑄 𝑅) (𝑇 𝑈)) (𝑄 𝑅))
2316, 19, 20, 22syl3anc 1477 . . . . . . 7 (𝜑 → ((𝑄 𝑅) (𝑇 𝑈)) (𝑄 𝑅))
2415, 23syl5eqbr 4839 . . . . . 6 (𝜑𝐸 (𝑄 𝑅))
25 breq1 4807 . . . . . 6 (𝐷 = 𝐸 → (𝐷 (𝑄 𝑅) ↔ 𝐸 (𝑄 𝑅)))
2624, 25syl5ibrcom 237 . . . . 5 (𝜑 → (𝐷 = 𝐸𝐷 (𝑄 𝑅)))
2726adantr 472 . . . 4 ((𝜑𝐷𝑄) → (𝐷 = 𝐸𝐷 (𝑄 𝑅)))
282adantr 472 . . . . 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 35469 . . . . . 6 (𝜑𝐷𝐴)
3231adantr 472 . . . . 5 ((𝜑𝐷𝑄) → 𝐷𝐴)
335adantr 472 . . . . 5 ((𝜑𝐷𝑄) → 𝑅𝐴)
344adantr 472 . . . . 5 ((𝜑𝐷𝑄) → 𝑄𝐴)
35 simpr 479 . . . . 5 ((𝜑𝐷𝑄) → 𝐷𝑄)
367, 8, 9hlatexch1 35202 . . . . 5 ((𝐾 ∈ HL ∧ (𝐷𝐴𝑅𝐴𝑄𝐴) ∧ 𝐷𝑄) → (𝐷 (𝑄 𝑅) → 𝑅 (𝑄 𝐷)))
3728, 32, 33, 34, 35, 36syl131anc 1490 . . . 4 ((𝜑𝐷𝑄) → (𝐷 (𝑄 𝑅) → 𝑅 (𝑄 𝐷)))
387, 8, 9hlatlej2 35183 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
392, 3, 4, 38syl3anc 1477 . . . . . . 7 (𝜑𝑄 (𝑃 𝑄))
401, 8, 9dalempjqeb 35452 . . . . . . . . 9 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
411, 8, 9dalemsjteb 35453 . . . . . . . . 9 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
4217, 7, 21latmle1 17297 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
4316, 40, 41, 42syl3anc 1477 . . . . . . . 8 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
4430, 43syl5eqbr 4839 . . . . . . 7 (𝜑𝐷 (𝑃 𝑄))
451, 9dalemqeb 35447 . . . . . . . 8 (𝜑𝑄 ∈ (Base‘𝐾))
4617, 9atbase 35097 . . . . . . . . 9 (𝐷𝐴𝐷 ∈ (Base‘𝐾))
4731, 46syl 17 . . . . . . . 8 (𝜑𝐷 ∈ (Base‘𝐾))
4817, 7, 8latjle12 17283 . . . . . . . 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 1479 . . . . . . 7 (𝜑 → ((𝑄 (𝑃 𝑄) ∧ 𝐷 (𝑃 𝑄)) ↔ (𝑄 𝐷) (𝑃 𝑄)))
5039, 44, 49mpbi2and 994 . . . . . 6 (𝜑 → (𝑄 𝐷) (𝑃 𝑄))
511, 9dalemreb 35448 . . . . . . 7 (𝜑𝑅 ∈ (Base‘𝐾))
5217, 8, 9hlatjcl 35174 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴𝐷𝐴) → (𝑄 𝐷) ∈ (Base‘𝐾))
532, 4, 31, 52syl3anc 1477 . . . . . . 7 (𝜑 → (𝑄 𝐷) ∈ (Base‘𝐾))
5417, 7lattr 17277 . . . . . . 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 1479 . . . . . 6 (𝜑 → ((𝑅 (𝑄 𝐷) ∧ (𝑄 𝐷) (𝑃 𝑄)) → 𝑅 (𝑃 𝑄)))
5650, 55mpan2d 712 . . . . 5 (𝜑 → (𝑅 (𝑄 𝐷) → 𝑅 (𝑃 𝑄)))
5756adantr 472 . . . 4 ((𝜑𝐷𝑄) → (𝑅 (𝑄 𝐷) → 𝑅 (𝑃 𝑄)))
5827, 37, 573syld 60 . . 3 ((𝜑𝐷𝑄) → (𝐷 = 𝐸𝑅 (𝑃 𝑄)))
5958necon3bd 2946 . 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 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  (class class class)co 6814  Basecbs 16079  lecple 16170  joincjn 17165  meetcmee 17166  Latclat 17266  Atomscatm 35071  HLchlt 35158  LPlanesclpl 35299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-llines 35305  df-lplanes 35306
This theorem is referenced by:  dalem4  35472  dalemdnee  35473
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