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Theorem dalem24 35482
 Description: Lemma for dath 35521. Show that auxiliary atom 𝐺 is outside of plane 𝑌. (Contributed by NM, 2-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))))
dalem23.m = (meet‘𝐾)
dalem23.o 𝑂 = (LPlanes‘𝐾)
dalem23.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem23.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem23.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
Assertion
Ref Expression
dalem24 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)

Proof of Theorem dalem24
StepHypRef Expression
1 dalem23.g . . . . 5 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
21oveq1i 6819 . . . 4 (𝐺 𝑌) = (((𝑐 𝑃) (𝑑 𝑆)) 𝑌)
3 dalem.ph . . . . . . . 8 (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)))))
43dalemkehl 35408 . . . . . . 7 (𝜑𝐾 ∈ HL)
5 hlol 35147 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OL)
64, 5syl 17 . . . . . 6 (𝜑𝐾 ∈ OL)
763ad2ant1 1128 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ OL)
843ad2ant1 1128 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
9 dalem.ps . . . . . . . 8 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
109dalemccea 35468 . . . . . . 7 (𝜓𝑐𝐴)
11103ad2ant3 1130 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
123dalempea 35411 . . . . . . 7 (𝜑𝑃𝐴)
13123ad2ant1 1128 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
14 eqid 2756 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
15 dalem.j . . . . . . 7 = (join‘𝐾)
16 dalem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
1714, 15, 16hlatjcl 35152 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
188, 11, 13, 17syl3anc 1477 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
199dalemddea 35469 . . . . . . 7 (𝜓𝑑𝐴)
20193ad2ant3 1130 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
213dalemsea 35414 . . . . . . 7 (𝜑𝑆𝐴)
22213ad2ant1 1128 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2314, 15, 16hlatjcl 35152 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
248, 20, 22, 23syl3anc 1477 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
25 dalem23.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
263, 25dalemyeb 35434 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐾))
27263ad2ant1 1128 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
28 dalem23.m . . . . . 6 = (meet‘𝐾)
2914, 28latmmdir 35021 . . . . 5 ((𝐾 ∈ OL ∧ ((𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → (((𝑐 𝑃) (𝑑 𝑆)) 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
307, 18, 24, 27, 29syl13anc 1479 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) (𝑑 𝑆)) 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
312, 30syl5eq 2802 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
3215, 16hlatjcom 35153 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) = (𝑃 𝑐))
338, 11, 13, 32syl3anc 1477 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) = (𝑃 𝑐))
3433oveq1d 6824 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) 𝑌) = ((𝑃 𝑐) 𝑌))
35 dalem.l . . . . . . . 8 = (le‘𝐾)
36 dalem23.y . . . . . . . 8 𝑌 = ((𝑃 𝑄) 𝑅)
373, 35, 15, 16, 25, 36dalemply 35439 . . . . . . 7 (𝜑𝑃 𝑌)
38373ad2ant1 1128 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 𝑌)
399dalem-ccly 35470 . . . . . . 7 (𝜓 → ¬ 𝑐 𝑌)
40393ad2ant3 1130 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑌)
4114, 35, 15, 28, 162atjm 35230 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑐𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑃 𝑌 ∧ ¬ 𝑐 𝑌)) → ((𝑃 𝑐) 𝑌) = 𝑃)
428, 13, 11, 27, 38, 40, 41syl132anc 1495 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑐) 𝑌) = 𝑃)
4334, 42eqtrd 2790 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) 𝑌) = 𝑃)
4415, 16hlatjcom 35153 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) = (𝑆 𝑑))
458, 20, 22, 44syl3anc 1477 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) = (𝑆 𝑑))
4645oveq1d 6824 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑 𝑆) 𝑌) = ((𝑆 𝑑) 𝑌))
47 dalem23.z . . . . . . . 8 𝑍 = ((𝑆 𝑇) 𝑈)
483, 35, 15, 16, 47dalemsly 35440 . . . . . . 7 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
49483adant3 1127 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
509dalem-ddly 35471 . . . . . . 7 (𝜓 → ¬ 𝑑 𝑌)
51503ad2ant3 1130 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
5214, 35, 15, 28, 162atjm 35230 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑑𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 𝑌 ∧ ¬ 𝑑 𝑌)) → ((𝑆 𝑑) 𝑌) = 𝑆)
538, 22, 20, 27, 49, 51, 52syl132anc 1495 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑆 𝑑) 𝑌) = 𝑆)
5446, 53eqtrd 2790 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑 𝑆) 𝑌) = 𝑆)
5543, 54oveq12d 6827 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)) = (𝑃 𝑆))
563, 35, 15, 16, 25, 36dalempnes 35436 . . . . 5 (𝜑𝑃𝑆)
57 hlatl 35146 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
584, 57syl 17 . . . . . 6 (𝜑𝐾 ∈ AtLat)
59 eqid 2756 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
6028, 59, 16atnem0 35104 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑆𝐴) → (𝑃𝑆 ↔ (𝑃 𝑆) = (0.‘𝐾)))
6158, 12, 21, 60syl3anc 1477 . . . . 5 (𝜑 → (𝑃𝑆 ↔ (𝑃 𝑆) = (0.‘𝐾)))
6256, 61mpbid 222 . . . 4 (𝜑 → (𝑃 𝑆) = (0.‘𝐾))
63623ad2ant1 1128 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) = (0.‘𝐾))
6431, 55, 633eqtrd 2794 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑌) = (0.‘𝐾))
65583ad2ant1 1128 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
663, 35, 15, 16, 9, 28, 25, 36, 47, 1dalem23 35481 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
6714, 35, 28, 59, 16atnle 35103 . . 3 ((𝐾 ∈ AtLat ∧ 𝐺𝐴𝑌 ∈ (Base‘𝐾)) → (¬ 𝐺 𝑌 ↔ (𝐺 𝑌) = (0.‘𝐾)))
6865, 66, 27, 67syl3anc 1477 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝐺 𝑌 ↔ (𝐺 𝑌) = (0.‘𝐾)))
6964, 68mpbird 247 1 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1628   ∈ wcel 2135   ≠ wne 2928   class class class wbr 4800  ‘cfv 6045  (class class class)co 6809  Basecbs 16055  lecple 16146  joincjn 17141  meetcmee 17142  0.cp0 17234  OLcol 34960  Atomscatm 35049  AtLatcal 35050  HLchlt 35136  LPlanesclpl 35277 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-preset 17125  df-poset 17143  df-plt 17155  df-lub 17171  df-glb 17172  df-join 17173  df-meet 17174  df-p0 17236  df-lat 17243  df-clat 17305  df-oposet 34962  df-ol 34964  df-oml 34965  df-covers 35052  df-ats 35053  df-atl 35084  df-cvlat 35108  df-hlat 35137  df-llines 35283  df-lplanes 35284 This theorem is referenced by:  dalem27  35484  dalem30  35487  dalem54  35511
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