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Theorem dalem24 38373
Description: Lemma for dath 38412. 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 7403 . . . 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 38299 . . . . . . 7 (𝜑𝐾 ∈ HL)
5 hlol 38036 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OL)
64, 5syl 17 . . . . . 6 (𝜑𝐾 ∈ OL)
763ad2ant1 1133 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ OL)
843ad2ant1 1133 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
9 dalem.ps . . . . . . . 8 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
109dalemccea 38359 . . . . . . 7 (𝜓𝑐𝐴)
11103ad2ant3 1135 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
123dalempea 38302 . . . . . . 7 (𝜑𝑃𝐴)
13123ad2ant1 1133 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
14 eqid 2731 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
15 dalem.j . . . . . . 7 = (join‘𝐾)
16 dalem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
1714, 15, 16hlatjcl 38042 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
188, 11, 13, 17syl3anc 1371 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
199dalemddea 38360 . . . . . . 7 (𝜓𝑑𝐴)
20193ad2ant3 1135 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
213dalemsea 38305 . . . . . . 7 (𝜑𝑆𝐴)
22213ad2ant1 1133 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2314, 15, 16hlatjcl 38042 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
248, 20, 22, 23syl3anc 1371 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
25 dalem23.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
263, 25dalemyeb 38325 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐾))
27263ad2ant1 1133 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
28 dalem23.m . . . . . 6 = (meet‘𝐾)
2914, 28latmmdir 37910 . . . . 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 1372 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) (𝑑 𝑆)) 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
312, 30eqtrid 2783 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
3215, 16hlatjcom 38043 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) = (𝑃 𝑐))
338, 11, 13, 32syl3anc 1371 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) = (𝑃 𝑐))
3433oveq1d 7408 . . . . 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 38330 . . . . . . 7 (𝜑𝑃 𝑌)
38373ad2ant1 1133 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 𝑌)
399dalem-ccly 38361 . . . . . . 7 (𝜓 → ¬ 𝑐 𝑌)
40393ad2ant3 1135 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑌)
4114, 35, 15, 28, 162atjm 38121 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑐𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑃 𝑌 ∧ ¬ 𝑐 𝑌)) → ((𝑃 𝑐) 𝑌) = 𝑃)
428, 13, 11, 27, 38, 40, 41syl132anc 1388 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑐) 𝑌) = 𝑃)
4334, 42eqtrd 2771 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) 𝑌) = 𝑃)
4415, 16hlatjcom 38043 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) = (𝑆 𝑑))
458, 20, 22, 44syl3anc 1371 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) = (𝑆 𝑑))
4645oveq1d 7408 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑 𝑆) 𝑌) = ((𝑆 𝑑) 𝑌))
47 dalem23.z . . . . . . . 8 𝑍 = ((𝑆 𝑇) 𝑈)
483, 35, 15, 16, 47dalemsly 38331 . . . . . . 7 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
49483adant3 1132 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
509dalem-ddly 38362 . . . . . . 7 (𝜓 → ¬ 𝑑 𝑌)
51503ad2ant3 1135 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
5214, 35, 15, 28, 162atjm 38121 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑑𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 𝑌 ∧ ¬ 𝑑 𝑌)) → ((𝑆 𝑑) 𝑌) = 𝑆)
538, 22, 20, 27, 49, 51, 52syl132anc 1388 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑆 𝑑) 𝑌) = 𝑆)
5446, 53eqtrd 2771 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑 𝑆) 𝑌) = 𝑆)
5543, 54oveq12d 7411 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)) = (𝑃 𝑆))
563, 35, 15, 16, 25, 36dalempnes 38327 . . . . 5 (𝜑𝑃𝑆)
57 hlatl 38035 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
584, 57syl 17 . . . . . 6 (𝜑𝐾 ∈ AtLat)
59 eqid 2731 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
6028, 59, 16atnem0 37993 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑆𝐴) → (𝑃𝑆 ↔ (𝑃 𝑆) = (0.‘𝐾)))
6158, 12, 21, 60syl3anc 1371 . . . . 5 (𝜑 → (𝑃𝑆 ↔ (𝑃 𝑆) = (0.‘𝐾)))
6256, 61mpbid 231 . . . 4 (𝜑 → (𝑃 𝑆) = (0.‘𝐾))
63623ad2ant1 1133 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) = (0.‘𝐾))
6431, 55, 633eqtrd 2775 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑌) = (0.‘𝐾))
65583ad2ant1 1133 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
663, 35, 15, 16, 9, 28, 25, 36, 47, 1dalem23 38372 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
6714, 35, 28, 59, 16atnle 37992 . . 3 ((𝐾 ∈ AtLat ∧ 𝐺𝐴𝑌 ∈ (Base‘𝐾)) → (¬ 𝐺 𝑌 ↔ (𝐺 𝑌) = (0.‘𝐾)))
6865, 66, 27, 67syl3anc 1371 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝐺 𝑌 ↔ (𝐺 𝑌) = (0.‘𝐾)))
6964, 68mpbird 256 1 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2939   class class class wbr 5141  cfv 6532  (class class class)co 7393  Basecbs 17126  lecple 17186  joincjn 18246  meetcmee 18247  0.cp0 18358  OLcol 37849  Atomscatm 37938  AtLatcal 37939  HLchlt 38025  LPlanesclpl 38168
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-proset 18230  df-poset 18248  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-lat 18367  df-clat 18434  df-oposet 37851  df-ol 37853  df-oml 37854  df-covers 37941  df-ats 37942  df-atl 37973  df-cvlat 37997  df-hlat 38026  df-llines 38174  df-lplanes 38175
This theorem is referenced by:  dalem27  38375  dalem30  38378  dalem54  38402
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