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Theorem dalem24 33799
Description: Lemma for dath 33838. 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 6532 . . . 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 33725 . . . . . . 7 (𝜑𝐾 ∈ HL)
5 hlol 33464 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OL)
64, 5syl 17 . . . . . 6 (𝜑𝐾 ∈ OL)
763ad2ant1 1074 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ OL)
843ad2ant1 1074 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
9 dalem.ps . . . . . . . 8 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
109dalemccea 33785 . . . . . . 7 (𝜓𝑐𝐴)
11103ad2ant3 1076 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
123dalempea 33728 . . . . . . 7 (𝜑𝑃𝐴)
13123ad2ant1 1074 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
14 eqid 2604 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
15 dalem.j . . . . . . 7 = (join‘𝐾)
16 dalem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
1714, 15, 16hlatjcl 33469 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
188, 11, 13, 17syl3anc 1317 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
199dalemddea 33786 . . . . . . 7 (𝜓𝑑𝐴)
20193ad2ant3 1076 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
213dalemsea 33731 . . . . . . 7 (𝜑𝑆𝐴)
22213ad2ant1 1074 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2314, 15, 16hlatjcl 33469 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
248, 20, 22, 23syl3anc 1317 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
25 dalem23.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
263, 25dalemyeb 33751 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐾))
27263ad2ant1 1074 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
28 dalem23.m . . . . . 6 = (meet‘𝐾)
2914, 28latmmdir 33338 . . . . 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 1319 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) (𝑑 𝑆)) 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
312, 30syl5eq 2650 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
3215, 16hlatjcom 33470 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) = (𝑃 𝑐))
338, 11, 13, 32syl3anc 1317 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) = (𝑃 𝑐))
3433oveq1d 6537 . . . . 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 33756 . . . . . . 7 (𝜑𝑃 𝑌)
38373ad2ant1 1074 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 𝑌)
399dalem-ccly 33787 . . . . . . 7 (𝜓 → ¬ 𝑐 𝑌)
40393ad2ant3 1076 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑌)
4114, 35, 15, 28, 162atjm 33547 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑐𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑃 𝑌 ∧ ¬ 𝑐 𝑌)) → ((𝑃 𝑐) 𝑌) = 𝑃)
428, 13, 11, 27, 38, 40, 41syl132anc 1335 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑐) 𝑌) = 𝑃)
4334, 42eqtrd 2638 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) 𝑌) = 𝑃)
4415, 16hlatjcom 33470 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) = (𝑆 𝑑))
458, 20, 22, 44syl3anc 1317 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) = (𝑆 𝑑))
4645oveq1d 6537 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑 𝑆) 𝑌) = ((𝑆 𝑑) 𝑌))
47 dalem23.z . . . . . . . 8 𝑍 = ((𝑆 𝑇) 𝑈)
483, 35, 15, 16, 47dalemsly 33757 . . . . . . 7 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
49483adant3 1073 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
509dalem-ddly 33788 . . . . . . 7 (𝜓 → ¬ 𝑑 𝑌)
51503ad2ant3 1076 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
5214, 35, 15, 28, 162atjm 33547 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑑𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 𝑌 ∧ ¬ 𝑑 𝑌)) → ((𝑆 𝑑) 𝑌) = 𝑆)
538, 22, 20, 27, 49, 51, 52syl132anc 1335 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑆 𝑑) 𝑌) = 𝑆)
5446, 53eqtrd 2638 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑 𝑆) 𝑌) = 𝑆)
5543, 54oveq12d 6540 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)) = (𝑃 𝑆))
563, 35, 15, 16, 25, 36dalempnes 33753 . . . . 5 (𝜑𝑃𝑆)
57 hlatl 33463 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
584, 57syl 17 . . . . . 6 (𝜑𝐾 ∈ AtLat)
59 eqid 2604 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
6028, 59, 16atnem0 33421 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑆𝐴) → (𝑃𝑆 ↔ (𝑃 𝑆) = (0.‘𝐾)))
6158, 12, 21, 60syl3anc 1317 . . . . 5 (𝜑 → (𝑃𝑆 ↔ (𝑃 𝑆) = (0.‘𝐾)))
6256, 61mpbid 220 . . . 4 (𝜑 → (𝑃 𝑆) = (0.‘𝐾))
63623ad2ant1 1074 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) = (0.‘𝐾))
6431, 55, 633eqtrd 2642 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑌) = (0.‘𝐾))
65583ad2ant1 1074 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
663, 35, 15, 16, 9, 28, 25, 36, 47, 1dalem23 33798 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
6714, 35, 28, 59, 16atnle 33420 . . 3 ((𝐾 ∈ AtLat ∧ 𝐺𝐴𝑌 ∈ (Base‘𝐾)) → (¬ 𝐺 𝑌 ↔ (𝐺 𝑌) = (0.‘𝐾)))
6865, 66, 27, 67syl3anc 1317 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝐺 𝑌 ↔ (𝐺 𝑌) = (0.‘𝐾)))
6964, 68mpbird 245 1 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774   class class class wbr 4572  cfv 5785  (class class class)co 6522  Basecbs 15636  lecple 15716  joincjn 16708  meetcmee 16709  0.cp0 16801  OLcol 33277  Atomscatm 33366  AtLatcal 33367  HLchlt 33453  LPlanesclpl 33594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-lat 16810  df-clat 16872  df-oposet 33279  df-ol 33281  df-oml 33282  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454  df-llines 33600  df-lplanes 33601
This theorem is referenced by:  dalem27  33801  dalem30  33804  dalem54  33828
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