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Theorem dalem24 39679
Description: Lemma for dath 39718. 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 7440 . . . 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 39605 . . . . . . 7 (𝜑𝐾 ∈ HL)
5 hlol 39342 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OL)
64, 5syl 17 . . . . . 6 (𝜑𝐾 ∈ OL)
763ad2ant1 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ OL)
843ad2ant1 1132 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
9 dalem.ps . . . . . . . 8 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
109dalemccea 39665 . . . . . . 7 (𝜓𝑐𝐴)
11103ad2ant3 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
123dalempea 39608 . . . . . . 7 (𝜑𝑃𝐴)
13123ad2ant1 1132 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
14 eqid 2734 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
15 dalem.j . . . . . . 7 = (join‘𝐾)
16 dalem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
1714, 15, 16hlatjcl 39348 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
188, 11, 13, 17syl3anc 1370 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
199dalemddea 39666 . . . . . . 7 (𝜓𝑑𝐴)
20193ad2ant3 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
213dalemsea 39611 . . . . . . 7 (𝜑𝑆𝐴)
22213ad2ant1 1132 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2314, 15, 16hlatjcl 39348 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
248, 20, 22, 23syl3anc 1370 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
25 dalem23.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
263, 25dalemyeb 39631 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐾))
27263ad2ant1 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
28 dalem23.m . . . . . 6 = (meet‘𝐾)
2914, 28latmmdir 39216 . . . . 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 1371 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) (𝑑 𝑆)) 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
312, 30eqtrid 2786 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑌) = (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)))
3215, 16hlatjcom 39349 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) = (𝑃 𝑐))
338, 11, 13, 32syl3anc 1370 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) = (𝑃 𝑐))
3433oveq1d 7445 . . . . 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 39636 . . . . . . 7 (𝜑𝑃 𝑌)
38373ad2ant1 1132 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 𝑌)
399dalem-ccly 39667 . . . . . . 7 (𝜓 → ¬ 𝑐 𝑌)
40393ad2ant3 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑐 𝑌)
4114, 35, 15, 28, 162atjm 39427 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑐𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑃 𝑌 ∧ ¬ 𝑐 𝑌)) → ((𝑃 𝑐) 𝑌) = 𝑃)
428, 13, 11, 27, 38, 40, 41syl132anc 1387 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑃 𝑐) 𝑌) = 𝑃)
4334, 42eqtrd 2774 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) 𝑌) = 𝑃)
4415, 16hlatjcom 39349 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) = (𝑆 𝑑))
458, 20, 22, 44syl3anc 1370 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) = (𝑆 𝑑))
4645oveq1d 7445 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑 𝑆) 𝑌) = ((𝑆 𝑑) 𝑌))
47 dalem23.z . . . . . . . 8 𝑍 = ((𝑆 𝑇) 𝑈)
483, 35, 15, 16, 47dalemsly 39637 . . . . . . 7 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
49483adant3 1131 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
509dalem-ddly 39668 . . . . . . 7 (𝜓 → ¬ 𝑑 𝑌)
51503ad2ant3 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
5214, 35, 15, 28, 162atjm 39427 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑑𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 𝑌 ∧ ¬ 𝑑 𝑌)) → ((𝑆 𝑑) 𝑌) = 𝑆)
538, 22, 20, 27, 49, 51, 52syl132anc 1387 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝑆 𝑑) 𝑌) = 𝑆)
5446, 53eqtrd 2774 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝑑 𝑆) 𝑌) = 𝑆)
5543, 54oveq12d 7448 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝑐 𝑃) 𝑌) ((𝑑 𝑆) 𝑌)) = (𝑃 𝑆))
563, 35, 15, 16, 25, 36dalempnes 39633 . . . . 5 (𝜑𝑃𝑆)
57 hlatl 39341 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
584, 57syl 17 . . . . . 6 (𝜑𝐾 ∈ AtLat)
59 eqid 2734 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
6028, 59, 16atnem0 39299 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑆𝐴) → (𝑃𝑆 ↔ (𝑃 𝑆) = (0.‘𝐾)))
6158, 12, 21, 60syl3anc 1370 . . . . 5 (𝜑 → (𝑃𝑆 ↔ (𝑃 𝑆) = (0.‘𝐾)))
6256, 61mpbid 232 . . . 4 (𝜑 → (𝑃 𝑆) = (0.‘𝐾))
63623ad2ant1 1132 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) = (0.‘𝐾))
6431, 55, 633eqtrd 2778 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝑌) = (0.‘𝐾))
65583ad2ant1 1132 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
663, 35, 15, 16, 9, 28, 25, 36, 47, 1dalem23 39678 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
6714, 35, 28, 59, 16atnle 39298 . . 3 ((𝐾 ∈ AtLat ∧ 𝐺𝐴𝑌 ∈ (Base‘𝐾)) → (¬ 𝐺 𝑌 ↔ (𝐺 𝑌) = (0.‘𝐾)))
6865, 66, 27, 67syl3anc 1370 . 2 ((𝜑𝑌 = 𝑍𝜓) → (¬ 𝐺 𝑌 ↔ (𝐺 𝑌) = (0.‘𝐾)))
6964, 68mpbird 257 1 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937   class class class wbr 5147  cfv 6562  (class class class)co 7430  Basecbs 17244  lecple 17304  joincjn 18368  meetcmee 18369  0.cp0 18480  OLcol 39155  Atomscatm 39244  AtLatcal 39245  HLchlt 39331  LPlanesclpl 39474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-proset 18351  df-poset 18370  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-p0 18482  df-lat 18489  df-clat 18556  df-oposet 39157  df-ol 39159  df-oml 39160  df-covers 39247  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332  df-llines 39480  df-lplanes 39481
This theorem is referenced by:  dalem27  39681  dalem30  39684  dalem54  39708
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