Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem21 Structured version   Visualization version   GIF version

Theorem dalem21 35501
Description: Lemma for dath 35543. Show that lines 𝑐𝑑 and 𝑃𝑆 intersect at an atom. (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))))
dalem21.m = (meet‘𝐾)
dalem21.o 𝑂 = (LPlanes‘𝐾)
dalem21.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem21.z 𝑍 = ((𝑆 𝑇) 𝑈)
Assertion
Ref Expression
dalem21 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ 𝐴)

Proof of Theorem dalem21
StepHypRef Expression
1 dalem.ph . . . 4 (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 . . 3 (𝜑𝐾 ∈ HL)
323ad2ant1 1128 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
4 dalem.l . . . 4 = (le‘𝐾)
5 dalem.j . . . 4 = (join‘𝐾)
6 dalem.a . . . 4 𝐴 = (Atoms‘𝐾)
7 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
81, 4, 5, 6, 7dalemcjden 35499 . . 3 ((𝜑𝜓) → (𝑐 𝑑) ∈ (LLines‘𝐾))
983adant2 1126 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) ∈ (LLines‘𝐾))
10 dalem21.o . . . 4 𝑂 = (LPlanes‘𝐾)
11 dalem21.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
121, 4, 5, 6, 10, 11dalempjsen 35460 . . 3 (𝜑 → (𝑃 𝑆) ∈ (LLines‘𝐾))
13123ad2ant1 1128 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) ∈ (LLines‘𝐾))
141, 4, 5, 6, 10, 11dalemply 35461 . . . . . . 7 (𝜑𝑃 𝑌)
1514adantr 472 . . . . . 6 ((𝜑𝑌 = 𝑍) → 𝑃 𝑌)
16 dalem21.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
171, 4, 5, 6, 16dalemsly 35462 . . . . . 6 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
181dalemkelat 35431 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
191, 6dalempeb 35446 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
201, 6dalemseb 35449 . . . . . . . 8 (𝜑𝑆 ∈ (Base‘𝐾))
211, 10dalemyeb 35456 . . . . . . . 8 (𝜑𝑌 ∈ (Base‘𝐾))
22 eqid 2760 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2322, 4, 5latjle12 17283 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2418, 19, 20, 21, 23syl13anc 1479 . . . . . . 7 (𝜑 → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2524adantr 472 . . . . . 6 ((𝜑𝑌 = 𝑍) → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2615, 17, 25mpbi2and 994 . . . . 5 ((𝜑𝑌 = 𝑍) → (𝑃 𝑆) 𝑌)
27263adant3 1127 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) 𝑌)
287dalem-ccly 35492 . . . . . . 7 (𝜓 → ¬ 𝑐 𝑌)
2928adantl 473 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 𝑌)
3018adantr 472 . . . . . . . 8 ((𝜑𝜓) → 𝐾 ∈ Lat)
317, 6dalemcceb 35496 . . . . . . . . 9 (𝜓𝑐 ∈ (Base‘𝐾))
3231adantl 473 . . . . . . . 8 ((𝜑𝜓) → 𝑐 ∈ (Base‘𝐾))
337dalemddea 35491 . . . . . . . . . 10 (𝜓𝑑𝐴)
3422, 6atbase 35097 . . . . . . . . . 10 (𝑑𝐴𝑑 ∈ (Base‘𝐾))
3533, 34syl 17 . . . . . . . . 9 (𝜓𝑑 ∈ (Base‘𝐾))
3635adantl 473 . . . . . . . 8 ((𝜑𝜓) → 𝑑 ∈ (Base‘𝐾))
3722, 4, 5latlej1 17281 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾)) → 𝑐 (𝑐 𝑑))
3830, 32, 36, 37syl3anc 1477 . . . . . . 7 ((𝜑𝜓) → 𝑐 (𝑐 𝑑))
39 eqid 2760 . . . . . . . . . 10 (LLines‘𝐾) = (LLines‘𝐾)
4022, 39llnbase 35316 . . . . . . . . 9 ((𝑐 𝑑) ∈ (LLines‘𝐾) → (𝑐 𝑑) ∈ (Base‘𝐾))
418, 40syl 17 . . . . . . . 8 ((𝜑𝜓) → (𝑐 𝑑) ∈ (Base‘𝐾))
4221adantr 472 . . . . . . . 8 ((𝜑𝜓) → 𝑌 ∈ (Base‘𝐾))
4322, 4lattr 17277 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ (𝑐 𝑑) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑐 (𝑐 𝑑) ∧ (𝑐 𝑑) 𝑌) → 𝑐 𝑌))
4430, 32, 41, 42, 43syl13anc 1479 . . . . . . 7 ((𝜑𝜓) → ((𝑐 (𝑐 𝑑) ∧ (𝑐 𝑑) 𝑌) → 𝑐 𝑌))
4538, 44mpand 713 . . . . . 6 ((𝜑𝜓) → ((𝑐 𝑑) 𝑌𝑐 𝑌))
4629, 45mtod 189 . . . . 5 ((𝜑𝜓) → ¬ (𝑐 𝑑) 𝑌)
47463adant2 1126 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ (𝑐 𝑑) 𝑌)
48 nbrne2 4824 . . . 4 (((𝑃 𝑆) 𝑌 ∧ ¬ (𝑐 𝑑) 𝑌) → (𝑃 𝑆) ≠ (𝑐 𝑑))
4927, 47, 48syl2anc 696 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) ≠ (𝑐 𝑑))
5049necomd 2987 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) ≠ (𝑃 𝑆))
51 hlatl 35168 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
522, 51syl 17 . . . . 5 (𝜑𝐾 ∈ AtLat)
5352adantr 472 . . . 4 ((𝜑𝜓) → 𝐾 ∈ AtLat)
541dalempea 35433 . . . . . . 7 (𝜑𝑃𝐴)
551dalemsea 35436 . . . . . . 7 (𝜑𝑆𝐴)
5622, 5, 6hlatjcl 35174 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
572, 54, 55, 56syl3anc 1477 . . . . . 6 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
5857adantr 472 . . . . 5 ((𝜑𝜓) → (𝑃 𝑆) ∈ (Base‘𝐾))
59 dalem21.m . . . . . 6 = (meet‘𝐾)
6022, 59latmcl 17273 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑐 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ (Base‘𝐾))
6130, 41, 58, 60syl3anc 1477 . . . 4 ((𝜑𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ (Base‘𝐾))
621, 4, 5, 6, 10, 11dalemcea 35467 . . . . 5 (𝜑𝐶𝐴)
6362adantr 472 . . . 4 ((𝜑𝜓) → 𝐶𝐴)
647dalemclccjdd 35495 . . . . . 6 (𝜓𝐶 (𝑐 𝑑))
6564adantl 473 . . . . 5 ((𝜑𝜓) → 𝐶 (𝑐 𝑑))
661dalemclpjs 35441 . . . . . 6 (𝜑𝐶 (𝑃 𝑆))
6766adantr 472 . . . . 5 ((𝜑𝜓) → 𝐶 (𝑃 𝑆))
681, 6dalemceb 35445 . . . . . . 7 (𝜑𝐶 ∈ (Base‘𝐾))
6968adantr 472 . . . . . 6 ((𝜑𝜓) → 𝐶 ∈ (Base‘𝐾))
7022, 4, 59latlem12 17299 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑐 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾))) → ((𝐶 (𝑐 𝑑) ∧ 𝐶 (𝑃 𝑆)) ↔ 𝐶 ((𝑐 𝑑) (𝑃 𝑆))))
7130, 69, 41, 58, 70syl13anc 1479 . . . . 5 ((𝜑𝜓) → ((𝐶 (𝑐 𝑑) ∧ 𝐶 (𝑃 𝑆)) ↔ 𝐶 ((𝑐 𝑑) (𝑃 𝑆))))
7265, 67, 71mpbi2and 994 . . . 4 ((𝜑𝜓) → 𝐶 ((𝑐 𝑑) (𝑃 𝑆)))
73 eqid 2760 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
7422, 4, 73, 6atlen0 35118 . . . 4 (((𝐾 ∈ AtLat ∧ ((𝑐 𝑑) (𝑃 𝑆)) ∈ (Base‘𝐾) ∧ 𝐶𝐴) ∧ 𝐶 ((𝑐 𝑑) (𝑃 𝑆))) → ((𝑐 𝑑) (𝑃 𝑆)) ≠ (0.‘𝐾))
7553, 61, 63, 72, 74syl31anc 1480 . . 3 ((𝜑𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ≠ (0.‘𝐾))
76753adant2 1126 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ≠ (0.‘𝐾))
7759, 73, 6, 392llnmat 35331 . 2 (((𝐾 ∈ HL ∧ (𝑐 𝑑) ∈ (LLines‘𝐾) ∧ (𝑃 𝑆) ∈ (LLines‘𝐾)) ∧ ((𝑐 𝑑) ≠ (𝑃 𝑆) ∧ ((𝑐 𝑑) (𝑃 𝑆)) ≠ (0.‘𝐾))) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ 𝐴)
783, 9, 13, 50, 76, 77syl32anc 1485 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ 𝐴)
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  0.cp0 17258  Latclat 17266  Atomscatm 35071  AtLatcal 35072  HLchlt 35158  LLinesclln 35298  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:  dalem22  35502
  Copyright terms: Public domain W3C validator