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Theorem dalem21 35502
Description: Lemma for dath 35544. 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 35431 . . 3 (𝜑𝐾 ∈ HL)
323ad2ant1 1127 . 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 35500 . . 3 ((𝜑𝜓) → (𝑐 𝑑) ∈ (LLines‘𝐾))
983adant2 1125 . 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 35461 . . 3 (𝜑 → (𝑃 𝑆) ∈ (LLines‘𝐾))
13123ad2ant1 1127 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) ∈ (LLines‘𝐾))
141, 4, 5, 6, 10, 11dalemply 35462 . . . . . . 7 (𝜑𝑃 𝑌)
1514adantr 466 . . . . . 6 ((𝜑𝑌 = 𝑍) → 𝑃 𝑌)
16 dalem21.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
171, 4, 5, 6, 16dalemsly 35463 . . . . . 6 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
181dalemkelat 35432 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
191, 6dalempeb 35447 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
201, 6dalemseb 35450 . . . . . . . 8 (𝜑𝑆 ∈ (Base‘𝐾))
211, 10dalemyeb 35457 . . . . . . . 8 (𝜑𝑌 ∈ (Base‘𝐾))
22 eqid 2771 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2322, 4, 5latjle12 17270 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2418, 19, 20, 21, 23syl13anc 1478 . . . . . . 7 (𝜑 → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2524adantr 466 . . . . . 6 ((𝜑𝑌 = 𝑍) → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2615, 17, 25mpbi2and 691 . . . . 5 ((𝜑𝑌 = 𝑍) → (𝑃 𝑆) 𝑌)
27263adant3 1126 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) 𝑌)
287dalem-ccly 35493 . . . . . . 7 (𝜓 → ¬ 𝑐 𝑌)
2928adantl 467 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 𝑌)
3018adantr 466 . . . . . . . 8 ((𝜑𝜓) → 𝐾 ∈ Lat)
317, 6dalemcceb 35497 . . . . . . . . 9 (𝜓𝑐 ∈ (Base‘𝐾))
3231adantl 467 . . . . . . . 8 ((𝜑𝜓) → 𝑐 ∈ (Base‘𝐾))
337dalemddea 35492 . . . . . . . . . 10 (𝜓𝑑𝐴)
3422, 6atbase 35098 . . . . . . . . . 10 (𝑑𝐴𝑑 ∈ (Base‘𝐾))
3533, 34syl 17 . . . . . . . . 9 (𝜓𝑑 ∈ (Base‘𝐾))
3635adantl 467 . . . . . . . 8 ((𝜑𝜓) → 𝑑 ∈ (Base‘𝐾))
3722, 4, 5latlej1 17268 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾)) → 𝑐 (𝑐 𝑑))
3830, 32, 36, 37syl3anc 1476 . . . . . . 7 ((𝜑𝜓) → 𝑐 (𝑐 𝑑))
39 eqid 2771 . . . . . . . . . 10 (LLines‘𝐾) = (LLines‘𝐾)
4022, 39llnbase 35317 . . . . . . . . 9 ((𝑐 𝑑) ∈ (LLines‘𝐾) → (𝑐 𝑑) ∈ (Base‘𝐾))
418, 40syl 17 . . . . . . . 8 ((𝜑𝜓) → (𝑐 𝑑) ∈ (Base‘𝐾))
4221adantr 466 . . . . . . . 8 ((𝜑𝜓) → 𝑌 ∈ (Base‘𝐾))
4322, 4lattr 17264 . . . . . . . 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 1478 . . . . . . 7 ((𝜑𝜓) → ((𝑐 (𝑐 𝑑) ∧ (𝑐 𝑑) 𝑌) → 𝑐 𝑌))
4538, 44mpand 675 . . . . . 6 ((𝜑𝜓) → ((𝑐 𝑑) 𝑌𝑐 𝑌))
4629, 45mtod 189 . . . . 5 ((𝜑𝜓) → ¬ (𝑐 𝑑) 𝑌)
47463adant2 1125 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ (𝑐 𝑑) 𝑌)
48 nbrne2 4806 . . . 4 (((𝑃 𝑆) 𝑌 ∧ ¬ (𝑐 𝑑) 𝑌) → (𝑃 𝑆) ≠ (𝑐 𝑑))
4927, 47, 48syl2anc 573 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) ≠ (𝑐 𝑑))
5049necomd 2998 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) ≠ (𝑃 𝑆))
51 hlatl 35169 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
522, 51syl 17 . . . . 5 (𝜑𝐾 ∈ AtLat)
5352adantr 466 . . . 4 ((𝜑𝜓) → 𝐾 ∈ AtLat)
541dalempea 35434 . . . . . . 7 (𝜑𝑃𝐴)
551dalemsea 35437 . . . . . . 7 (𝜑𝑆𝐴)
5622, 5, 6hlatjcl 35175 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
572, 54, 55, 56syl3anc 1476 . . . . . 6 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
5857adantr 466 . . . . 5 ((𝜑𝜓) → (𝑃 𝑆) ∈ (Base‘𝐾))
59 dalem21.m . . . . . 6 = (meet‘𝐾)
6022, 59latmcl 17260 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑐 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ (Base‘𝐾))
6130, 41, 58, 60syl3anc 1476 . . . 4 ((𝜑𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ (Base‘𝐾))
621, 4, 5, 6, 10, 11dalemcea 35468 . . . . 5 (𝜑𝐶𝐴)
6362adantr 466 . . . 4 ((𝜑𝜓) → 𝐶𝐴)
647dalemclccjdd 35496 . . . . . 6 (𝜓𝐶 (𝑐 𝑑))
6564adantl 467 . . . . 5 ((𝜑𝜓) → 𝐶 (𝑐 𝑑))
661dalemclpjs 35442 . . . . . 6 (𝜑𝐶 (𝑃 𝑆))
6766adantr 466 . . . . 5 ((𝜑𝜓) → 𝐶 (𝑃 𝑆))
681, 6dalemceb 35446 . . . . . . 7 (𝜑𝐶 ∈ (Base‘𝐾))
6968adantr 466 . . . . . 6 ((𝜑𝜓) → 𝐶 ∈ (Base‘𝐾))
7022, 4, 59latlem12 17286 . . . . . 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 1478 . . . . 5 ((𝜑𝜓) → ((𝐶 (𝑐 𝑑) ∧ 𝐶 (𝑃 𝑆)) ↔ 𝐶 ((𝑐 𝑑) (𝑃 𝑆))))
7265, 67, 71mpbi2and 691 . . . 4 ((𝜑𝜓) → 𝐶 ((𝑐 𝑑) (𝑃 𝑆)))
73 eqid 2771 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
7422, 4, 73, 6atlen0 35119 . . . 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 1479 . . 3 ((𝜑𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ≠ (0.‘𝐾))
76753adant2 1125 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ≠ (0.‘𝐾))
7759, 73, 6, 392llnmat 35332 . 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 1484 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  0.cp0 17245  Latclat 17253  Atomscatm 35072  AtLatcal 35073  HLchlt 35159  LLinesclln 35299  LPlanesclpl 35300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307
This theorem is referenced by:  dalem22  35503
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