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Theorem dalem21 33794
Description: Lemma for dath 33836. 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 33723 . . 3 (𝜑𝐾 ∈ HL)
323ad2ant1 1074 . 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 33792 . . 3 ((𝜑𝜓) → (𝑐 𝑑) ∈ (LLines‘𝐾))
983adant2 1072 . 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 33753 . . 3 (𝜑 → (𝑃 𝑆) ∈ (LLines‘𝐾))
13123ad2ant1 1074 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) ∈ (LLines‘𝐾))
141, 4, 5, 6, 10, 11dalemply 33754 . . . . . . 7 (𝜑𝑃 𝑌)
1514adantr 479 . . . . . 6 ((𝜑𝑌 = 𝑍) → 𝑃 𝑌)
16 dalem21.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
171, 4, 5, 6, 16dalemsly 33755 . . . . . 6 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
181dalemkelat 33724 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
191, 6dalempeb 33739 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
201, 6dalemseb 33742 . . . . . . . 8 (𝜑𝑆 ∈ (Base‘𝐾))
211, 10dalemyeb 33749 . . . . . . . 8 (𝜑𝑌 ∈ (Base‘𝐾))
22 eqid 2609 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2322, 4, 5latjle12 16831 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2418, 19, 20, 21, 23syl13anc 1319 . . . . . . 7 (𝜑 → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2524adantr 479 . . . . . 6 ((𝜑𝑌 = 𝑍) → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2615, 17, 25mpbi2and 957 . . . . 5 ((𝜑𝑌 = 𝑍) → (𝑃 𝑆) 𝑌)
27263adant3 1073 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) 𝑌)
287dalem-ccly 33785 . . . . . . 7 (𝜓 → ¬ 𝑐 𝑌)
2928adantl 480 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 𝑌)
3018adantr 479 . . . . . . . 8 ((𝜑𝜓) → 𝐾 ∈ Lat)
317, 6dalemcceb 33789 . . . . . . . . 9 (𝜓𝑐 ∈ (Base‘𝐾))
3231adantl 480 . . . . . . . 8 ((𝜑𝜓) → 𝑐 ∈ (Base‘𝐾))
337dalemddea 33784 . . . . . . . . . 10 (𝜓𝑑𝐴)
3422, 6atbase 33390 . . . . . . . . . 10 (𝑑𝐴𝑑 ∈ (Base‘𝐾))
3533, 34syl 17 . . . . . . . . 9 (𝜓𝑑 ∈ (Base‘𝐾))
3635adantl 480 . . . . . . . 8 ((𝜑𝜓) → 𝑑 ∈ (Base‘𝐾))
3722, 4, 5latlej1 16829 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾)) → 𝑐 (𝑐 𝑑))
3830, 32, 36, 37syl3anc 1317 . . . . . . 7 ((𝜑𝜓) → 𝑐 (𝑐 𝑑))
39 eqid 2609 . . . . . . . . . 10 (LLines‘𝐾) = (LLines‘𝐾)
4022, 39llnbase 33609 . . . . . . . . 9 ((𝑐 𝑑) ∈ (LLines‘𝐾) → (𝑐 𝑑) ∈ (Base‘𝐾))
418, 40syl 17 . . . . . . . 8 ((𝜑𝜓) → (𝑐 𝑑) ∈ (Base‘𝐾))
4221adantr 479 . . . . . . . 8 ((𝜑𝜓) → 𝑌 ∈ (Base‘𝐾))
4322, 4lattr 16825 . . . . . . . 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 1319 . . . . . . 7 ((𝜑𝜓) → ((𝑐 (𝑐 𝑑) ∧ (𝑐 𝑑) 𝑌) → 𝑐 𝑌))
4538, 44mpand 706 . . . . . 6 ((𝜑𝜓) → ((𝑐 𝑑) 𝑌𝑐 𝑌))
4629, 45mtod 187 . . . . 5 ((𝜑𝜓) → ¬ (𝑐 𝑑) 𝑌)
47463adant2 1072 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ (𝑐 𝑑) 𝑌)
48 nbrne2 4597 . . . 4 (((𝑃 𝑆) 𝑌 ∧ ¬ (𝑐 𝑑) 𝑌) → (𝑃 𝑆) ≠ (𝑐 𝑑))
4927, 47, 48syl2anc 690 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) ≠ (𝑐 𝑑))
5049necomd 2836 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) ≠ (𝑃 𝑆))
51 hlatl 33461 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
522, 51syl 17 . . . . 5 (𝜑𝐾 ∈ AtLat)
5352adantr 479 . . . 4 ((𝜑𝜓) → 𝐾 ∈ AtLat)
541dalempea 33726 . . . . . . 7 (𝜑𝑃𝐴)
551dalemsea 33729 . . . . . . 7 (𝜑𝑆𝐴)
5622, 5, 6hlatjcl 33467 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
572, 54, 55, 56syl3anc 1317 . . . . . 6 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
5857adantr 479 . . . . 5 ((𝜑𝜓) → (𝑃 𝑆) ∈ (Base‘𝐾))
59 dalem21.m . . . . . 6 = (meet‘𝐾)
6022, 59latmcl 16821 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑐 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ (Base‘𝐾))
6130, 41, 58, 60syl3anc 1317 . . . 4 ((𝜑𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ (Base‘𝐾))
621, 4, 5, 6, 10, 11dalemcea 33760 . . . . 5 (𝜑𝐶𝐴)
6362adantr 479 . . . 4 ((𝜑𝜓) → 𝐶𝐴)
647dalemclccjdd 33788 . . . . . 6 (𝜓𝐶 (𝑐 𝑑))
6564adantl 480 . . . . 5 ((𝜑𝜓) → 𝐶 (𝑐 𝑑))
661dalemclpjs 33734 . . . . . 6 (𝜑𝐶 (𝑃 𝑆))
6766adantr 479 . . . . 5 ((𝜑𝜓) → 𝐶 (𝑃 𝑆))
681, 6dalemceb 33738 . . . . . . 7 (𝜑𝐶 ∈ (Base‘𝐾))
6968adantr 479 . . . . . 6 ((𝜑𝜓) → 𝐶 ∈ (Base‘𝐾))
7022, 4, 59latlem12 16847 . . . . . 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 1319 . . . . 5 ((𝜑𝜓) → ((𝐶 (𝑐 𝑑) ∧ 𝐶 (𝑃 𝑆)) ↔ 𝐶 ((𝑐 𝑑) (𝑃 𝑆))))
7265, 67, 71mpbi2and 957 . . . 4 ((𝜑𝜓) → 𝐶 ((𝑐 𝑑) (𝑃 𝑆)))
73 eqid 2609 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
7422, 4, 73, 6atlen0 33411 . . . 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 1320 . . 3 ((𝜑𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ≠ (0.‘𝐾))
76753adant2 1072 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ≠ (0.‘𝐾))
7759, 73, 6, 392llnmat 33624 . 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 1325 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  meetcmee 16714  0.cp0 16806  Latclat 16814  Atomscatm 33364  AtLatcal 33365  HLchlt 33451  LLinesclln 33591  LPlanesclpl 33592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-lat 16815  df-clat 16877  df-oposet 33277  df-ol 33279  df-oml 33280  df-covers 33367  df-ats 33368  df-atl 33399  df-cvlat 33423  df-hlat 33452  df-llines 33598  df-lplanes 33599
This theorem is referenced by:  dalem22  33795
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