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Theorem dalem21 39676
Description: Lemma for dath 39718. 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 39605 . . 3 (𝜑𝐾 ∈ HL)
323ad2ant1 1132 . 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 39674 . . 3 ((𝜑𝜓) → (𝑐 𝑑) ∈ (LLines‘𝐾))
983adant2 1130 . 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 39635 . . 3 (𝜑 → (𝑃 𝑆) ∈ (LLines‘𝐾))
13123ad2ant1 1132 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) ∈ (LLines‘𝐾))
141, 4, 5, 6, 10, 11dalemply 39636 . . . . . . 7 (𝜑𝑃 𝑌)
1514adantr 480 . . . . . 6 ((𝜑𝑌 = 𝑍) → 𝑃 𝑌)
16 dalem21.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
171, 4, 5, 6, 16dalemsly 39637 . . . . . 6 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
181dalemkelat 39606 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
191, 6dalempeb 39621 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
201, 6dalemseb 39624 . . . . . . . 8 (𝜑𝑆 ∈ (Base‘𝐾))
211, 10dalemyeb 39631 . . . . . . . 8 (𝜑𝑌 ∈ (Base‘𝐾))
22 eqid 2734 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2322, 4, 5latjle12 18507 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2418, 19, 20, 21, 23syl13anc 1371 . . . . . . 7 (𝜑 → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2524adantr 480 . . . . . 6 ((𝜑𝑌 = 𝑍) → ((𝑃 𝑌𝑆 𝑌) ↔ (𝑃 𝑆) 𝑌))
2615, 17, 25mpbi2and 712 . . . . 5 ((𝜑𝑌 = 𝑍) → (𝑃 𝑆) 𝑌)
27263adant3 1131 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) 𝑌)
287dalem-ccly 39667 . . . . . . 7 (𝜓 → ¬ 𝑐 𝑌)
2928adantl 481 . . . . . 6 ((𝜑𝜓) → ¬ 𝑐 𝑌)
3018adantr 480 . . . . . . . 8 ((𝜑𝜓) → 𝐾 ∈ Lat)
317, 6dalemcceb 39671 . . . . . . . . 9 (𝜓𝑐 ∈ (Base‘𝐾))
3231adantl 481 . . . . . . . 8 ((𝜑𝜓) → 𝑐 ∈ (Base‘𝐾))
337dalemddea 39666 . . . . . . . . . 10 (𝜓𝑑𝐴)
3422, 6atbase 39270 . . . . . . . . . 10 (𝑑𝐴𝑑 ∈ (Base‘𝐾))
3533, 34syl 17 . . . . . . . . 9 (𝜓𝑑 ∈ (Base‘𝐾))
3635adantl 481 . . . . . . . 8 ((𝜑𝜓) → 𝑑 ∈ (Base‘𝐾))
3722, 4, 5latlej1 18505 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾)) → 𝑐 (𝑐 𝑑))
3830, 32, 36, 37syl3anc 1370 . . . . . . 7 ((𝜑𝜓) → 𝑐 (𝑐 𝑑))
39 eqid 2734 . . . . . . . . . 10 (LLines‘𝐾) = (LLines‘𝐾)
4022, 39llnbase 39491 . . . . . . . . 9 ((𝑐 𝑑) ∈ (LLines‘𝐾) → (𝑐 𝑑) ∈ (Base‘𝐾))
418, 40syl 17 . . . . . . . 8 ((𝜑𝜓) → (𝑐 𝑑) ∈ (Base‘𝐾))
4221adantr 480 . . . . . . . 8 ((𝜑𝜓) → 𝑌 ∈ (Base‘𝐾))
4322, 4lattr 18501 . . . . . . . 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 1371 . . . . . . 7 ((𝜑𝜓) → ((𝑐 (𝑐 𝑑) ∧ (𝑐 𝑑) 𝑌) → 𝑐 𝑌))
4538, 44mpand 695 . . . . . 6 ((𝜑𝜓) → ((𝑐 𝑑) 𝑌𝑐 𝑌))
4629, 45mtod 198 . . . . 5 ((𝜑𝜓) → ¬ (𝑐 𝑑) 𝑌)
47463adant2 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ (𝑐 𝑑) 𝑌)
48 nbrne2 5167 . . . 4 (((𝑃 𝑆) 𝑌 ∧ ¬ (𝑐 𝑑) 𝑌) → (𝑃 𝑆) ≠ (𝑐 𝑑))
4927, 47, 48syl2anc 584 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑆) ≠ (𝑐 𝑑))
5049necomd 2993 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) ≠ (𝑃 𝑆))
51 hlatl 39341 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
522, 51syl 17 . . . . 5 (𝜑𝐾 ∈ AtLat)
5352adantr 480 . . . 4 ((𝜑𝜓) → 𝐾 ∈ AtLat)
541dalempea 39608 . . . . . . 7 (𝜑𝑃𝐴)
551dalemsea 39611 . . . . . . 7 (𝜑𝑆𝐴)
5622, 5, 6hlatjcl 39348 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
572, 54, 55, 56syl3anc 1370 . . . . . 6 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
5857adantr 480 . . . . 5 ((𝜑𝜓) → (𝑃 𝑆) ∈ (Base‘𝐾))
59 dalem21.m . . . . . 6 = (meet‘𝐾)
6022, 59latmcl 18497 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑐 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ (Base‘𝐾))
6130, 41, 58, 60syl3anc 1370 . . . 4 ((𝜑𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ (Base‘𝐾))
621, 4, 5, 6, 10, 11dalemcea 39642 . . . . 5 (𝜑𝐶𝐴)
6362adantr 480 . . . 4 ((𝜑𝜓) → 𝐶𝐴)
647dalemclccjdd 39670 . . . . . 6 (𝜓𝐶 (𝑐 𝑑))
6564adantl 481 . . . . 5 ((𝜑𝜓) → 𝐶 (𝑐 𝑑))
661dalemclpjs 39616 . . . . . 6 (𝜑𝐶 (𝑃 𝑆))
6766adantr 480 . . . . 5 ((𝜑𝜓) → 𝐶 (𝑃 𝑆))
681, 6dalemceb 39620 . . . . . . 7 (𝜑𝐶 ∈ (Base‘𝐾))
6968adantr 480 . . . . . 6 ((𝜑𝜓) → 𝐶 ∈ (Base‘𝐾))
7022, 4, 59latlem12 18523 . . . . . 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 1371 . . . . 5 ((𝜑𝜓) → ((𝐶 (𝑐 𝑑) ∧ 𝐶 (𝑃 𝑆)) ↔ 𝐶 ((𝑐 𝑑) (𝑃 𝑆))))
7265, 67, 71mpbi2and 712 . . . 4 ((𝜑𝜓) → 𝐶 ((𝑐 𝑑) (𝑃 𝑆)))
73 eqid 2734 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
7422, 4, 73, 6atlen0 39291 . . . 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 1372 . . 3 ((𝜑𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ≠ (0.‘𝐾))
76753adant2 1130 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ≠ (0.‘𝐾))
7759, 73, 6, 392llnmat 39506 . 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 1377 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑑) (𝑃 𝑆)) ∈ 𝐴)
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  Latclat 18488  Atomscatm 39244  AtLatcal 39245  HLchlt 39331  LLinesclln 39473  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:  dalem22  39677
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