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Theorem 4atexlemcnd 40055
Description: Lemma for 4atexlem7 40058. (Contributed by NM, 24-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))
4thatlem0.l = (le‘𝐾)
4thatlem0.j = (join‘𝐾)
4thatlem0.m = (meet‘𝐾)
4thatlem0.a 𝐴 = (Atoms‘𝐾)
4thatlem0.h 𝐻 = (LHyp‘𝐾)
4thatlem0.u 𝑈 = ((𝑃 𝑄) 𝑊)
4thatlem0.v 𝑉 = ((𝑃 𝑆) 𝑊)
4thatlem0.c 𝐶 = ((𝑄 𝑇) (𝑃 𝑆))
4thatlem0.d 𝐷 = ((𝑅 𝑇) (𝑃 𝑆))
Assertion
Ref Expression
4atexlemcnd (𝜑𝐶𝐷)

Proof of Theorem 4atexlemcnd
StepHypRef Expression
1 4thatlem.ph . . . 4 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))
2 4thatlem0.l . . . 4 = (le‘𝐾)
3 4thatlem0.j . . . 4 = (join‘𝐾)
4 4thatlem0.m . . . 4 = (meet‘𝐾)
5 4thatlem0.a . . . 4 𝐴 = (Atoms‘𝐾)
6 4thatlem0.h . . . 4 𝐻 = (LHyp‘𝐾)
7 4thatlem0.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
8 4thatlem0.v . . . 4 𝑉 = ((𝑃 𝑆) 𝑊)
91, 2, 3, 4, 5, 6, 7, 84atexlemtlw 40050 . . 3 (𝜑𝑇 𝑊)
10 4thatlem0.c . . . 4 𝐶 = ((𝑄 𝑇) (𝑃 𝑆))
111, 2, 3, 4, 5, 6, 7, 8, 104atexlemnclw 40053 . . 3 (𝜑 → ¬ 𝐶 𝑊)
12 nbrne2 5168 . . 3 ((𝑇 𝑊 ∧ ¬ 𝐶 𝑊) → 𝑇𝐶)
139, 11, 12syl2anc 584 . 2 (𝜑𝑇𝐶)
1414atexlemk 40030 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
1514atexlemq 40034 . . . . . . . . 9 (𝜑𝑄𝐴)
1614atexlemt 40036 . . . . . . . . 9 (𝜑𝑇𝐴)
173, 5hlatjcom 39350 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) = (𝑇 𝑄))
1814, 15, 16, 17syl3anc 1370 . . . . . . . 8 (𝜑 → (𝑄 𝑇) = (𝑇 𝑄))
19 simp221 1313 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅𝐴)
201, 19sylbi 217 . . . . . . . . 9 (𝜑𝑅𝐴)
213, 5hlatjcom 39350 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑇𝐴) → (𝑅 𝑇) = (𝑇 𝑅))
2214, 20, 16, 21syl3anc 1370 . . . . . . . 8 (𝜑 → (𝑅 𝑇) = (𝑇 𝑅))
2318, 22oveq12d 7449 . . . . . . 7 (𝜑 → ((𝑄 𝑇) (𝑅 𝑇)) = ((𝑇 𝑄) (𝑇 𝑅)))
2414atexlemkc 40041 . . . . . . . . 9 (𝜑𝐾 ∈ CvLat)
2514atexlemp 40033 . . . . . . . . 9 (𝜑𝑃𝐴)
2614atexlempnq 40038 . . . . . . . . 9 (𝜑𝑃𝑄)
27 simp223 1315 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑃 𝑅) = (𝑄 𝑅))
281, 27sylbi 217 . . . . . . . . 9 (𝜑 → (𝑃 𝑅) = (𝑄 𝑅))
295, 3cvlsupr6 39329 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑄)
3029necomd 2994 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑄𝑅)
3124, 25, 15, 20, 26, 28, 30syl132anc 1387 . . . . . . . 8 (𝜑𝑄𝑅)
321, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 40051 . . . . . . . . 9 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
335, 3cvlsupr7 39330 . . . . . . . . . . . 12 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑅 𝑄))
3424, 25, 15, 20, 26, 28, 33syl132anc 1387 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) = (𝑅 𝑄))
353, 5hlatjcom 39350 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) = (𝑅 𝑄))
3614, 15, 20, 35syl3anc 1370 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑅) = (𝑅 𝑄))
3734, 36eqtr4d 2778 . . . . . . . . . 10 (𝜑 → (𝑃 𝑄) = (𝑄 𝑅))
3837breq2d 5160 . . . . . . . . 9 (𝜑 → (𝑇 (𝑃 𝑄) ↔ 𝑇 (𝑄 𝑅)))
3932, 38mtbid 324 . . . . . . . 8 (𝜑 → ¬ 𝑇 (𝑄 𝑅))
402, 3, 4, 52llnma2 39772 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑇𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 (𝑄 𝑅))) → ((𝑇 𝑄) (𝑇 𝑅)) = 𝑇)
4114, 15, 20, 16, 31, 39, 40syl132anc 1387 . . . . . . 7 (𝜑 → ((𝑇 𝑄) (𝑇 𝑅)) = 𝑇)
4223, 41eqtr2d 2776 . . . . . 6 (𝜑𝑇 = ((𝑄 𝑇) (𝑅 𝑇)))
4342adantr 480 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝑇 = ((𝑄 𝑇) (𝑅 𝑇)))
4414atexlemkl 40040 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
451, 3, 54atexlemqtb 40044 . . . . . . . . . 10 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
461, 3, 54atexlempsb 40043 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
47 eqid 2735 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
4847, 2, 4latmle1 18522 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
4944, 45, 46, 48syl3anc 1370 . . . . . . . . 9 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
5010, 49eqbrtrid 5183 . . . . . . . 8 (𝜑𝐶 (𝑄 𝑇))
5150adantr 480 . . . . . . 7 ((𝜑𝐶 = 𝐷) → 𝐶 (𝑄 𝑇))
52 simpr 484 . . . . . . . 8 ((𝜑𝐶 = 𝐷) → 𝐶 = 𝐷)
53 4thatlem0.d . . . . . . . . . 10 𝐷 = ((𝑅 𝑇) (𝑃 𝑆))
5447, 3, 5hlatjcl 39349 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑇𝐴) → (𝑅 𝑇) ∈ (Base‘𝐾))
5514, 20, 16, 54syl3anc 1370 . . . . . . . . . . 11 (𝜑 → (𝑅 𝑇) ∈ (Base‘𝐾))
5647, 2, 4latmle1 18522 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑅 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑅 𝑇) (𝑃 𝑆)) (𝑅 𝑇))
5744, 55, 46, 56syl3anc 1370 . . . . . . . . . 10 (𝜑 → ((𝑅 𝑇) (𝑃 𝑆)) (𝑅 𝑇))
5853, 57eqbrtrid 5183 . . . . . . . . 9 (𝜑𝐷 (𝑅 𝑇))
5958adantr 480 . . . . . . . 8 ((𝜑𝐶 = 𝐷) → 𝐷 (𝑅 𝑇))
6052, 59eqbrtrd 5170 . . . . . . 7 ((𝜑𝐶 = 𝐷) → 𝐶 (𝑅 𝑇))
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 40052 . . . . . . . . . 10 (𝜑𝐶𝐴)
6247, 5atbase 39271 . . . . . . . . . 10 (𝐶𝐴𝐶 ∈ (Base‘𝐾))
6361, 62syl 17 . . . . . . . . 9 (𝜑𝐶 ∈ (Base‘𝐾))
6447, 2, 4latlem12 18524 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑅 𝑇) ∈ (Base‘𝐾))) → ((𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑇)) ↔ 𝐶 ((𝑄 𝑇) (𝑅 𝑇))))
6544, 63, 45, 55, 64syl13anc 1371 . . . . . . . 8 (𝜑 → ((𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑇)) ↔ 𝐶 ((𝑄 𝑇) (𝑅 𝑇))))
6665adantr 480 . . . . . . 7 ((𝜑𝐶 = 𝐷) → ((𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑇)) ↔ 𝐶 ((𝑄 𝑇) (𝑅 𝑇))))
6751, 60, 66mpbi2and 712 . . . . . 6 ((𝜑𝐶 = 𝐷) → 𝐶 ((𝑄 𝑇) (𝑅 𝑇)))
68 hlatl 39342 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
6914, 68syl 17 . . . . . . . 8 (𝜑𝐾 ∈ AtLat)
7042, 16eqeltrrd 2840 . . . . . . . 8 (𝜑 → ((𝑄 𝑇) (𝑅 𝑇)) ∈ 𝐴)
712, 5atcmp 39293 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝐶𝐴 ∧ ((𝑄 𝑇) (𝑅 𝑇)) ∈ 𝐴) → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7269, 61, 70, 71syl3anc 1370 . . . . . . 7 (𝜑 → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7372adantr 480 . . . . . 6 ((𝜑𝐶 = 𝐷) → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7467, 73mpbid 232 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝐶 = ((𝑄 𝑇) (𝑅 𝑇)))
7543, 74eqtr4d 2778 . . . 4 ((𝜑𝐶 = 𝐷) → 𝑇 = 𝐶)
7675ex 412 . . 3 (𝜑 → (𝐶 = 𝐷𝑇 = 𝐶))
7776necon3d 2959 . 2 (𝜑 → (𝑇𝐶𝐶𝐷))
7813, 77mpd 15 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Latclat 18489  Atomscatm 39245  AtLatcal 39246  CvLatclc 39247  HLchlt 39332  LHypclh 39967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lhyp 39971
This theorem is referenced by:  4atexlemex4  40056
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