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Theorem 4atexlemcnd 34877
Description: Lemma for 4atexlem7 34880. (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 34872 . . 3 (𝜑𝑇 𝑊)
10 4thatlem0.c . . . 4 𝐶 = ((𝑄 𝑇) (𝑃 𝑆))
111, 2, 3, 4, 5, 6, 7, 8, 104atexlemnclw 34875 . . 3 (𝜑 → ¬ 𝐶 𝑊)
12 nbrne2 4643 . . 3 ((𝑇 𝑊 ∧ ¬ 𝐶 𝑊) → 𝑇𝐶)
139, 11, 12syl2anc 692 . 2 (𝜑𝑇𝐶)
1414atexlemk 34852 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
1514atexlemq 34856 . . . . . . . . 9 (𝜑𝑄𝐴)
1614atexlemt 34858 . . . . . . . . 9 (𝜑𝑇𝐴)
173, 5hlatjcom 34173 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) = (𝑇 𝑄))
1814, 15, 16, 17syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑄 𝑇) = (𝑇 𝑄))
19 simp221 1200 . . . . . . . . . 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 207 . . . . . . . . 9 (𝜑𝑅𝐴)
213, 5hlatjcom 34173 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑇𝐴) → (𝑅 𝑇) = (𝑇 𝑅))
2214, 20, 16, 21syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑅 𝑇) = (𝑇 𝑅))
2318, 22oveq12d 6633 . . . . . . 7 (𝜑 → ((𝑄 𝑇) (𝑅 𝑇)) = ((𝑇 𝑄) (𝑇 𝑅)))
2414atexlemkc 34863 . . . . . . . . 9 (𝜑𝐾 ∈ CvLat)
2514atexlemp 34855 . . . . . . . . 9 (𝜑𝑃𝐴)
2614atexlempnq 34860 . . . . . . . . 9 (𝜑𝑃𝑄)
27 simp223 1202 . . . . . . . . . 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 207 . . . . . . . . 9 (𝜑 → (𝑃 𝑅) = (𝑄 𝑅))
295, 3cvlsupr6 34153 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑄)
3029necomd 2845 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑄𝑅)
3124, 25, 15, 20, 26, 28, 30syl132anc 1341 . . . . . . . 8 (𝜑𝑄𝑅)
321, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 34873 . . . . . . . . 9 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
335, 3cvlsupr7 34154 . . . . . . . . . . . 12 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑅 𝑄))
3424, 25, 15, 20, 26, 28, 33syl132anc 1341 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) = (𝑅 𝑄))
353, 5hlatjcom 34173 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) = (𝑅 𝑄))
3614, 15, 20, 35syl3anc 1323 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑅) = (𝑅 𝑄))
3734, 36eqtr4d 2658 . . . . . . . . . 10 (𝜑 → (𝑃 𝑄) = (𝑄 𝑅))
3837breq2d 4635 . . . . . . . . 9 (𝜑 → (𝑇 (𝑃 𝑄) ↔ 𝑇 (𝑄 𝑅)))
3932, 38mtbid 314 . . . . . . . 8 (𝜑 → ¬ 𝑇 (𝑄 𝑅))
402, 3, 4, 52llnma2 34594 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑇𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 (𝑄 𝑅))) → ((𝑇 𝑄) (𝑇 𝑅)) = 𝑇)
4114, 15, 20, 16, 31, 39, 40syl132anc 1341 . . . . . . 7 (𝜑 → ((𝑇 𝑄) (𝑇 𝑅)) = 𝑇)
4223, 41eqtr2d 2656 . . . . . 6 (𝜑𝑇 = ((𝑄 𝑇) (𝑅 𝑇)))
4342adantr 481 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝑇 = ((𝑄 𝑇) (𝑅 𝑇)))
4414atexlemkl 34862 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
451, 3, 54atexlemqtb 34866 . . . . . . . . . 10 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
461, 3, 54atexlempsb 34865 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
47 eqid 2621 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
4847, 2, 4latmle1 17016 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
4944, 45, 46, 48syl3anc 1323 . . . . . . . . 9 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
5010, 49syl5eqbr 4658 . . . . . . . 8 (𝜑𝐶 (𝑄 𝑇))
5150adantr 481 . . . . . . 7 ((𝜑𝐶 = 𝐷) → 𝐶 (𝑄 𝑇))
52 simpr 477 . . . . . . . 8 ((𝜑𝐶 = 𝐷) → 𝐶 = 𝐷)
53 4thatlem0.d . . . . . . . . . 10 𝐷 = ((𝑅 𝑇) (𝑃 𝑆))
5447, 3, 5hlatjcl 34172 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑇𝐴) → (𝑅 𝑇) ∈ (Base‘𝐾))
5514, 20, 16, 54syl3anc 1323 . . . . . . . . . . 11 (𝜑 → (𝑅 𝑇) ∈ (Base‘𝐾))
5647, 2, 4latmle1 17016 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑅 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑅 𝑇) (𝑃 𝑆)) (𝑅 𝑇))
5744, 55, 46, 56syl3anc 1323 . . . . . . . . . 10 (𝜑 → ((𝑅 𝑇) (𝑃 𝑆)) (𝑅 𝑇))
5853, 57syl5eqbr 4658 . . . . . . . . 9 (𝜑𝐷 (𝑅 𝑇))
5958adantr 481 . . . . . . . 8 ((𝜑𝐶 = 𝐷) → 𝐷 (𝑅 𝑇))
6052, 59eqbrtrd 4645 . . . . . . 7 ((𝜑𝐶 = 𝐷) → 𝐶 (𝑅 𝑇))
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 34874 . . . . . . . . . 10 (𝜑𝐶𝐴)
6247, 5atbase 34095 . . . . . . . . . 10 (𝐶𝐴𝐶 ∈ (Base‘𝐾))
6361, 62syl 17 . . . . . . . . 9 (𝜑𝐶 ∈ (Base‘𝐾))
6447, 2, 4latlem12 17018 . . . . . . . . 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 1325 . . . . . . . 8 (𝜑 → ((𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑇)) ↔ 𝐶 ((𝑄 𝑇) (𝑅 𝑇))))
6665adantr 481 . . . . . . 7 ((𝜑𝐶 = 𝐷) → ((𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑇)) ↔ 𝐶 ((𝑄 𝑇) (𝑅 𝑇))))
6751, 60, 66mpbi2and 955 . . . . . 6 ((𝜑𝐶 = 𝐷) → 𝐶 ((𝑄 𝑇) (𝑅 𝑇)))
68 hlatl 34166 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
6914, 68syl 17 . . . . . . . 8 (𝜑𝐾 ∈ AtLat)
7042, 16eqeltrrd 2699 . . . . . . . 8 (𝜑 → ((𝑄 𝑇) (𝑅 𝑇)) ∈ 𝐴)
712, 5atcmp 34117 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝐶𝐴 ∧ ((𝑄 𝑇) (𝑅 𝑇)) ∈ 𝐴) → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7269, 61, 70, 71syl3anc 1323 . . . . . . 7 (𝜑 → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7372adantr 481 . . . . . 6 ((𝜑𝐶 = 𝐷) → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7467, 73mpbid 222 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝐶 = ((𝑄 𝑇) (𝑅 𝑇)))
7543, 74eqtr4d 2658 . . . 4 ((𝜑𝐶 = 𝐷) → 𝑇 = 𝐶)
7675ex 450 . . 3 (𝜑 → (𝐶 = 𝐷𝑇 = 𝐶))
7776necon3d 2811 . 2 (𝜑 → (𝑇𝐶𝐶𝐷))
7813, 77mpd 15 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  lecple 15888  joincjn 16884  meetcmee 16885  Latclat 16985  Atomscatm 34069  AtLatcal 34070  CvLatclc 34071  HLchlt 34156  LHypclh 34789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-preset 16868  df-poset 16886  df-plt 16898  df-lub 16914  df-glb 16915  df-join 16916  df-meet 16917  df-p0 16979  df-p1 16980  df-lat 16986  df-clat 17048  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-llines 34303  df-lplanes 34304  df-lhyp 34793
This theorem is referenced by:  4atexlemex4  34878
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