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Theorem 4atexlemnclw 34875
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))
Assertion
Ref Expression
4atexlemnclw (𝜑 → ¬ 𝐶 𝑊)

Proof of Theorem 4atexlemnclw
StepHypRef Expression
1 4thatlem0.c . . . 4 𝐶 = ((𝑄 𝑇) (𝑃 𝑆))
2 4thatlem.ph . . . . . 6 (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))))
324atexlemkl 34862 . . . . 5 (𝜑𝐾 ∈ Lat)
4 4thatlem0.j . . . . . 6 = (join‘𝐾)
5 4thatlem0.a . . . . . 6 𝐴 = (Atoms‘𝐾)
62, 4, 54atexlemqtb 34866 . . . . 5 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
72, 4, 54atexlempsb 34865 . . . . 5 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
8 eqid 2621 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
9 4thatlem0.l . . . . . 6 = (le‘𝐾)
10 4thatlem0.m . . . . . 6 = (meet‘𝐾)
118, 9, 10latmle1 17016 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
123, 6, 7, 11syl3anc 1323 . . . 4 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
131, 12syl5eqbr 4658 . . 3 (𝜑𝐶 (𝑄 𝑇))
14 simp13r 1175 . . . . 5 ((((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))) → ¬ 𝑄 𝑊)
152, 14sylbi 207 . . . 4 (𝜑 → ¬ 𝑄 𝑊)
1624atexlemkc 34863 . . . . . 6 (𝜑𝐾 ∈ CvLat)
17 4thatlem0.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
18 4thatlem0.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
19 4thatlem0.v . . . . . . 7 𝑉 = ((𝑃 𝑆) 𝑊)
202, 9, 4, 10, 5, 17, 18, 194atexlemv 34870 . . . . . 6 (𝜑𝑉𝐴)
2124atexlemq 34856 . . . . . 6 (𝜑𝑄𝐴)
2224atexlemt 34858 . . . . . 6 (𝜑𝑇𝐴)
232, 9, 4, 10, 5, 17, 184atexlemu 34869 . . . . . . 7 (𝜑𝑈𝐴)
242, 9, 4, 10, 5, 17, 18, 194atexlemunv 34871 . . . . . . 7 (𝜑𝑈𝑉)
2524atexlemutvt 34859 . . . . . . 7 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
265, 4cvlsupr6 34153 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇𝑉)
2726necomd 2845 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑉𝑇)
2816, 23, 20, 22, 24, 25, 27syl132anc 1341 . . . . . 6 (𝜑𝑉𝑇)
299, 4, 5cvlatexch2 34143 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑉𝐴𝑄𝐴𝑇𝐴) ∧ 𝑉𝑇) → (𝑉 (𝑄 𝑇) → 𝑄 (𝑉 𝑇)))
3016, 20, 21, 22, 28, 29syl131anc 1336 . . . . 5 (𝜑 → (𝑉 (𝑄 𝑇) → 𝑄 (𝑉 𝑇)))
312, 174atexlemwb 34864 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘𝐾))
328, 9, 10latmle2 17017 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) 𝑊)
333, 7, 31, 32syl3anc 1323 . . . . . . . 8 (𝜑 → ((𝑃 𝑆) 𝑊) 𝑊)
3419, 33syl5eqbr 4658 . . . . . . 7 (𝜑𝑉 𝑊)
352, 9, 4, 10, 5, 17, 18, 194atexlemtlw 34872 . . . . . . 7 (𝜑𝑇 𝑊)
368, 5atbase 34095 . . . . . . . . 9 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
3720, 36syl 17 . . . . . . . 8 (𝜑𝑉 ∈ (Base‘𝐾))
388, 5atbase 34095 . . . . . . . . 9 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
3922, 38syl 17 . . . . . . . 8 (𝜑𝑇 ∈ (Base‘𝐾))
408, 9, 4latjle12 17002 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑉 𝑊𝑇 𝑊) ↔ (𝑉 𝑇) 𝑊))
413, 37, 39, 31, 40syl13anc 1325 . . . . . . 7 (𝜑 → ((𝑉 𝑊𝑇 𝑊) ↔ (𝑉 𝑇) 𝑊))
4234, 35, 41mpbi2and 955 . . . . . 6 (𝜑 → (𝑉 𝑇) 𝑊)
438, 5atbase 34095 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
4421, 43syl 17 . . . . . . 7 (𝜑𝑄 ∈ (Base‘𝐾))
4524atexlemk 34852 . . . . . . . 8 (𝜑𝐾 ∈ HL)
468, 4, 5hlatjcl 34172 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑇𝐴) → (𝑉 𝑇) ∈ (Base‘𝐾))
4745, 20, 22, 46syl3anc 1323 . . . . . . 7 (𝜑 → (𝑉 𝑇) ∈ (Base‘𝐾))
488, 9lattr 16996 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑉 𝑇) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 (𝑉 𝑇) ∧ (𝑉 𝑇) 𝑊) → 𝑄 𝑊))
493, 44, 47, 31, 48syl13anc 1325 . . . . . 6 (𝜑 → ((𝑄 (𝑉 𝑇) ∧ (𝑉 𝑇) 𝑊) → 𝑄 𝑊))
5042, 49mpan2d 709 . . . . 5 (𝜑 → (𝑄 (𝑉 𝑇) → 𝑄 𝑊))
5130, 50syld 47 . . . 4 (𝜑 → (𝑉 (𝑄 𝑇) → 𝑄 𝑊))
5215, 51mtod 189 . . 3 (𝜑 → ¬ 𝑉 (𝑄 𝑇))
53 nbrne2 4643 . . 3 ((𝐶 (𝑄 𝑇) ∧ ¬ 𝑉 (𝑄 𝑇)) → 𝐶𝑉)
5413, 52, 53syl2anc 692 . 2 (𝜑𝐶𝑉)
5524atexlemw 34853 . . . 4 (𝜑𝑊𝐻)
5645, 55jca 554 . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
5724atexlempw 34854 . . 3 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5824atexlems 34857 . . 3 (𝜑𝑆𝐴)
592, 9, 4, 10, 5, 17, 18, 19, 14atexlemc 34874 . . 3 (𝜑𝐶𝐴)
602, 9, 4, 54atexlempns 34867 . . 3 (𝜑𝑃𝑆)
618, 9, 10latmle2 17017 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
623, 6, 7, 61syl3anc 1323 . . . 4 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
631, 62syl5eqbr 4658 . . 3 (𝜑𝐶 (𝑃 𝑆))
649, 4, 10, 5, 17, 19lhpat3 34851 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑆𝐴𝐶𝐴) ∧ (𝑃𝑆𝐶 (𝑃 𝑆))) → (¬ 𝐶 𝑊𝐶𝑉))
6556, 57, 58, 59, 60, 63, 64syl222anc 1339 . 2 (𝜑 → (¬ 𝐶 𝑊𝐶𝑉))
6654, 65mpbird 247 1 (𝜑 → ¬ 𝐶 𝑊)
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  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:  4atexlemex2  34876  4atexlemcnd  34877
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