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Theorem 4atexlemnclw 35879
Description: Lemma for 4atexlem7 35884. (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 35866 . . . . 5 (𝜑𝐾 ∈ Lat)
4 4thatlem0.j . . . . . 6 = (join‘𝐾)
5 4thatlem0.a . . . . . 6 𝐴 = (Atoms‘𝐾)
62, 4, 54atexlemqtb 35870 . . . . 5 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
72, 4, 54atexlempsb 35869 . . . . 5 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
8 eqid 2771 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
9 4thatlem0.l . . . . . 6 = (le‘𝐾)
10 4thatlem0.m . . . . . 6 = (meet‘𝐾)
118, 9, 10latmle1 17284 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
123, 6, 7, 11syl3anc 1476 . . . 4 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
131, 12syl5eqbr 4822 . . 3 (𝜑𝐶 (𝑄 𝑇))
14 simp13r 1373 . . . . 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 35867 . . . . . 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 35874 . . . . . 6 (𝜑𝑉𝐴)
2124atexlemq 35860 . . . . . 6 (𝜑𝑄𝐴)
2224atexlemt 35862 . . . . . 6 (𝜑𝑇𝐴)
232, 9, 4, 10, 5, 17, 184atexlemu 35873 . . . . . . 7 (𝜑𝑈𝐴)
242, 9, 4, 10, 5, 17, 18, 194atexlemunv 35875 . . . . . . 7 (𝜑𝑈𝑉)
2524atexlemutvt 35863 . . . . . . 7 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
265, 4cvlsupr6 35156 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇𝑉)
2726necomd 2998 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑉𝑇)
2816, 23, 20, 22, 24, 25, 27syl132anc 1494 . . . . . 6 (𝜑𝑉𝑇)
299, 4, 5cvlatexch2 35146 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑉𝐴𝑄𝐴𝑇𝐴) ∧ 𝑉𝑇) → (𝑉 (𝑄 𝑇) → 𝑄 (𝑉 𝑇)))
3016, 20, 21, 22, 28, 29syl131anc 1489 . . . . 5 (𝜑 → (𝑉 (𝑄 𝑇) → 𝑄 (𝑉 𝑇)))
312, 174atexlemwb 35868 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘𝐾))
328, 9, 10latmle2 17285 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) 𝑊)
333, 7, 31, 32syl3anc 1476 . . . . . . . 8 (𝜑 → ((𝑃 𝑆) 𝑊) 𝑊)
3419, 33syl5eqbr 4822 . . . . . . 7 (𝜑𝑉 𝑊)
352, 9, 4, 10, 5, 17, 18, 194atexlemtlw 35876 . . . . . . 7 (𝜑𝑇 𝑊)
368, 5atbase 35098 . . . . . . . . 9 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
3720, 36syl 17 . . . . . . . 8 (𝜑𝑉 ∈ (Base‘𝐾))
388, 5atbase 35098 . . . . . . . . 9 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
3922, 38syl 17 . . . . . . . 8 (𝜑𝑇 ∈ (Base‘𝐾))
408, 9, 4latjle12 17270 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑉 𝑊𝑇 𝑊) ↔ (𝑉 𝑇) 𝑊))
413, 37, 39, 31, 40syl13anc 1478 . . . . . . 7 (𝜑 → ((𝑉 𝑊𝑇 𝑊) ↔ (𝑉 𝑇) 𝑊))
4234, 35, 41mpbi2and 691 . . . . . 6 (𝜑 → (𝑉 𝑇) 𝑊)
438, 5atbase 35098 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
4421, 43syl 17 . . . . . . 7 (𝜑𝑄 ∈ (Base‘𝐾))
4524atexlemk 35856 . . . . . . . 8 (𝜑𝐾 ∈ HL)
468, 4, 5hlatjcl 35176 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑇𝐴) → (𝑉 𝑇) ∈ (Base‘𝐾))
4745, 20, 22, 46syl3anc 1476 . . . . . . 7 (𝜑 → (𝑉 𝑇) ∈ (Base‘𝐾))
488, 9lattr 17264 . . . . . . 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 1478 . . . . . 6 (𝜑 → ((𝑄 (𝑉 𝑇) ∧ (𝑉 𝑇) 𝑊) → 𝑄 𝑊))
5042, 49mpan2d 674 . . . . 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 4807 . . 3 ((𝐶 (𝑄 𝑇) ∧ ¬ 𝑉 (𝑄 𝑇)) → 𝐶𝑉)
5413, 52, 53syl2anc 573 . 2 (𝜑𝐶𝑉)
5524atexlemw 35857 . . . 4 (𝜑𝑊𝐻)
5645, 55jca 501 . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
5724atexlempw 35858 . . 3 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5824atexlems 35861 . . 3 (𝜑𝑆𝐴)
592, 9, 4, 10, 5, 17, 18, 19, 14atexlemc 35878 . . 3 (𝜑𝐶𝐴)
602, 9, 4, 54atexlempns 35871 . . 3 (𝜑𝑃𝑆)
618, 9, 10latmle2 17285 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
623, 6, 7, 61syl3anc 1476 . . . 4 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
631, 62syl5eqbr 4822 . . 3 (𝜑𝐶 (𝑃 𝑆))
649, 4, 10, 5, 17, 19lhpat3 35855 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑆𝐴𝐶𝐴) ∧ (𝑃𝑆𝐶 (𝑃 𝑆))) → (¬ 𝐶 𝑊𝐶𝑉))
6556, 57, 58, 59, 60, 63, 64syl222anc 1492 . 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 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4787  cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Latclat 17253  Atomscatm 35072  CvLatclc 35074  HLchlt 35159  LHypclh 35793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35307  df-lplanes 35308  df-lhyp 35797
This theorem is referenced by:  4atexlemex2  35880  4atexlemcnd  35881
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