Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atexlemnclw Structured version   Visualization version   GIF version

Theorem 4atexlemnclw 38310
Description: Lemma for 4atexlem7 38315. (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 38297 . . . . 5 (𝜑𝐾 ∈ Lat)
4 4thatlem0.j . . . . . 6 = (join‘𝐾)
5 4thatlem0.a . . . . . 6 𝐴 = (Atoms‘𝐾)
62, 4, 54atexlemqtb 38301 . . . . 5 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
72, 4, 54atexlempsb 38300 . . . . 5 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
8 eqid 2736 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
9 4thatlem0.l . . . . . 6 = (le‘𝐾)
10 4thatlem0.m . . . . . 6 = (meet‘𝐾)
118, 9, 10latmle1 18256 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
123, 6, 7, 11syl3anc 1370 . . . 4 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
131, 12eqbrtrid 5121 . . 3 (𝜑𝐶 (𝑄 𝑇))
14 simp13r 1288 . . . . 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 216 . . . 4 (𝜑 → ¬ 𝑄 𝑊)
1624atexlemkc 38298 . . . . . 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 38305 . . . . . 6 (𝜑𝑉𝐴)
2124atexlemq 38291 . . . . . 6 (𝜑𝑄𝐴)
2224atexlemt 38293 . . . . . 6 (𝜑𝑇𝐴)
232, 9, 4, 10, 5, 17, 184atexlemu 38304 . . . . . . 7 (𝜑𝑈𝐴)
242, 9, 4, 10, 5, 17, 18, 194atexlemunv 38306 . . . . . . 7 (𝜑𝑈𝑉)
2524atexlemutvt 38294 . . . . . . 7 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
265, 4cvlsupr6 37586 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇𝑉)
2726necomd 2996 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑉𝑇)
2816, 23, 20, 22, 24, 25, 27syl132anc 1387 . . . . . 6 (𝜑𝑉𝑇)
299, 4, 5cvlatexch2 37576 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑉𝐴𝑄𝐴𝑇𝐴) ∧ 𝑉𝑇) → (𝑉 (𝑄 𝑇) → 𝑄 (𝑉 𝑇)))
3016, 20, 21, 22, 28, 29syl131anc 1382 . . . . 5 (𝜑 → (𝑉 (𝑄 𝑇) → 𝑄 (𝑉 𝑇)))
312, 174atexlemwb 38299 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘𝐾))
328, 9, 10latmle2 18257 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) 𝑊)
333, 7, 31, 32syl3anc 1370 . . . . . . . 8 (𝜑 → ((𝑃 𝑆) 𝑊) 𝑊)
3419, 33eqbrtrid 5121 . . . . . . 7 (𝜑𝑉 𝑊)
352, 9, 4, 10, 5, 17, 18, 194atexlemtlw 38307 . . . . . . 7 (𝜑𝑇 𝑊)
368, 5atbase 37528 . . . . . . . . 9 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
3720, 36syl 17 . . . . . . . 8 (𝜑𝑉 ∈ (Base‘𝐾))
388, 5atbase 37528 . . . . . . . . 9 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
3922, 38syl 17 . . . . . . . 8 (𝜑𝑇 ∈ (Base‘𝐾))
408, 9, 4latjle12 18242 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑉 𝑊𝑇 𝑊) ↔ (𝑉 𝑇) 𝑊))
413, 37, 39, 31, 40syl13anc 1371 . . . . . . 7 (𝜑 → ((𝑉 𝑊𝑇 𝑊) ↔ (𝑉 𝑇) 𝑊))
4234, 35, 41mpbi2and 709 . . . . . 6 (𝜑 → (𝑉 𝑇) 𝑊)
438, 5atbase 37528 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
4421, 43syl 17 . . . . . . 7 (𝜑𝑄 ∈ (Base‘𝐾))
4524atexlemk 38287 . . . . . . . 8 (𝜑𝐾 ∈ HL)
468, 4, 5hlatjcl 37606 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑇𝐴) → (𝑉 𝑇) ∈ (Base‘𝐾))
4745, 20, 22, 46syl3anc 1370 . . . . . . 7 (𝜑 → (𝑉 𝑇) ∈ (Base‘𝐾))
488, 9lattr 18236 . . . . . . 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 1371 . . . . . 6 (𝜑 → ((𝑄 (𝑉 𝑇) ∧ (𝑉 𝑇) 𝑊) → 𝑄 𝑊))
5042, 49mpan2d 691 . . . . 5 (𝜑 → (𝑄 (𝑉 𝑇) → 𝑄 𝑊))
5130, 50syld 47 . . . 4 (𝜑 → (𝑉 (𝑄 𝑇) → 𝑄 𝑊))
5215, 51mtod 197 . . 3 (𝜑 → ¬ 𝑉 (𝑄 𝑇))
53 nbrne2 5106 . . 3 ((𝐶 (𝑄 𝑇) ∧ ¬ 𝑉 (𝑄 𝑇)) → 𝐶𝑉)
5413, 52, 53syl2anc 584 . 2 (𝜑𝐶𝑉)
5524atexlemw 38288 . . . 4 (𝜑𝑊𝐻)
5645, 55jca 512 . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
5724atexlempw 38289 . . 3 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5824atexlems 38292 . . 3 (𝜑𝑆𝐴)
592, 9, 4, 10, 5, 17, 18, 19, 14atexlemc 38309 . . 3 (𝜑𝐶𝐴)
602, 9, 4, 54atexlempns 38302 . . 3 (𝜑𝑃𝑆)
618, 9, 10latmle2 18257 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
623, 6, 7, 61syl3anc 1370 . . . 4 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑆))
631, 62eqbrtrid 5121 . . 3 (𝜑𝐶 (𝑃 𝑆))
649, 4, 10, 5, 17, 19lhpat3 38286 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑆𝐴𝐶𝐴) ∧ (𝑃𝑆𝐶 (𝑃 𝑆))) → (¬ 𝐶 𝑊𝐶𝑉))
6556, 57, 58, 59, 60, 63, 64syl222anc 1385 . 2 (𝜑 → (¬ 𝐶 𝑊𝐶𝑉))
6654, 65mpbird 256 1 (𝜑 → ¬ 𝐶 𝑊)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wne 2940   class class class wbr 5086  cfv 6465  (class class class)co 7316  Basecbs 16986  lecple 17043  joincjn 18103  meetcmee 18104  Latclat 18223  Atomscatm 37502  CvLatclc 37504  HLchlt 37589  LHypclh 38224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7273  df-ov 7319  df-oprab 7320  df-proset 18087  df-poset 18105  df-plt 18122  df-lub 18138  df-glb 18139  df-join 18140  df-meet 18141  df-p0 18217  df-p1 18218  df-lat 18224  df-clat 18291  df-oposet 37415  df-ol 37417  df-oml 37418  df-covers 37505  df-ats 37506  df-atl 37537  df-cvlat 37561  df-hlat 37590  df-llines 37738  df-lplanes 37739  df-lhyp 38228
This theorem is referenced by:  4atexlemex2  38311  4atexlemcnd  38312
  Copyright terms: Public domain W3C validator