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

Theorem 4atexlemtlw 40050
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)
Assertion
Ref Expression
4atexlemtlw (𝜑𝑇 𝑊)

Proof of Theorem 4atexlemtlw
StepHypRef Expression
1 eqid 2735 . 2 (Base‘𝐾) = (Base‘𝐾)
2 4thatlem0.l . 2 = (le‘𝐾)
3 4thatlem.ph . . 3 (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))))
434atexlemkl 40040 . 2 (𝜑𝐾 ∈ Lat)
534atexlemt 40036 . . 3 (𝜑𝑇𝐴)
6 4thatlem0.a . . . 4 𝐴 = (Atoms‘𝐾)
71, 6atbase 39271 . . 3 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
85, 7syl 17 . 2 (𝜑𝑇 ∈ (Base‘𝐾))
934atexlemk 40030 . . 3 (𝜑𝐾 ∈ HL)
10 4thatlem0.j . . . 4 = (join‘𝐾)
11 4thatlem0.m . . . 4 = (meet‘𝐾)
12 4thatlem0.h . . . 4 𝐻 = (LHyp‘𝐾)
13 4thatlem0.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
143, 2, 10, 11, 6, 12, 134atexlemu 40047 . . 3 (𝜑𝑈𝐴)
15 4thatlem0.v . . . 4 𝑉 = ((𝑃 𝑆) 𝑊)
163, 2, 10, 11, 6, 12, 13, 154atexlemv 40048 . . 3 (𝜑𝑉𝐴)
171, 10, 6hlatjcl 39349 . . 3 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) ∈ (Base‘𝐾))
189, 14, 16, 17syl3anc 1370 . 2 (𝜑 → (𝑈 𝑉) ∈ (Base‘𝐾))
193, 124atexlemwb 40042 . 2 (𝜑𝑊 ∈ (Base‘𝐾))
2034atexlemkc 40041 . . 3 (𝜑𝐾 ∈ CvLat)
213, 2, 10, 11, 6, 12, 13, 154atexlemunv 40049 . . 3 (𝜑𝑈𝑉)
2234atexlemutvt 40037 . . 3 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
236, 2, 10cvlsupr4 39327 . . 3 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇 (𝑈 𝑉))
2420, 14, 16, 5, 21, 22, 23syl132anc 1387 . 2 (𝜑𝑇 (𝑈 𝑉))
2534atexlemp 40033 . . . . . 6 (𝜑𝑃𝐴)
2634atexlemq 40034 . . . . . 6 (𝜑𝑄𝐴)
271, 10, 6hlatjcl 39349 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
289, 25, 26, 27syl3anc 1370 . . . . 5 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
291, 2, 11latmle2 18523 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
304, 28, 19, 29syl3anc 1370 . . . 4 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
3113, 30eqbrtrid 5183 . . 3 (𝜑𝑈 𝑊)
323, 10, 64atexlempsb 40043 . . . . 5 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
331, 2, 11latmle2 18523 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) 𝑊)
344, 32, 19, 33syl3anc 1370 . . . 4 (𝜑 → ((𝑃 𝑆) 𝑊) 𝑊)
3515, 34eqbrtrid 5183 . . 3 (𝜑𝑉 𝑊)
361, 6atbase 39271 . . . . 5 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
3714, 36syl 17 . . . 4 (𝜑𝑈 ∈ (Base‘𝐾))
381, 6atbase 39271 . . . . 5 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
3916, 38syl 17 . . . 4 (𝜑𝑉 ∈ (Base‘𝐾))
401, 2, 10latjle12 18508 . . . 4 ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑈 𝑊𝑉 𝑊) ↔ (𝑈 𝑉) 𝑊))
414, 37, 39, 19, 40syl13anc 1371 . . 3 (𝜑 → ((𝑈 𝑊𝑉 𝑊) ↔ (𝑈 𝑉) 𝑊))
4231, 35, 41mpbi2and 712 . 2 (𝜑 → (𝑈 𝑉) 𝑊)
431, 2, 4, 8, 18, 19, 24, 42lattrd 18504 1 (𝜑𝑇 𝑊)
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  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-lhyp 39971
This theorem is referenced by:  4atexlemntlpq  40051  4atexlemnclw  40053  4atexlemcnd  40055
  Copyright terms: Public domain W3C validator