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Theorem 4atexlemtlw 35875
Description: Lemma for 4atexlem7 35883. (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 2771 . 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 35865 . 2 (𝜑𝐾 ∈ Lat)
534atexlemt 35861 . . 3 (𝜑𝑇𝐴)
6 4thatlem0.a . . . 4 𝐴 = (Atoms‘𝐾)
71, 6atbase 35098 . . 3 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
85, 7syl 17 . 2 (𝜑𝑇 ∈ (Base‘𝐾))
934atexlemk 35855 . . 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 35872 . . 3 (𝜑𝑈𝐴)
15 4thatlem0.v . . . 4 𝑉 = ((𝑃 𝑆) 𝑊)
163, 2, 10, 11, 6, 12, 13, 154atexlemv 35873 . . 3 (𝜑𝑉𝐴)
171, 10, 6hlatjcl 35175 . . 3 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) ∈ (Base‘𝐾))
189, 14, 16, 17syl3anc 1476 . 2 (𝜑 → (𝑈 𝑉) ∈ (Base‘𝐾))
193, 124atexlemwb 35867 . 2 (𝜑𝑊 ∈ (Base‘𝐾))
2034atexlemkc 35866 . . 3 (𝜑𝐾 ∈ CvLat)
213, 2, 10, 11, 6, 12, 13, 154atexlemunv 35874 . . 3 (𝜑𝑈𝑉)
2234atexlemutvt 35862 . . 3 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
236, 2, 10cvlsupr4 35154 . . 3 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇 (𝑈 𝑉))
2420, 14, 16, 5, 21, 22, 23syl132anc 1494 . 2 (𝜑𝑇 (𝑈 𝑉))
2534atexlemp 35858 . . . . . 6 (𝜑𝑃𝐴)
2634atexlemq 35859 . . . . . 6 (𝜑𝑄𝐴)
271, 10, 6hlatjcl 35175 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
289, 25, 26, 27syl3anc 1476 . . . . 5 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
291, 2, 11latmle2 17285 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
304, 28, 19, 29syl3anc 1476 . . . 4 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
3113, 30syl5eqbr 4821 . . 3 (𝜑𝑈 𝑊)
323, 10, 64atexlempsb 35868 . . . . 5 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
331, 2, 11latmle2 17285 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) 𝑊)
344, 32, 19, 33syl3anc 1476 . . . 4 (𝜑 → ((𝑃 𝑆) 𝑊) 𝑊)
3515, 34syl5eqbr 4821 . . 3 (𝜑𝑉 𝑊)
361, 6atbase 35098 . . . . 5 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
3714, 36syl 17 . . . 4 (𝜑𝑈 ∈ (Base‘𝐾))
381, 6atbase 35098 . . . . 5 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
3916, 38syl 17 . . . 4 (𝜑𝑉 ∈ (Base‘𝐾))
401, 2, 10latjle12 17270 . . . 4 ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑈 𝑊𝑉 𝑊) ↔ (𝑈 𝑉) 𝑊))
414, 37, 39, 19, 40syl13anc 1478 . . 3 (𝜑 → ((𝑈 𝑊𝑉 𝑊) ↔ (𝑈 𝑉) 𝑊))
4231, 35, 41mpbi2and 691 . 2 (𝜑 → (𝑈 𝑉) 𝑊)
431, 2, 4, 8, 18, 19, 24, 42lattrd 17266 1 (𝜑𝑇 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 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Latclat 17253  Atomscatm 35072  CvLatclc 35074  HLchlt 35159  LHypclh 35792
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 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  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-lhyp 35796
This theorem is referenced by:  4atexlemntlpq  35876  4atexlemnclw  35878  4atexlemcnd  35880
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