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Theorem 4atexlemtlw 34371
 Description: Lemma for 4atexlem7 34379. (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 2610 . 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 34361 . 2 (𝜑𝐾 ∈ Lat)
534atexlemt 34357 . . 3 (𝜑𝑇𝐴)
6 4thatlem0.a . . . 4 𝐴 = (Atoms‘𝐾)
71, 6atbase 33594 . . 3 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
85, 7syl 17 . 2 (𝜑𝑇 ∈ (Base‘𝐾))
934atexlemk 34351 . . 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 34368 . . 3 (𝜑𝑈𝐴)
15 4thatlem0.v . . . 4 𝑉 = ((𝑃 𝑆) 𝑊)
163, 2, 10, 11, 6, 12, 13, 154atexlemv 34369 . . 3 (𝜑𝑉𝐴)
171, 10, 6hlatjcl 33671 . . 3 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) ∈ (Base‘𝐾))
189, 14, 16, 17syl3anc 1318 . 2 (𝜑 → (𝑈 𝑉) ∈ (Base‘𝐾))
193, 124atexlemwb 34363 . 2 (𝜑𝑊 ∈ (Base‘𝐾))
2034atexlemkc 34362 . . 3 (𝜑𝐾 ∈ CvLat)
213, 2, 10, 11, 6, 12, 13, 154atexlemunv 34370 . . 3 (𝜑𝑈𝑉)
2234atexlemutvt 34358 . . 3 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
236, 2, 10cvlsupr4 33650 . . 3 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇 (𝑈 𝑉))
2420, 14, 16, 5, 21, 22, 23syl132anc 1336 . 2 (𝜑𝑇 (𝑈 𝑉))
2534atexlemp 34354 . . . . . 6 (𝜑𝑃𝐴)
2634atexlemq 34355 . . . . . 6 (𝜑𝑄𝐴)
271, 10, 6hlatjcl 33671 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
289, 25, 26, 27syl3anc 1318 . . . . 5 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
291, 2, 11latmle2 16900 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
304, 28, 19, 29syl3anc 1318 . . . 4 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
3113, 30syl5eqbr 4618 . . 3 (𝜑𝑈 𝑊)
323, 10, 64atexlempsb 34364 . . . . 5 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
331, 2, 11latmle2 16900 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) 𝑊)
344, 32, 19, 33syl3anc 1318 . . . 4 (𝜑 → ((𝑃 𝑆) 𝑊) 𝑊)
3515, 34syl5eqbr 4618 . . 3 (𝜑𝑉 𝑊)
361, 6atbase 33594 . . . . 5 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
3714, 36syl 17 . . . 4 (𝜑𝑈 ∈ (Base‘𝐾))
381, 6atbase 33594 . . . . 5 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
3916, 38syl 17 . . . 4 (𝜑𝑉 ∈ (Base‘𝐾))
401, 2, 10latjle12 16885 . . . 4 ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑈 𝑊𝑉 𝑊) ↔ (𝑈 𝑉) 𝑊))
414, 37, 39, 19, 40syl13anc 1320 . . 3 (𝜑 → ((𝑈 𝑊𝑉 𝑊) ↔ (𝑈 𝑉) 𝑊))
4231, 35, 41mpbi2and 958 . 2 (𝜑 → (𝑈 𝑉) 𝑊)
431, 2, 4, 8, 18, 19, 24, 42lattrd 16881 1 (𝜑𝑇 𝑊)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868  Atomscatm 33568  CvLatclc 33570  HLchlt 33655  LHypclh 34288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-lhyp 34292 This theorem is referenced by:  4atexlemntlpq  34372  4atexlemnclw  34374  4atexlemcnd  34376
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