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Theorem 4atexlemunv 34370
 Description: Lemma for 4atexlem7 34379. (Contributed by NM, 21-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
4atexlemunv (𝜑𝑈𝑉)

Proof of Theorem 4atexlemunv
StepHypRef Expression
1 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))))
214atexlemnslpq 34360 . 2 (𝜑 → ¬ 𝑆 (𝑃 𝑄))
314atexlemk 34351 . . . . . . 7 (𝜑𝐾 ∈ HL)
414atexlemp 34354 . . . . . . 7 (𝜑𝑃𝐴)
514atexlems 34356 . . . . . . 7 (𝜑𝑆𝐴)
6 4thatlem0.l . . . . . . . 8 = (le‘𝐾)
7 4thatlem0.j . . . . . . . 8 = (join‘𝐾)
8 4thatlem0.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatlej2 33680 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑆 (𝑃 𝑆))
103, 4, 5, 9syl3anc 1318 . . . . . 6 (𝜑𝑆 (𝑃 𝑆))
1110adantr 480 . . . . 5 ((𝜑𝑈 = 𝑉) → 𝑆 (𝑃 𝑆))
12 4thatlem0.v . . . . . . . . 9 𝑉 = ((𝑃 𝑆) 𝑊)
1314atexlemkl 34361 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
141, 7, 84atexlempsb 34364 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
15 4thatlem0.h . . . . . . . . . . 11 𝐻 = (LHyp‘𝐾)
161, 154atexlemwb 34363 . . . . . . . . . 10 (𝜑𝑊 ∈ (Base‘𝐾))
17 eqid 2610 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
18 4thatlem0.m . . . . . . . . . . 11 = (meet‘𝐾)
1917, 6, 18latmle1 16899 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
2013, 14, 16, 19syl3anc 1318 . . . . . . . . 9 (𝜑 → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
2112, 20syl5eqbr 4618 . . . . . . . 8 (𝜑𝑉 (𝑃 𝑆))
2214atexlemkc 34362 . . . . . . . . 9 (𝜑𝐾 ∈ CvLat)
23 4thatlem0.u . . . . . . . . . 10 𝑈 = ((𝑃 𝑄) 𝑊)
241, 6, 7, 18, 8, 15, 23, 124atexlemv 34369 . . . . . . . . 9 (𝜑𝑉𝐴)
2517, 6, 18latmle2 16900 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) 𝑊)
2613, 14, 16, 25syl3anc 1318 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑆) 𝑊) 𝑊)
2712, 26syl5eqbr 4618 . . . . . . . . . 10 (𝜑𝑉 𝑊)
2814atexlempw 34353 . . . . . . . . . . 11 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2928simprd 478 . . . . . . . . . 10 (𝜑 → ¬ 𝑃 𝑊)
30 nbrne2 4603 . . . . . . . . . 10 ((𝑉 𝑊 ∧ ¬ 𝑃 𝑊) → 𝑉𝑃)
3127, 29, 30syl2anc 691 . . . . . . . . 9 (𝜑𝑉𝑃)
326, 7, 8cvlatexchb1 33639 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑉𝐴𝑆𝐴𝑃𝐴) ∧ 𝑉𝑃) → (𝑉 (𝑃 𝑆) ↔ (𝑃 𝑉) = (𝑃 𝑆)))
3322, 24, 5, 4, 31, 32syl131anc 1331 . . . . . . . 8 (𝜑 → (𝑉 (𝑃 𝑆) ↔ (𝑃 𝑉) = (𝑃 𝑆)))
3421, 33mpbid 221 . . . . . . 7 (𝜑 → (𝑃 𝑉) = (𝑃 𝑆))
3534adantr 480 . . . . . 6 ((𝜑𝑈 = 𝑉) → (𝑃 𝑉) = (𝑃 𝑆))
36 oveq2 6557 . . . . . . . 8 (𝑈 = 𝑉 → (𝑃 𝑈) = (𝑃 𝑉))
3736eqcomd 2616 . . . . . . 7 (𝑈 = 𝑉 → (𝑃 𝑉) = (𝑃 𝑈))
3814atexlemq 34355 . . . . . . . . . . 11 (𝜑𝑄𝐴)
3917, 7, 8hlatjcl 33671 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
403, 4, 38, 39syl3anc 1318 . . . . . . . . . 10 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
4117, 6, 18latmle1 16899 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
4213, 40, 16, 41syl3anc 1318 . . . . . . . . 9 (𝜑 → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
4323, 42syl5eqbr 4618 . . . . . . . 8 (𝜑𝑈 (𝑃 𝑄))
441, 6, 7, 18, 8, 15, 234atexlemu 34368 . . . . . . . . 9 (𝜑𝑈𝐴)
4517, 6, 18latmle2 16900 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
4613, 40, 16, 45syl3anc 1318 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
4723, 46syl5eqbr 4618 . . . . . . . . . 10 (𝜑𝑈 𝑊)
48 nbrne2 4603 . . . . . . . . . 10 ((𝑈 𝑊 ∧ ¬ 𝑃 𝑊) → 𝑈𝑃)
4947, 29, 48syl2anc 691 . . . . . . . . 9 (𝜑𝑈𝑃)
506, 7, 8cvlatexchb1 33639 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑄𝐴𝑃𝐴) ∧ 𝑈𝑃) → (𝑈 (𝑃 𝑄) ↔ (𝑃 𝑈) = (𝑃 𝑄)))
5122, 44, 38, 4, 49, 50syl131anc 1331 . . . . . . . 8 (𝜑 → (𝑈 (𝑃 𝑄) ↔ (𝑃 𝑈) = (𝑃 𝑄)))
5243, 51mpbid 221 . . . . . . 7 (𝜑 → (𝑃 𝑈) = (𝑃 𝑄))
5337, 52sylan9eqr 2666 . . . . . 6 ((𝜑𝑈 = 𝑉) → (𝑃 𝑉) = (𝑃 𝑄))
5435, 53eqtr3d 2646 . . . . 5 ((𝜑𝑈 = 𝑉) → (𝑃 𝑆) = (𝑃 𝑄))
5511, 54breqtrd 4609 . . . 4 ((𝜑𝑈 = 𝑉) → 𝑆 (𝑃 𝑄))
5655ex 449 . . 3 (𝜑 → (𝑈 = 𝑉𝑆 (𝑃 𝑄)))
5756necon3bd 2796 . 2 (𝜑 → (¬ 𝑆 (𝑃 𝑄) → 𝑈𝑉))
582, 57mpd 15 1 (𝜑𝑈𝑉)
 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:  4atexlemtlw  34371  4atexlemntlpq  34372  4atexlemc  34373  4atexlemnclw  34374
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