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Theorem 4atexlemc 34373
 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)
4thatlem0.c 𝐶 = ((𝑄 𝑇) (𝑃 𝑆))
Assertion
Ref Expression
4atexlemc (𝜑𝐶𝐴)

Proof of Theorem 4atexlemc
StepHypRef Expression
1 4thatlem0.c . . 3 𝐶 = ((𝑄 𝑇) (𝑃 𝑆))
2 4thatlem.ph . . . . 5 (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 34361 . . . 4 (𝜑𝐾 ∈ Lat)
4 4thatlem0.j . . . . 5 = (join‘𝐾)
5 4thatlem0.a . . . . 5 𝐴 = (Atoms‘𝐾)
62, 4, 54atexlemqtb 34365 . . . 4 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
72, 4, 54atexlempsb 34364 . . . 4 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
8 eqid 2610 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
9 4thatlem0.m . . . . 5 = (meet‘𝐾)
108, 9latmcom 16898 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) = ((𝑃 𝑆) (𝑄 𝑇)))
113, 6, 7, 10syl3anc 1318 . . 3 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) = ((𝑃 𝑆) (𝑄 𝑇)))
121, 11syl5eq 2656 . 2 (𝜑𝐶 = ((𝑃 𝑆) (𝑄 𝑇)))
1324atexlemk 34351 . . 3 (𝜑𝐾 ∈ HL)
1424atexlemp 34354 . . 3 (𝜑𝑃𝐴)
1524atexlems 34356 . . 3 (𝜑𝑆𝐴)
1624atexlemq 34355 . . 3 (𝜑𝑄𝐴)
1724atexlemt 34357 . . 3 (𝜑𝑇𝐴)
18 4thatlem0.l . . . 4 = (le‘𝐾)
192, 18, 4, 54atexlempns 34366 . . 3 (𝜑𝑃𝑆)
20 4thatlem0.h . . . . 5 𝐻 = (LHyp‘𝐾)
21 4thatlem0.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
22 4thatlem0.v . . . . 5 𝑉 = ((𝑃 𝑆) 𝑊)
232, 18, 4, 9, 5, 20, 21, 224atexlemntlpq 34372 . . . 4 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
2418, 4, 5atnlej2 33684 . . . . 5 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑇 (𝑃 𝑄)) → 𝑇𝑄)
2524necomd 2837 . . . 4 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑇 (𝑃 𝑄)) → 𝑄𝑇)
2613, 17, 14, 16, 23, 25syl131anc 1331 . . 3 (𝜑𝑄𝑇)
2724atexlempnq 34359 . . . 4 (𝜑𝑃𝑄)
2824atexlemnslpq 34360 . . . 4 (𝜑 → ¬ 𝑆 (𝑃 𝑄))
2918, 4, 54atlem0ae 33898 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑄 (𝑃 𝑆))
3013, 14, 16, 15, 27, 28, 29syl132anc 1336 . . 3 (𝜑 → ¬ 𝑄 (𝑃 𝑆))
318, 5atbase 33594 . . . . 5 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
3217, 31syl 17 . . . 4 (𝜑𝑇 ∈ (Base‘𝐾))
332, 18, 4, 9, 5, 20, 214atexlemu 34368 . . . . 5 (𝜑𝑈𝐴)
342, 18, 4, 9, 5, 20, 21, 224atexlemv 34369 . . . . 5 (𝜑𝑉𝐴)
358, 4, 5hlatjcl 33671 . . . . 5 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) ∈ (Base‘𝐾))
3613, 33, 34, 35syl3anc 1318 . . . 4 (𝜑 → (𝑈 𝑉) ∈ (Base‘𝐾))
378, 5atbase 33594 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3816, 37syl 17 . . . . 5 (𝜑𝑄 ∈ (Base‘𝐾))
398, 4latjcl 16874 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾))
403, 7, 38, 39syl3anc 1318 . . . 4 (𝜑 → ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾))
4124atexlemkc 34362 . . . . 5 (𝜑𝐾 ∈ CvLat)
422, 18, 4, 9, 5, 20, 21, 224atexlemunv 34370 . . . . 5 (𝜑𝑈𝑉)
4324atexlemutvt 34358 . . . . 5 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
445, 18, 4cvlsupr4 33650 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇 (𝑈 𝑉))
4541, 33, 34, 17, 42, 43, 44syl132anc 1336 . . . 4 (𝜑𝑇 (𝑈 𝑉))
468, 4, 5hlatjcl 33671 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
4713, 14, 16, 46syl3anc 1318 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
482, 204atexlemwb 34363 . . . . . . . 8 (𝜑𝑊 ∈ (Base‘𝐾))
498, 18, 9latmle1 16899 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
503, 47, 48, 49syl3anc 1318 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
5121, 50syl5eqbr 4618 . . . . . 6 (𝜑𝑈 (𝑃 𝑄))
528, 18, 9latmle1 16899 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
533, 7, 48, 52syl3anc 1318 . . . . . . 7 (𝜑 → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
5422, 53syl5eqbr 4618 . . . . . 6 (𝜑𝑉 (𝑃 𝑆))
558, 5atbase 33594 . . . . . . . 8 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
5633, 55syl 17 . . . . . . 7 (𝜑𝑈 ∈ (Base‘𝐾))
578, 5atbase 33594 . . . . . . . 8 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
5834, 57syl 17 . . . . . . 7 (𝜑𝑉 ∈ (Base‘𝐾))
598, 18, 4latjlej12 16890 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) ∧ (𝑉 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾))) → ((𝑈 (𝑃 𝑄) ∧ 𝑉 (𝑃 𝑆)) → (𝑈 𝑉) ((𝑃 𝑄) (𝑃 𝑆))))
603, 56, 47, 58, 7, 59syl122anc 1327 . . . . . 6 (𝜑 → ((𝑈 (𝑃 𝑄) ∧ 𝑉 (𝑃 𝑆)) → (𝑈 𝑉) ((𝑃 𝑄) (𝑃 𝑆))))
6151, 54, 60mp2and 711 . . . . 5 (𝜑 → (𝑈 𝑉) ((𝑃 𝑄) (𝑃 𝑆)))
624, 5hlatjass 33674 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
6313, 14, 16, 15, 62syl13anc 1320 . . . . . 6 (𝜑 → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
648, 5atbase 33594 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
6514, 64syl 17 . . . . . . 7 (𝜑𝑃 ∈ (Base‘𝐾))
668, 5atbase 33594 . . . . . . . 8 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
6715, 66syl 17 . . . . . . 7 (𝜑𝑆 ∈ (Base‘𝐾))
688, 4latj32 16920 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → ((𝑃 𝑄) 𝑆) = ((𝑃 𝑆) 𝑄))
693, 65, 38, 67, 68syl13anc 1320 . . . . . 6 (𝜑 → ((𝑃 𝑄) 𝑆) = ((𝑃 𝑆) 𝑄))
708, 4latjjdi 16926 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (𝑃 (𝑄 𝑆)) = ((𝑃 𝑄) (𝑃 𝑆)))
713, 65, 38, 67, 70syl13anc 1320 . . . . . 6 (𝜑 → (𝑃 (𝑄 𝑆)) = ((𝑃 𝑄) (𝑃 𝑆)))
7263, 69, 713eqtr3rd 2653 . . . . 5 (𝜑 → ((𝑃 𝑄) (𝑃 𝑆)) = ((𝑃 𝑆) 𝑄))
7361, 72breqtrd 4609 . . . 4 (𝜑 → (𝑈 𝑉) ((𝑃 𝑆) 𝑄))
748, 18, 3, 32, 36, 40, 45, 73lattrd 16881 . . 3 (𝜑𝑇 ((𝑃 𝑆) 𝑄))
7518, 4, 9, 52atmat 33865 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) ∧ (𝑄𝐴𝑇𝐴𝑃𝑆) ∧ (𝑄𝑇 ∧ ¬ 𝑄 (𝑃 𝑆) ∧ 𝑇 ((𝑃 𝑆) 𝑄))) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴)
7613, 14, 15, 16, 17, 19, 26, 30, 74, 75syl333anc 1350 . 2 (𝜑 → ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴)
7712, 76eqeltrd 2688 1 (𝜑𝐶𝐴)
 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-llines 33802  df-lplanes 33803  df-lhyp 34292 This theorem is referenced by:  4atexlemnclw  34374  4atexlemex2  34375  4atexlemcnd  34376
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