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Theorem 4atexlemc 38532
Description: Lemma for 4atexlem7 38538. (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 38520 . . . 4 (𝜑𝐾 ∈ Lat)
4 4thatlem0.j . . . . 5 = (join‘𝐾)
5 4thatlem0.a . . . . 5 𝐴 = (Atoms‘𝐾)
62, 4, 54atexlemqtb 38524 . . . 4 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
72, 4, 54atexlempsb 38523 . . . 4 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
8 eqid 2736 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
9 4thatlem0.m . . . . 5 = (meet‘𝐾)
108, 9latmcom 18352 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) = ((𝑃 𝑆) (𝑄 𝑇)))
113, 6, 7, 10syl3anc 1371 . . 3 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) = ((𝑃 𝑆) (𝑄 𝑇)))
121, 11eqtrid 2788 . 2 (𝜑𝐶 = ((𝑃 𝑆) (𝑄 𝑇)))
1324atexlemk 38510 . . 3 (𝜑𝐾 ∈ HL)
1424atexlemp 38513 . . 3 (𝜑𝑃𝐴)
1524atexlems 38515 . . 3 (𝜑𝑆𝐴)
1624atexlemq 38514 . . 3 (𝜑𝑄𝐴)
1724atexlemt 38516 . . 3 (𝜑𝑇𝐴)
18 4thatlem0.l . . . 4 = (le‘𝐾)
192, 18, 4, 54atexlempns 38525 . . 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 38531 . . . 4 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
2418, 4, 5atnlej2 37843 . . . . 5 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑇 (𝑃 𝑄)) → 𝑇𝑄)
2524necomd 2999 . . . 4 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑇 (𝑃 𝑄)) → 𝑄𝑇)
2613, 17, 14, 16, 23, 25syl131anc 1383 . . 3 (𝜑𝑄𝑇)
2724atexlempnq 38518 . . . 4 (𝜑𝑃𝑄)
2824atexlemnslpq 38519 . . . 4 (𝜑 → ¬ 𝑆 (𝑃 𝑄))
2918, 4, 54atlem0ae 38057 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑄 (𝑃 𝑆))
3013, 14, 16, 15, 27, 28, 29syl132anc 1388 . . 3 (𝜑 → ¬ 𝑄 (𝑃 𝑆))
318, 5atbase 37751 . . . . 5 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
3217, 31syl 17 . . . 4 (𝜑𝑇 ∈ (Base‘𝐾))
332, 18, 4, 9, 5, 20, 214atexlemu 38527 . . . . 5 (𝜑𝑈𝐴)
342, 18, 4, 9, 5, 20, 21, 224atexlemv 38528 . . . . 5 (𝜑𝑉𝐴)
358, 4, 5hlatjcl 37829 . . . . 5 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) ∈ (Base‘𝐾))
3613, 33, 34, 35syl3anc 1371 . . . 4 (𝜑 → (𝑈 𝑉) ∈ (Base‘𝐾))
378, 5atbase 37751 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3816, 37syl 17 . . . . 5 (𝜑𝑄 ∈ (Base‘𝐾))
398, 4latjcl 18328 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾))
403, 7, 38, 39syl3anc 1371 . . . 4 (𝜑 → ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾))
4124atexlemkc 38521 . . . . 5 (𝜑𝐾 ∈ CvLat)
422, 18, 4, 9, 5, 20, 21, 224atexlemunv 38529 . . . . 5 (𝜑𝑈𝑉)
4324atexlemutvt 38517 . . . . 5 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
445, 18, 4cvlsupr4 37807 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇 (𝑈 𝑉))
4541, 33, 34, 17, 42, 43, 44syl132anc 1388 . . . 4 (𝜑𝑇 (𝑈 𝑉))
468, 4, 5hlatjcl 37829 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
4713, 14, 16, 46syl3anc 1371 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
482, 204atexlemwb 38522 . . . . . . . 8 (𝜑𝑊 ∈ (Base‘𝐾))
498, 18, 9latmle1 18353 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
503, 47, 48, 49syl3anc 1371 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
5121, 50eqbrtrid 5140 . . . . . 6 (𝜑𝑈 (𝑃 𝑄))
528, 18, 9latmle1 18353 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
533, 7, 48, 52syl3anc 1371 . . . . . . 7 (𝜑 → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
5422, 53eqbrtrid 5140 . . . . . 6 (𝜑𝑉 (𝑃 𝑆))
558, 5atbase 37751 . . . . . . . 8 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
5633, 55syl 17 . . . . . . 7 (𝜑𝑈 ∈ (Base‘𝐾))
578, 5atbase 37751 . . . . . . . 8 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
5834, 57syl 17 . . . . . . 7 (𝜑𝑉 ∈ (Base‘𝐾))
598, 18, 4latjlej12 18344 . . . . . . 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 1379 . . . . . 6 (𝜑 → ((𝑈 (𝑃 𝑄) ∧ 𝑉 (𝑃 𝑆)) → (𝑈 𝑉) ((𝑃 𝑄) (𝑃 𝑆))))
6151, 54, 60mp2and 697 . . . . 5 (𝜑 → (𝑈 𝑉) ((𝑃 𝑄) (𝑃 𝑆)))
624, 5hlatjass 37832 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
6313, 14, 16, 15, 62syl13anc 1372 . . . . . 6 (𝜑 → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
648, 5atbase 37751 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
6514, 64syl 17 . . . . . . 7 (𝜑𝑃 ∈ (Base‘𝐾))
668, 5atbase 37751 . . . . . . . 8 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
6715, 66syl 17 . . . . . . 7 (𝜑𝑆 ∈ (Base‘𝐾))
688, 4latj32 18374 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → ((𝑃 𝑄) 𝑆) = ((𝑃 𝑆) 𝑄))
693, 65, 38, 67, 68syl13anc 1372 . . . . . 6 (𝜑 → ((𝑃 𝑄) 𝑆) = ((𝑃 𝑆) 𝑄))
708, 4latjjdi 18380 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (𝑃 (𝑄 𝑆)) = ((𝑃 𝑄) (𝑃 𝑆)))
713, 65, 38, 67, 70syl13anc 1372 . . . . . 6 (𝜑 → (𝑃 (𝑄 𝑆)) = ((𝑃 𝑄) (𝑃 𝑆)))
7263, 69, 713eqtr3rd 2785 . . . . 5 (𝜑 → ((𝑃 𝑄) (𝑃 𝑆)) = ((𝑃 𝑆) 𝑄))
7361, 72breqtrd 5131 . . . 4 (𝜑 → (𝑈 𝑉) ((𝑃 𝑆) 𝑄))
748, 18, 3, 32, 36, 40, 45, 73lattrd 18335 . . 3 (𝜑𝑇 ((𝑃 𝑆) 𝑄))
7518, 4, 9, 52atmat 38024 . . 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 1402 . 2 (𝜑 → ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴)
7712, 76eqeltrd 2838 1 (𝜑𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Latclat 18320  Atomscatm 37725  CvLatclc 37727  HLchlt 37812  LHypclh 38447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lhyp 38451
This theorem is referenced by:  4atexlemnclw  38533  4atexlemex2  38534  4atexlemcnd  38535
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