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Theorem 4atexlemc 35878
Description: Lemma for 4atexlem7 35884. (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 35866 . . . 4 (𝜑𝐾 ∈ Lat)
4 4thatlem0.j . . . . 5 = (join‘𝐾)
5 4thatlem0.a . . . . 5 𝐴 = (Atoms‘𝐾)
62, 4, 54atexlemqtb 35870 . . . 4 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
72, 4, 54atexlempsb 35869 . . . 4 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
8 eqid 2771 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
9 4thatlem0.m . . . . 5 = (meet‘𝐾)
108, 9latmcom 17284 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) = ((𝑃 𝑆) (𝑄 𝑇)))
113, 6, 7, 10syl3anc 1476 . . 3 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) = ((𝑃 𝑆) (𝑄 𝑇)))
121, 11syl5eq 2817 . 2 (𝜑𝐶 = ((𝑃 𝑆) (𝑄 𝑇)))
1324atexlemk 35856 . . 3 (𝜑𝐾 ∈ HL)
1424atexlemp 35859 . . 3 (𝜑𝑃𝐴)
1524atexlems 35861 . . 3 (𝜑𝑆𝐴)
1624atexlemq 35860 . . 3 (𝜑𝑄𝐴)
1724atexlemt 35862 . . 3 (𝜑𝑇𝐴)
18 4thatlem0.l . . . 4 = (le‘𝐾)
192, 18, 4, 54atexlempns 35871 . . 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 35877 . . . 4 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
2418, 4, 5atnlej2 35189 . . . . 5 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑇 (𝑃 𝑄)) → 𝑇𝑄)
2524necomd 2998 . . . 4 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑇 (𝑃 𝑄)) → 𝑄𝑇)
2613, 17, 14, 16, 23, 25syl131anc 1489 . . 3 (𝜑𝑄𝑇)
2724atexlempnq 35864 . . . 4 (𝜑𝑃𝑄)
2824atexlemnslpq 35865 . . . 4 (𝜑 → ¬ 𝑆 (𝑃 𝑄))
2918, 4, 54atlem0ae 35403 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑄 (𝑃 𝑆))
3013, 14, 16, 15, 27, 28, 29syl132anc 1494 . . 3 (𝜑 → ¬ 𝑄 (𝑃 𝑆))
318, 5atbase 35099 . . . . 5 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
3217, 31syl 17 . . . 4 (𝜑𝑇 ∈ (Base‘𝐾))
332, 18, 4, 9, 5, 20, 214atexlemu 35873 . . . . 5 (𝜑𝑈𝐴)
342, 18, 4, 9, 5, 20, 21, 224atexlemv 35874 . . . . 5 (𝜑𝑉𝐴)
358, 4, 5hlatjcl 35176 . . . . 5 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) ∈ (Base‘𝐾))
3613, 33, 34, 35syl3anc 1476 . . . 4 (𝜑 → (𝑈 𝑉) ∈ (Base‘𝐾))
378, 5atbase 35099 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3816, 37syl 17 . . . . 5 (𝜑𝑄 ∈ (Base‘𝐾))
398, 4latjcl 17260 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾))
403, 7, 38, 39syl3anc 1476 . . . 4 (𝜑 → ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾))
4124atexlemkc 35867 . . . . 5 (𝜑𝐾 ∈ CvLat)
422, 18, 4, 9, 5, 20, 21, 224atexlemunv 35875 . . . . 5 (𝜑𝑈𝑉)
4324atexlemutvt 35863 . . . . 5 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
445, 18, 4cvlsupr4 35155 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇 (𝑈 𝑉))
4541, 33, 34, 17, 42, 43, 44syl132anc 1494 . . . 4 (𝜑𝑇 (𝑈 𝑉))
468, 4, 5hlatjcl 35176 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
4713, 14, 16, 46syl3anc 1476 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
482, 204atexlemwb 35868 . . . . . . . 8 (𝜑𝑊 ∈ (Base‘𝐾))
498, 18, 9latmle1 17285 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
503, 47, 48, 49syl3anc 1476 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
5121, 50syl5eqbr 4822 . . . . . 6 (𝜑𝑈 (𝑃 𝑄))
528, 18, 9latmle1 17285 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
533, 7, 48, 52syl3anc 1476 . . . . . . 7 (𝜑 → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
5422, 53syl5eqbr 4822 . . . . . 6 (𝜑𝑉 (𝑃 𝑆))
558, 5atbase 35099 . . . . . . . 8 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
5633, 55syl 17 . . . . . . 7 (𝜑𝑈 ∈ (Base‘𝐾))
578, 5atbase 35099 . . . . . . . 8 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
5834, 57syl 17 . . . . . . 7 (𝜑𝑉 ∈ (Base‘𝐾))
598, 18, 4latjlej12 17276 . . . . . . 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 1485 . . . . . 6 (𝜑 → ((𝑈 (𝑃 𝑄) ∧ 𝑉 (𝑃 𝑆)) → (𝑈 𝑉) ((𝑃 𝑄) (𝑃 𝑆))))
6151, 54, 60mp2and 673 . . . . 5 (𝜑 → (𝑈 𝑉) ((𝑃 𝑄) (𝑃 𝑆)))
624, 5hlatjass 35179 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
6313, 14, 16, 15, 62syl13anc 1478 . . . . . 6 (𝜑 → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
648, 5atbase 35099 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
6514, 64syl 17 . . . . . . 7 (𝜑𝑃 ∈ (Base‘𝐾))
668, 5atbase 35099 . . . . . . . 8 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
6715, 66syl 17 . . . . . . 7 (𝜑𝑆 ∈ (Base‘𝐾))
688, 4latj32 17306 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → ((𝑃 𝑄) 𝑆) = ((𝑃 𝑆) 𝑄))
693, 65, 38, 67, 68syl13anc 1478 . . . . . 6 (𝜑 → ((𝑃 𝑄) 𝑆) = ((𝑃 𝑆) 𝑄))
708, 4latjjdi 17312 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (𝑃 (𝑄 𝑆)) = ((𝑃 𝑄) (𝑃 𝑆)))
713, 65, 38, 67, 70syl13anc 1478 . . . . . 6 (𝜑 → (𝑃 (𝑄 𝑆)) = ((𝑃 𝑄) (𝑃 𝑆)))
7263, 69, 713eqtr3rd 2814 . . . . 5 (𝜑 → ((𝑃 𝑄) (𝑃 𝑆)) = ((𝑃 𝑆) 𝑄))
7361, 72breqtrd 4813 . . . 4 (𝜑 → (𝑈 𝑉) ((𝑃 𝑆) 𝑄))
748, 18, 3, 32, 36, 40, 45, 73lattrd 17267 . . 3 (𝜑𝑇 ((𝑃 𝑆) 𝑄))
7518, 4, 9, 52atmat 35370 . . 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 1508 . 2 (𝜑 → ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴)
7712, 76eqeltrd 2850 1 (𝜑𝐶𝐴)
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 4787  cfv 6032  (class class class)co 6794  Basecbs 16065  lecple 16157  joincjn 17153  meetcmee 17154  Latclat 17254  Atomscatm 35073  CvLatclc 35075  HLchlt 35160  LHypclh 35793
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 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  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 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-preset 17137  df-poset 17155  df-plt 17167  df-lub 17183  df-glb 17184  df-join 17185  df-meet 17186  df-p0 17248  df-p1 17249  df-lat 17255  df-clat 17317  df-oposet 34986  df-ol 34988  df-oml 34989  df-covers 35076  df-ats 35077  df-atl 35108  df-cvlat 35132  df-hlat 35161  df-llines 35307  df-lplanes 35308  df-lhyp 35797
This theorem is referenced by:  4atexlemnclw  35879  4atexlemex2  35880  4atexlemcnd  35881
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