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Theorem 4atexlemc 38928
Description: Lemma for 4atexlem7 38934. (Contributed by NM, 24-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph (πœ‘ ↔ (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))))
4thatlem0.l ≀ = (leβ€˜πΎ)
4thatlem0.j ∨ = (joinβ€˜πΎ)
4thatlem0.m ∧ = (meetβ€˜πΎ)
4thatlem0.a 𝐴 = (Atomsβ€˜πΎ)
4thatlem0.h 𝐻 = (LHypβ€˜πΎ)
4thatlem0.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
4thatlem0.v 𝑉 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
4thatlem0.c 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
Assertion
Ref Expression
4atexlemc (πœ‘ β†’ 𝐢 ∈ 𝐴)

Proof of Theorem 4atexlemc
StepHypRef Expression
1 4thatlem0.c . . 3 𝐢 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
2 4thatlem.ph . . . . 5 (πœ‘ ↔ (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))))
324atexlemkl 38916 . . . 4 (πœ‘ β†’ 𝐾 ∈ Lat)
4 4thatlem0.j . . . . 5 ∨ = (joinβ€˜πΎ)
5 4thatlem0.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
62, 4, 54atexlemqtb 38920 . . . 4 (πœ‘ β†’ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
72, 4, 54atexlempsb 38919 . . . 4 (πœ‘ β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
8 eqid 2732 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
9 4thatlem0.m . . . . 5 ∧ = (meetβ€˜πΎ)
108, 9latmcom 18412 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ)) β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
113, 6, 7, 10syl3anc 1371 . . 3 (πœ‘ β†’ ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
121, 11eqtrid 2784 . 2 (πœ‘ β†’ 𝐢 = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
1324atexlemk 38906 . . 3 (πœ‘ β†’ 𝐾 ∈ HL)
1424atexlemp 38909 . . 3 (πœ‘ β†’ 𝑃 ∈ 𝐴)
1524atexlems 38911 . . 3 (πœ‘ β†’ 𝑆 ∈ 𝐴)
1624atexlemq 38910 . . 3 (πœ‘ β†’ 𝑄 ∈ 𝐴)
1724atexlemt 38912 . . 3 (πœ‘ β†’ 𝑇 ∈ 𝐴)
18 4thatlem0.l . . . 4 ≀ = (leβ€˜πΎ)
192, 18, 4, 54atexlempns 38921 . . 3 (πœ‘ β†’ 𝑃 β‰  𝑆)
20 4thatlem0.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
21 4thatlem0.u . . . . 5 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
22 4thatlem0.v . . . . 5 𝑉 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
232, 18, 4, 9, 5, 20, 21, 224atexlemntlpq 38927 . . . 4 (πœ‘ β†’ Β¬ 𝑇 ≀ (𝑃 ∨ 𝑄))
2418, 4, 5atnlej2 38239 . . . . 5 ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑇 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑇 β‰  𝑄)
2524necomd 2996 . . . 4 ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑇 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑄 β‰  𝑇)
2613, 17, 14, 16, 23, 25syl131anc 1383 . . 3 (πœ‘ β†’ 𝑄 β‰  𝑇)
2724atexlempnq 38914 . . . 4 (πœ‘ β†’ 𝑃 β‰  𝑄)
2824atexlemnslpq 38915 . . . 4 (πœ‘ β†’ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))
2918, 4, 54atlem0ae 38453 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑆))
3013, 14, 16, 15, 27, 28, 29syl132anc 1388 . . 3 (πœ‘ β†’ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑆))
318, 5atbase 38147 . . . . 5 (𝑇 ∈ 𝐴 β†’ 𝑇 ∈ (Baseβ€˜πΎ))
3217, 31syl 17 . . . 4 (πœ‘ β†’ 𝑇 ∈ (Baseβ€˜πΎ))
332, 18, 4, 9, 5, 20, 214atexlemu 38923 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ 𝐴)
342, 18, 4, 9, 5, 20, 21, 224atexlemv 38924 . . . . 5 (πœ‘ β†’ 𝑉 ∈ 𝐴)
358, 4, 5hlatjcl 38225 . . . . 5 ((𝐾 ∈ HL ∧ π‘ˆ ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) β†’ (π‘ˆ ∨ 𝑉) ∈ (Baseβ€˜πΎ))
3613, 33, 34, 35syl3anc 1371 . . . 4 (πœ‘ β†’ (π‘ˆ ∨ 𝑉) ∈ (Baseβ€˜πΎ))
378, 5atbase 38147 . . . . . 6 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
3816, 37syl 17 . . . . 5 (πœ‘ β†’ 𝑄 ∈ (Baseβ€˜πΎ))
398, 4latjcl 18388 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Baseβ€˜πΎ))
403, 7, 38, 39syl3anc 1371 . . . 4 (πœ‘ β†’ ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Baseβ€˜πΎ))
4124atexlemkc 38917 . . . . 5 (πœ‘ β†’ 𝐾 ∈ CvLat)
422, 18, 4, 9, 5, 20, 21, 224atexlemunv 38925 . . . . 5 (πœ‘ β†’ π‘ˆ β‰  𝑉)
4324atexlemutvt 38913 . . . . 5 (πœ‘ β†’ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))
445, 18, 4cvlsupr4 38203 . . . . 5 ((𝐾 ∈ CvLat ∧ (π‘ˆ ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (π‘ˆ β‰  𝑉 ∧ (π‘ˆ ∨ 𝑇) = (𝑉 ∨ 𝑇))) β†’ 𝑇 ≀ (π‘ˆ ∨ 𝑉))
4541, 33, 34, 17, 42, 43, 44syl132anc 1388 . . . 4 (πœ‘ β†’ 𝑇 ≀ (π‘ˆ ∨ 𝑉))
468, 4, 5hlatjcl 38225 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
4713, 14, 16, 46syl3anc 1371 . . . . . . . 8 (πœ‘ β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
482, 204atexlemwb 38918 . . . . . . . 8 (πœ‘ β†’ π‘Š ∈ (Baseβ€˜πΎ))
498, 18, 9latmle1 18413 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑄) ∧ π‘Š) ≀ (𝑃 ∨ 𝑄))
503, 47, 48, 49syl3anc 1371 . . . . . . 7 (πœ‘ β†’ ((𝑃 ∨ 𝑄) ∧ π‘Š) ≀ (𝑃 ∨ 𝑄))
5121, 50eqbrtrid 5182 . . . . . 6 (πœ‘ β†’ π‘ˆ ≀ (𝑃 ∨ 𝑄))
528, 18, 9latmle1 18413 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑆) ∧ π‘Š) ≀ (𝑃 ∨ 𝑆))
533, 7, 48, 52syl3anc 1371 . . . . . . 7 (πœ‘ β†’ ((𝑃 ∨ 𝑆) ∧ π‘Š) ≀ (𝑃 ∨ 𝑆))
5422, 53eqbrtrid 5182 . . . . . 6 (πœ‘ β†’ 𝑉 ≀ (𝑃 ∨ 𝑆))
558, 5atbase 38147 . . . . . . . 8 (π‘ˆ ∈ 𝐴 β†’ π‘ˆ ∈ (Baseβ€˜πΎ))
5633, 55syl 17 . . . . . . 7 (πœ‘ β†’ π‘ˆ ∈ (Baseβ€˜πΎ))
578, 5atbase 38147 . . . . . . . 8 (𝑉 ∈ 𝐴 β†’ 𝑉 ∈ (Baseβ€˜πΎ))
5834, 57syl 17 . . . . . . 7 (πœ‘ β†’ 𝑉 ∈ (Baseβ€˜πΎ))
598, 18, 4latjlej12 18404 . . . . . . 7 ((𝐾 ∈ Lat ∧ (π‘ˆ ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) ∧ (𝑉 ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))) β†’ ((π‘ˆ ≀ (𝑃 ∨ 𝑄) ∧ 𝑉 ≀ (𝑃 ∨ 𝑆)) β†’ (π‘ˆ ∨ 𝑉) ≀ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆))))
603, 56, 47, 58, 7, 59syl122anc 1379 . . . . . 6 (πœ‘ β†’ ((π‘ˆ ≀ (𝑃 ∨ 𝑄) ∧ 𝑉 ≀ (𝑃 ∨ 𝑆)) β†’ (π‘ˆ ∨ 𝑉) ≀ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆))))
6151, 54, 60mp2and 697 . . . . 5 (πœ‘ β†’ (π‘ˆ ∨ 𝑉) ≀ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
624, 5hlatjass 38228 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
6313, 14, 16, 15, 62syl13anc 1372 . . . . . 6 (πœ‘ β†’ ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
648, 5atbase 38147 . . . . . . . 8 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
6514, 64syl 17 . . . . . . 7 (πœ‘ β†’ 𝑃 ∈ (Baseβ€˜πΎ))
668, 5atbase 38147 . . . . . . . 8 (𝑆 ∈ 𝐴 β†’ 𝑆 ∈ (Baseβ€˜πΎ))
6715, 66syl 17 . . . . . . 7 (πœ‘ β†’ 𝑆 ∈ (Baseβ€˜πΎ))
688, 4latj32 18434 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑆 ∈ (Baseβ€˜πΎ))) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑆) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
693, 65, 38, 67, 68syl13anc 1372 . . . . . 6 (πœ‘ β†’ ((𝑃 ∨ 𝑄) ∨ 𝑆) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
708, 4latjjdi 18440 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑆 ∈ (Baseβ€˜πΎ))) β†’ (𝑃 ∨ (𝑄 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
713, 65, 38, 67, 70syl13anc 1372 . . . . . 6 (πœ‘ β†’ (𝑃 ∨ (𝑄 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
7263, 69, 713eqtr3rd 2781 . . . . 5 (πœ‘ β†’ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
7361, 72breqtrd 5173 . . . 4 (πœ‘ β†’ (π‘ˆ ∨ 𝑉) ≀ ((𝑃 ∨ 𝑆) ∨ 𝑄))
748, 18, 3, 32, 36, 40, 45, 73lattrd 18395 . . 3 (πœ‘ β†’ 𝑇 ≀ ((𝑃 ∨ 𝑆) ∨ 𝑄))
7518, 4, 9, 52atmat 38420 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑆) ∧ (𝑄 β‰  𝑇 ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑆) ∧ 𝑇 ≀ ((𝑃 ∨ 𝑆) ∨ 𝑄))) β†’ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ 𝐴)
7613, 14, 15, 16, 17, 19, 26, 30, 74, 75syl333anc 1402 . 2 (πœ‘ β†’ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ 𝐴)
7712, 76eqeltrd 2833 1 (πœ‘ β†’ 𝐢 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  meetcmee 18261  Latclat 18380  Atomscatm 38121  CvLatclc 38123  HLchlt 38208  LHypclh 38843
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-p1 18375  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-llines 38357  df-lplanes 38358  df-lhyp 38847
This theorem is referenced by:  4atexlemnclw  38929  4atexlemex2  38930  4atexlemcnd  38931
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