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Theorem 4atexlemcnd 38535
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))
4thatlem0.d 𝐷 = ((𝑅 𝑇) (𝑃 𝑆))
Assertion
Ref Expression
4atexlemcnd (𝜑𝐶𝐷)

Proof of Theorem 4atexlemcnd
StepHypRef Expression
1 4thatlem.ph . . . 4 (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))))
2 4thatlem0.l . . . 4 = (le‘𝐾)
3 4thatlem0.j . . . 4 = (join‘𝐾)
4 4thatlem0.m . . . 4 = (meet‘𝐾)
5 4thatlem0.a . . . 4 𝐴 = (Atoms‘𝐾)
6 4thatlem0.h . . . 4 𝐻 = (LHyp‘𝐾)
7 4thatlem0.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
8 4thatlem0.v . . . 4 𝑉 = ((𝑃 𝑆) 𝑊)
91, 2, 3, 4, 5, 6, 7, 84atexlemtlw 38530 . . 3 (𝜑𝑇 𝑊)
10 4thatlem0.c . . . 4 𝐶 = ((𝑄 𝑇) (𝑃 𝑆))
111, 2, 3, 4, 5, 6, 7, 8, 104atexlemnclw 38533 . . 3 (𝜑 → ¬ 𝐶 𝑊)
12 nbrne2 5125 . . 3 ((𝑇 𝑊 ∧ ¬ 𝐶 𝑊) → 𝑇𝐶)
139, 11, 12syl2anc 584 . 2 (𝜑𝑇𝐶)
1414atexlemk 38510 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
1514atexlemq 38514 . . . . . . . . 9 (𝜑𝑄𝐴)
1614atexlemt 38516 . . . . . . . . 9 (𝜑𝑇𝐴)
173, 5hlatjcom 37830 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) = (𝑇 𝑄))
1814, 15, 16, 17syl3anc 1371 . . . . . . . 8 (𝜑 → (𝑄 𝑇) = (𝑇 𝑄))
19 simp221 1314 . . . . . . . . . 10 ((((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))) → 𝑅𝐴)
201, 19sylbi 216 . . . . . . . . 9 (𝜑𝑅𝐴)
213, 5hlatjcom 37830 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑇𝐴) → (𝑅 𝑇) = (𝑇 𝑅))
2214, 20, 16, 21syl3anc 1371 . . . . . . . 8 (𝜑 → (𝑅 𝑇) = (𝑇 𝑅))
2318, 22oveq12d 7375 . . . . . . 7 (𝜑 → ((𝑄 𝑇) (𝑅 𝑇)) = ((𝑇 𝑄) (𝑇 𝑅)))
2414atexlemkc 38521 . . . . . . . . 9 (𝜑𝐾 ∈ CvLat)
2514atexlemp 38513 . . . . . . . . 9 (𝜑𝑃𝐴)
2614atexlempnq 38518 . . . . . . . . 9 (𝜑𝑃𝑄)
27 simp223 1316 . . . . . . . . . 10 ((((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))) → (𝑃 𝑅) = (𝑄 𝑅))
281, 27sylbi 216 . . . . . . . . 9 (𝜑 → (𝑃 𝑅) = (𝑄 𝑅))
295, 3cvlsupr6 37809 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑄)
3029necomd 2999 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑄𝑅)
3124, 25, 15, 20, 26, 28, 30syl132anc 1388 . . . . . . . 8 (𝜑𝑄𝑅)
321, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 38531 . . . . . . . . 9 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
335, 3cvlsupr7 37810 . . . . . . . . . . . 12 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑅 𝑄))
3424, 25, 15, 20, 26, 28, 33syl132anc 1388 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) = (𝑅 𝑄))
353, 5hlatjcom 37830 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) = (𝑅 𝑄))
3614, 15, 20, 35syl3anc 1371 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑅) = (𝑅 𝑄))
3734, 36eqtr4d 2779 . . . . . . . . . 10 (𝜑 → (𝑃 𝑄) = (𝑄 𝑅))
3837breq2d 5117 . . . . . . . . 9 (𝜑 → (𝑇 (𝑃 𝑄) ↔ 𝑇 (𝑄 𝑅)))
3932, 38mtbid 323 . . . . . . . 8 (𝜑 → ¬ 𝑇 (𝑄 𝑅))
402, 3, 4, 52llnma2 38252 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑇𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 (𝑄 𝑅))) → ((𝑇 𝑄) (𝑇 𝑅)) = 𝑇)
4114, 15, 20, 16, 31, 39, 40syl132anc 1388 . . . . . . 7 (𝜑 → ((𝑇 𝑄) (𝑇 𝑅)) = 𝑇)
4223, 41eqtr2d 2777 . . . . . 6 (𝜑𝑇 = ((𝑄 𝑇) (𝑅 𝑇)))
4342adantr 481 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝑇 = ((𝑄 𝑇) (𝑅 𝑇)))
4414atexlemkl 38520 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
451, 3, 54atexlemqtb 38524 . . . . . . . . . 10 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
461, 3, 54atexlempsb 38523 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
47 eqid 2736 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
4847, 2, 4latmle1 18353 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
4944, 45, 46, 48syl3anc 1371 . . . . . . . . 9 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
5010, 49eqbrtrid 5140 . . . . . . . 8 (𝜑𝐶 (𝑄 𝑇))
5150adantr 481 . . . . . . 7 ((𝜑𝐶 = 𝐷) → 𝐶 (𝑄 𝑇))
52 simpr 485 . . . . . . . 8 ((𝜑𝐶 = 𝐷) → 𝐶 = 𝐷)
53 4thatlem0.d . . . . . . . . . 10 𝐷 = ((𝑅 𝑇) (𝑃 𝑆))
5447, 3, 5hlatjcl 37829 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑇𝐴) → (𝑅 𝑇) ∈ (Base‘𝐾))
5514, 20, 16, 54syl3anc 1371 . . . . . . . . . . 11 (𝜑 → (𝑅 𝑇) ∈ (Base‘𝐾))
5647, 2, 4latmle1 18353 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑅 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑅 𝑇) (𝑃 𝑆)) (𝑅 𝑇))
5744, 55, 46, 56syl3anc 1371 . . . . . . . . . 10 (𝜑 → ((𝑅 𝑇) (𝑃 𝑆)) (𝑅 𝑇))
5853, 57eqbrtrid 5140 . . . . . . . . 9 (𝜑𝐷 (𝑅 𝑇))
5958adantr 481 . . . . . . . 8 ((𝜑𝐶 = 𝐷) → 𝐷 (𝑅 𝑇))
6052, 59eqbrtrd 5127 . . . . . . 7 ((𝜑𝐶 = 𝐷) → 𝐶 (𝑅 𝑇))
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 38532 . . . . . . . . . 10 (𝜑𝐶𝐴)
6247, 5atbase 37751 . . . . . . . . . 10 (𝐶𝐴𝐶 ∈ (Base‘𝐾))
6361, 62syl 17 . . . . . . . . 9 (𝜑𝐶 ∈ (Base‘𝐾))
6447, 2, 4latlem12 18355 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑅 𝑇) ∈ (Base‘𝐾))) → ((𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑇)) ↔ 𝐶 ((𝑄 𝑇) (𝑅 𝑇))))
6544, 63, 45, 55, 64syl13anc 1372 . . . . . . . 8 (𝜑 → ((𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑇)) ↔ 𝐶 ((𝑄 𝑇) (𝑅 𝑇))))
6665adantr 481 . . . . . . 7 ((𝜑𝐶 = 𝐷) → ((𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑇)) ↔ 𝐶 ((𝑄 𝑇) (𝑅 𝑇))))
6751, 60, 66mpbi2and 710 . . . . . 6 ((𝜑𝐶 = 𝐷) → 𝐶 ((𝑄 𝑇) (𝑅 𝑇)))
68 hlatl 37822 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
6914, 68syl 17 . . . . . . . 8 (𝜑𝐾 ∈ AtLat)
7042, 16eqeltrrd 2839 . . . . . . . 8 (𝜑 → ((𝑄 𝑇) (𝑅 𝑇)) ∈ 𝐴)
712, 5atcmp 37773 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝐶𝐴 ∧ ((𝑄 𝑇) (𝑅 𝑇)) ∈ 𝐴) → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7269, 61, 70, 71syl3anc 1371 . . . . . . 7 (𝜑 → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7372adantr 481 . . . . . 6 ((𝜑𝐶 = 𝐷) → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7467, 73mpbid 231 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝐶 = ((𝑄 𝑇) (𝑅 𝑇)))
7543, 74eqtr4d 2779 . . . 4 ((𝜑𝐶 = 𝐷) → 𝑇 = 𝐶)
7675ex 413 . . 3 (𝜑 → (𝐶 = 𝐷𝑇 = 𝐶))
7776necon3d 2964 . 2 (𝜑 → (𝑇𝐶𝐶𝐷))
7813, 77mpd 15 1 (𝜑𝐶𝐷)
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  AtLatcal 37726  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:  4atexlemex4  38536
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