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Theorem 4atexlemcnd 37244
Description: Lemma for 4atexlem7 37247. (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 37239 . . 3 (𝜑𝑇 𝑊)
10 4thatlem0.c . . . 4 𝐶 = ((𝑄 𝑇) (𝑃 𝑆))
111, 2, 3, 4, 5, 6, 7, 8, 104atexlemnclw 37242 . . 3 (𝜑 → ¬ 𝐶 𝑊)
12 nbrne2 5062 . . 3 ((𝑇 𝑊 ∧ ¬ 𝐶 𝑊) → 𝑇𝐶)
139, 11, 12syl2anc 586 . 2 (𝜑𝑇𝐶)
1414atexlemk 37219 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
1514atexlemq 37223 . . . . . . . . 9 (𝜑𝑄𝐴)
1614atexlemt 37225 . . . . . . . . 9 (𝜑𝑇𝐴)
173, 5hlatjcom 36540 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) = (𝑇 𝑄))
1814, 15, 16, 17syl3anc 1367 . . . . . . . 8 (𝜑 → (𝑄 𝑇) = (𝑇 𝑄))
19 simp221 1310 . . . . . . . . . 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 219 . . . . . . . . 9 (𝜑𝑅𝐴)
213, 5hlatjcom 36540 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑇𝐴) → (𝑅 𝑇) = (𝑇 𝑅))
2214, 20, 16, 21syl3anc 1367 . . . . . . . 8 (𝜑 → (𝑅 𝑇) = (𝑇 𝑅))
2318, 22oveq12d 7151 . . . . . . 7 (𝜑 → ((𝑄 𝑇) (𝑅 𝑇)) = ((𝑇 𝑄) (𝑇 𝑅)))
2414atexlemkc 37230 . . . . . . . . 9 (𝜑𝐾 ∈ CvLat)
2514atexlemp 37222 . . . . . . . . 9 (𝜑𝑃𝐴)
2614atexlempnq 37227 . . . . . . . . 9 (𝜑𝑃𝑄)
27 simp223 1312 . . . . . . . . . 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 219 . . . . . . . . 9 (𝜑 → (𝑃 𝑅) = (𝑄 𝑅))
295, 3cvlsupr6 36519 . . . . . . . . . 10 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑄)
3029necomd 3061 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑄𝑅)
3124, 25, 15, 20, 26, 28, 30syl132anc 1384 . . . . . . . 8 (𝜑𝑄𝑅)
321, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 37240 . . . . . . . . 9 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
335, 3cvlsupr7 36520 . . . . . . . . . . . 12 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑅 𝑄))
3424, 25, 15, 20, 26, 28, 33syl132anc 1384 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) = (𝑅 𝑄))
353, 5hlatjcom 36540 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) = (𝑅 𝑄))
3614, 15, 20, 35syl3anc 1367 . . . . . . . . . . 11 (𝜑 → (𝑄 𝑅) = (𝑅 𝑄))
3734, 36eqtr4d 2858 . . . . . . . . . 10 (𝜑 → (𝑃 𝑄) = (𝑄 𝑅))
3837breq2d 5054 . . . . . . . . 9 (𝜑 → (𝑇 (𝑃 𝑄) ↔ 𝑇 (𝑄 𝑅)))
3932, 38mtbid 326 . . . . . . . 8 (𝜑 → ¬ 𝑇 (𝑄 𝑅))
402, 3, 4, 52llnma2 36961 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑇𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 (𝑄 𝑅))) → ((𝑇 𝑄) (𝑇 𝑅)) = 𝑇)
4114, 15, 20, 16, 31, 39, 40syl132anc 1384 . . . . . . 7 (𝜑 → ((𝑇 𝑄) (𝑇 𝑅)) = 𝑇)
4223, 41eqtr2d 2856 . . . . . 6 (𝜑𝑇 = ((𝑄 𝑇) (𝑅 𝑇)))
4342adantr 483 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝑇 = ((𝑄 𝑇) (𝑅 𝑇)))
4414atexlemkl 37229 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
451, 3, 54atexlemqtb 37233 . . . . . . . . . 10 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
461, 3, 54atexlempsb 37232 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
47 eqid 2820 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
4847, 2, 4latmle1 17665 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
4944, 45, 46, 48syl3anc 1367 . . . . . . . . 9 (𝜑 → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑇))
5010, 49eqbrtrid 5077 . . . . . . . 8 (𝜑𝐶 (𝑄 𝑇))
5150adantr 483 . . . . . . 7 ((𝜑𝐶 = 𝐷) → 𝐶 (𝑄 𝑇))
52 simpr 487 . . . . . . . 8 ((𝜑𝐶 = 𝐷) → 𝐶 = 𝐷)
53 4thatlem0.d . . . . . . . . . 10 𝐷 = ((𝑅 𝑇) (𝑃 𝑆))
5447, 3, 5hlatjcl 36539 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑇𝐴) → (𝑅 𝑇) ∈ (Base‘𝐾))
5514, 20, 16, 54syl3anc 1367 . . . . . . . . . . 11 (𝜑 → (𝑅 𝑇) ∈ (Base‘𝐾))
5647, 2, 4latmle1 17665 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑅 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑅 𝑇) (𝑃 𝑆)) (𝑅 𝑇))
5744, 55, 46, 56syl3anc 1367 . . . . . . . . . 10 (𝜑 → ((𝑅 𝑇) (𝑃 𝑆)) (𝑅 𝑇))
5853, 57eqbrtrid 5077 . . . . . . . . 9 (𝜑𝐷 (𝑅 𝑇))
5958adantr 483 . . . . . . . 8 ((𝜑𝐶 = 𝐷) → 𝐷 (𝑅 𝑇))
6052, 59eqbrtrd 5064 . . . . . . 7 ((𝜑𝐶 = 𝐷) → 𝐶 (𝑅 𝑇))
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 37241 . . . . . . . . . 10 (𝜑𝐶𝐴)
6247, 5atbase 36461 . . . . . . . . . 10 (𝐶𝐴𝐶 ∈ (Base‘𝐾))
6361, 62syl 17 . . . . . . . . 9 (𝜑𝐶 ∈ (Base‘𝐾))
6447, 2, 4latlem12 17667 . . . . . . . . 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 1368 . . . . . . . 8 (𝜑 → ((𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑇)) ↔ 𝐶 ((𝑄 𝑇) (𝑅 𝑇))))
6665adantr 483 . . . . . . 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 36532 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
6914, 68syl 17 . . . . . . . 8 (𝜑𝐾 ∈ AtLat)
7042, 16eqeltrrd 2912 . . . . . . . 8 (𝜑 → ((𝑄 𝑇) (𝑅 𝑇)) ∈ 𝐴)
712, 5atcmp 36483 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝐶𝐴 ∧ ((𝑄 𝑇) (𝑅 𝑇)) ∈ 𝐴) → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7269, 61, 70, 71syl3anc 1367 . . . . . . 7 (𝜑 → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7372adantr 483 . . . . . 6 ((𝜑𝐶 = 𝐷) → (𝐶 ((𝑄 𝑇) (𝑅 𝑇)) ↔ 𝐶 = ((𝑄 𝑇) (𝑅 𝑇))))
7467, 73mpbid 234 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝐶 = ((𝑄 𝑇) (𝑅 𝑇)))
7543, 74eqtr4d 2858 . . . 4 ((𝜑𝐶 = 𝐷) → 𝑇 = 𝐶)
7675ex 415 . . 3 (𝜑 → (𝐶 = 𝐷𝑇 = 𝐶))
7776necon3d 3027 . 2 (𝜑 → (𝑇𝐶𝐶𝐷))
7813, 77mpd 15 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wne 3006   class class class wbr 5042  cfv 6331  (class class class)co 7133  Basecbs 16462  lecple 16551  joincjn 17533  meetcmee 17534  Latclat 17634  Atomscatm 36435  AtLatcal 36436  CvLatclc 36437  HLchlt 36522  LHypclh 37156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-proset 17517  df-poset 17535  df-plt 17547  df-lub 17563  df-glb 17564  df-join 17565  df-meet 17566  df-p0 17628  df-p1 17629  df-lat 17635  df-clat 17697  df-oposet 36348  df-ol 36350  df-oml 36351  df-covers 36438  df-ats 36439  df-atl 36470  df-cvlat 36494  df-hlat 36523  df-llines 36670  df-lplanes 36671  df-lhyp 37160
This theorem is referenced by:  4atexlemex4  37245
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