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Theorem 4atexlemswapqr 40065
Description: Lemma for 4atexlem7 40077. Swap 𝑄 and 𝑅, so that theorems involving 𝐶 can be reused for 𝐷. Note that 𝑈 must be expanded because it involves 𝑄. (Contributed by NM, 25-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))))
4thatlemslps.l = (le‘𝐾)
4thatlemslps.j = (join‘𝐾)
4thatlemslps.a 𝐴 = (Atoms‘𝐾)
4thatlemsw.u 𝑈 = ((𝑃 𝑄) 𝑊)
Assertion
Ref Expression
4atexlemswapqr (𝜑 → (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑆𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)) ∧ (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅))))

Proof of Theorem 4atexlemswapqr
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 simp11 1204 . . . 4 ((((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))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
31, 2sylbi 217 . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
414atexlempw 40051 . . 3 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5 simp22 1208 . . . . 5 ((((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 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
6 3simpa 1149 . . . . 5 ((𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
75, 6syl 17 . . . 4 ((((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 ∧ ¬ 𝑅 𝑊))
81, 7sylbi 217 . . 3 (𝜑 → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
93, 4, 83jca 1129 . 2 (𝜑 → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)))
1014atexlems 40054 . . 3 (𝜑𝑆𝐴)
1114atexlemq 40053 . . . 4 (𝜑𝑄𝐴)
12 simp13r 1290 . . . . 5 ((((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))) → ¬ 𝑄 𝑊)
131, 12sylbi 217 . . . 4 (𝜑 → ¬ 𝑄 𝑊)
1414atexlemkc 40060 . . . . 5 (𝜑𝐾 ∈ CvLat)
1514atexlemp 40052 . . . . 5 (𝜑𝑃𝐴)
168simpld 494 . . . . 5 (𝜑𝑅𝐴)
1714atexlempnq 40057 . . . . 5 (𝜑𝑃𝑄)
18 simp223 1317 . . . . . 6 ((((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))
191, 18sylbi 217 . . . . 5 (𝜑 → (𝑃 𝑅) = (𝑄 𝑅))
20 4thatlemslps.a . . . . . 6 𝐴 = (Atoms‘𝐾)
21 4thatlemslps.j . . . . . 6 = (join‘𝐾)
2220, 21cvlsupr7 39349 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑅 𝑄))
2314, 15, 11, 16, 17, 19, 22syl132anc 1390 . . . 4 (𝜑 → (𝑃 𝑄) = (𝑅 𝑄))
2411, 13, 233jca 1129 . . 3 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)))
2514atexlemt 40055 . . . 4 (𝜑𝑇𝐴)
26 4thatlemsw.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
2720, 21cvlsupr8 39350 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑃 𝑅))
2814, 15, 11, 16, 17, 19, 27syl132anc 1390 . . . . . . . 8 (𝜑 → (𝑃 𝑄) = (𝑃 𝑅))
2928oveq1d 7446 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) = ((𝑃 𝑅) 𝑊))
3026, 29eqtrid 2789 . . . . . 6 (𝜑𝑈 = ((𝑃 𝑅) 𝑊))
3130oveq1d 7446 . . . . 5 (𝜑 → (𝑈 𝑇) = (((𝑃 𝑅) 𝑊) 𝑇))
3214atexlemutvt 40056 . . . . 5 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
3331, 32eqtr3d 2779 . . . 4 (𝜑 → (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))
3425, 33jca 511 . . 3 (𝜑 → (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇)))
3510, 24, 343jca 1129 . 2 (𝜑 → (𝑆𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)) ∧ (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))))
3620, 21cvlsupr5 39347 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑃)
3736necomd 2996 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑃𝑅)
3814, 15, 11, 16, 17, 19, 37syl132anc 1390 . . 3 (𝜑𝑃𝑅)
3914atexlemnslpq 40058 . . . 4 (𝜑 → ¬ 𝑆 (𝑃 𝑄))
4028eqcomd 2743 . . . . 5 (𝜑 → (𝑃 𝑅) = (𝑃 𝑄))
4140breq2d 5155 . . . 4 (𝜑 → (𝑆 (𝑃 𝑅) ↔ 𝑆 (𝑃 𝑄)))
4239, 41mtbird 325 . . 3 (𝜑 → ¬ 𝑆 (𝑃 𝑅))
4338, 42jca 511 . 2 (𝜑 → (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅)))
449, 35, 433jca 1129 1 (𝜑 → (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑆𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)) ∧ (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  Atomscatm 39264  CvLatclc 39266  HLchlt 39351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352
This theorem is referenced by:  4atexlemex4  40075
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