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Theorem 4atexlemswapqr 40046
Description: Lemma for 4atexlem7 40058. 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 1202 . . . 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 40032 . . 3 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5 simp22 1206 . . . . 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 1147 . . . . 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 1127 . 2 (𝜑 → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)))
1014atexlems 40035 . . 3 (𝜑𝑆𝐴)
1114atexlemq 40034 . . . 4 (𝜑𝑄𝐴)
12 simp13r 1288 . . . . 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 40041 . . . . 5 (𝜑𝐾 ∈ CvLat)
1514atexlemp 40033 . . . . 5 (𝜑𝑃𝐴)
168simpld 494 . . . . 5 (𝜑𝑅𝐴)
1714atexlempnq 40038 . . . . 5 (𝜑𝑃𝑄)
18 simp223 1315 . . . . . 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 39330 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑅 𝑄))
2314, 15, 11, 16, 17, 19, 22syl132anc 1387 . . . 4 (𝜑 → (𝑃 𝑄) = (𝑅 𝑄))
2411, 13, 233jca 1127 . . 3 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)))
2514atexlemt 40036 . . . 4 (𝜑𝑇𝐴)
26 4thatlemsw.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
2720, 21cvlsupr8 39331 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑃 𝑅))
2814, 15, 11, 16, 17, 19, 27syl132anc 1387 . . . . . . . 8 (𝜑 → (𝑃 𝑄) = (𝑃 𝑅))
2928oveq1d 7446 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) = ((𝑃 𝑅) 𝑊))
3026, 29eqtrid 2787 . . . . . 6 (𝜑𝑈 = ((𝑃 𝑅) 𝑊))
3130oveq1d 7446 . . . . 5 (𝜑 → (𝑈 𝑇) = (((𝑃 𝑅) 𝑊) 𝑇))
3214atexlemutvt 40037 . . . . 5 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
3331, 32eqtr3d 2777 . . . 4 (𝜑 → (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))
3425, 33jca 511 . . 3 (𝜑 → (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇)))
3510, 24, 343jca 1127 . 2 (𝜑 → (𝑆𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)) ∧ (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))))
3620, 21cvlsupr5 39328 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑃)
3736necomd 2994 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑃𝑅)
3814, 15, 11, 16, 17, 19, 37syl132anc 1387 . . 3 (𝜑𝑃𝑅)
3914atexlemnslpq 40039 . . . 4 (𝜑 → ¬ 𝑆 (𝑃 𝑄))
4028eqcomd 2741 . . . . 5 (𝜑 → (𝑃 𝑅) = (𝑃 𝑄))
4140breq2d 5160 . . . 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 1127 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 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  lecple 17305  joincjn 18369  Atomscatm 39245  CvLatclc 39247  HLchlt 39332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333
This theorem is referenced by:  4atexlemex4  40056
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