Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atexlemswapqr Structured version   Visualization version   GIF version

Theorem 4atexlemswapqr 36751
Description: Lemma for 4atexlem7 36763. 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 1196 . . . 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 218 . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
414atexlempw 36737 . . 3 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5 simp22 1200 . . . . 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 1141 . . . . 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 218 . . 3 (𝜑 → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
93, 4, 83jca 1121 . 2 (𝜑 → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)))
1014atexlems 36740 . . 3 (𝜑𝑆𝐴)
1114atexlemq 36739 . . . 4 (𝜑𝑄𝐴)
12 simp13r 1282 . . . . 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 218 . . . 4 (𝜑 → ¬ 𝑄 𝑊)
1414atexlemkc 36746 . . . . 5 (𝜑𝐾 ∈ CvLat)
1514atexlemp 36738 . . . . 5 (𝜑𝑃𝐴)
168simpld 495 . . . . 5 (𝜑𝑅𝐴)
1714atexlempnq 36743 . . . . 5 (𝜑𝑃𝑄)
18 simp223 1309 . . . . . 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 218 . . . . 5 (𝜑 → (𝑃 𝑅) = (𝑄 𝑅))
20 4thatlemslps.a . . . . . 6 𝐴 = (Atoms‘𝐾)
21 4thatlemslps.j . . . . . 6 = (join‘𝐾)
2220, 21cvlsupr7 36036 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑅 𝑄))
2314, 15, 11, 16, 17, 19, 22syl132anc 1381 . . . 4 (𝜑 → (𝑃 𝑄) = (𝑅 𝑄))
2411, 13, 233jca 1121 . . 3 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)))
2514atexlemt 36741 . . . 4 (𝜑𝑇𝐴)
26 4thatlemsw.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
2720, 21cvlsupr8 36037 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑃 𝑅))
2814, 15, 11, 16, 17, 19, 27syl132anc 1381 . . . . . . . 8 (𝜑 → (𝑃 𝑄) = (𝑃 𝑅))
2928oveq1d 7038 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) = ((𝑃 𝑅) 𝑊))
3026, 29syl5eq 2845 . . . . . 6 (𝜑𝑈 = ((𝑃 𝑅) 𝑊))
3130oveq1d 7038 . . . . 5 (𝜑 → (𝑈 𝑇) = (((𝑃 𝑅) 𝑊) 𝑇))
3214atexlemutvt 36742 . . . . 5 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
3331, 32eqtr3d 2835 . . . 4 (𝜑 → (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))
3425, 33jca 512 . . 3 (𝜑 → (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇)))
3510, 24, 343jca 1121 . 2 (𝜑 → (𝑆𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)) ∧ (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))))
3620, 21cvlsupr5 36034 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑃)
3736necomd 3041 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑃𝑅)
3814, 15, 11, 16, 17, 19, 37syl132anc 1381 . . 3 (𝜑𝑃𝑅)
3914atexlemnslpq 36744 . . . 4 (𝜑 → ¬ 𝑆 (𝑃 𝑄))
4028eqcomd 2803 . . . . 5 (𝜑 → (𝑃 𝑅) = (𝑃 𝑄))
4140breq2d 4980 . . . 4 (𝜑 → (𝑆 (𝑃 𝑅) ↔ 𝑆 (𝑃 𝑄)))
4239, 41mtbird 326 . . 3 (𝜑 → ¬ 𝑆 (𝑃 𝑅))
4338, 42jca 512 . 2 (𝜑 → (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅)))
449, 35, 433jca 1121 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 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  wne 2986   class class class wbr 4968  cfv 6232  (class class class)co 7023  lecple 16405  joincjn 17387  Atomscatm 35951  CvLatclc 35953  HLchlt 36038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-proset 17371  df-poset 17389  df-plt 17401  df-lub 17417  df-glb 17418  df-join 17419  df-meet 17420  df-p0 17482  df-lat 17489  df-covers 35954  df-ats 35955  df-atl 35986  df-cvlat 36010  df-hlat 36039
This theorem is referenced by:  4atexlemex4  36761
  Copyright terms: Public domain W3C validator