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

Theorem 4atexlemswapqr 37399
 Description: Lemma for 4atexlem7 37411. 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 1200 . . . 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 220 . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
414atexlempw 37385 . . 3 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5 simp22 1204 . . . . 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 1145 . . . . 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 220 . . 3 (𝜑 → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
93, 4, 83jca 1125 . 2 (𝜑 → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)))
1014atexlems 37388 . . 3 (𝜑𝑆𝐴)
1114atexlemq 37387 . . . 4 (𝜑𝑄𝐴)
12 simp13r 1286 . . . . 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 220 . . . 4 (𝜑 → ¬ 𝑄 𝑊)
1414atexlemkc 37394 . . . . 5 (𝜑𝐾 ∈ CvLat)
1514atexlemp 37386 . . . . 5 (𝜑𝑃𝐴)
168simpld 498 . . . . 5 (𝜑𝑅𝐴)
1714atexlempnq 37391 . . . . 5 (𝜑𝑃𝑄)
18 simp223 1313 . . . . . 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 220 . . . . 5 (𝜑 → (𝑃 𝑅) = (𝑄 𝑅))
20 4thatlemslps.a . . . . . 6 𝐴 = (Atoms‘𝐾)
21 4thatlemslps.j . . . . . 6 = (join‘𝐾)
2220, 21cvlsupr7 36684 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑅 𝑄))
2314, 15, 11, 16, 17, 19, 22syl132anc 1385 . . . 4 (𝜑 → (𝑃 𝑄) = (𝑅 𝑄))
2411, 13, 233jca 1125 . . 3 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)))
2514atexlemt 37389 . . . 4 (𝜑𝑇𝐴)
26 4thatlemsw.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
2720, 21cvlsupr8 36685 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑃 𝑅))
2814, 15, 11, 16, 17, 19, 27syl132anc 1385 . . . . . . . 8 (𝜑 → (𝑃 𝑄) = (𝑃 𝑅))
2928oveq1d 7155 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) = ((𝑃 𝑅) 𝑊))
3026, 29syl5eq 2845 . . . . . 6 (𝜑𝑈 = ((𝑃 𝑅) 𝑊))
3130oveq1d 7155 . . . . 5 (𝜑 → (𝑈 𝑇) = (((𝑃 𝑅) 𝑊) 𝑇))
3214atexlemutvt 37390 . . . . 5 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
3331, 32eqtr3d 2835 . . . 4 (𝜑 → (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))
3425, 33jca 515 . . 3 (𝜑 → (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇)))
3510, 24, 343jca 1125 . 2 (𝜑 → (𝑆𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑅 𝑄)) ∧ (𝑇𝐴 ∧ (((𝑃 𝑅) 𝑊) 𝑇) = (𝑉 𝑇))))
3620, 21cvlsupr5 36682 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑃)
3736necomd 3042 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑃𝑅)
3814, 15, 11, 16, 17, 19, 37syl132anc 1385 . . 3 (𝜑𝑃𝑅)
3914atexlemnslpq 37392 . . . 4 (𝜑 → ¬ 𝑆 (𝑃 𝑄))
4028eqcomd 2804 . . . . 5 (𝜑 → (𝑃 𝑅) = (𝑃 𝑄))
4140breq2d 5043 . . . 4 (𝜑 → (𝑆 (𝑃 𝑅) ↔ 𝑆 (𝑃 𝑄)))
4239, 41mtbird 328 . . 3 (𝜑 → ¬ 𝑆 (𝑃 𝑅))
4338, 42jca 515 . 2 (𝜑 → (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅)))
449, 35, 433jca 1125 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   class class class wbr 5031  ‘cfv 6327  (class class class)co 7140  lecple 16571  joincjn 17553  Atomscatm 36599  CvLatclc 36601  HLchlt 36686 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7098  df-ov 7143  df-oprab 7144  df-proset 17537  df-poset 17555  df-plt 17567  df-lub 17583  df-glb 17584  df-join 17585  df-meet 17586  df-p0 17648  df-lat 17655  df-covers 36602  df-ats 36603  df-atl 36634  df-cvlat 36658  df-hlat 36687 This theorem is referenced by:  4atexlemex4  37409
 Copyright terms: Public domain W3C validator