Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlema2N Structured version   Visualization version   GIF version

Theorem cdlema2N 38967
Description: A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlema2.b 𝐵 = (Base‘𝐾)
cdlema2.l = (le‘𝐾)
cdlema2.j = (join‘𝐾)
cdlema2.m = (meet‘𝐾)
cdlema2.z 0 = (0.‘𝐾)
cdlema2.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cdlema2N (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (𝑅 𝑋) = 0 )

Proof of Theorem cdlema2N
StepHypRef Expression
1 simp3ll 1243 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝑅𝑃)
2 simp3rl 1245 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝑃 𝑋)
3 simp3rr 1246 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → ¬ 𝑄 𝑋)
4 simp3lr 1244 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝑅 (𝑃 𝑄))
52, 3, 43jca 1127 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄)))
6 cdlema2.b . . . . . 6 𝐵 = (Base‘𝐾)
7 cdlema2.l . . . . . 6 = (le‘𝐾)
8 cdlema2.j . . . . . 6 = (join‘𝐾)
9 cdlema2.a . . . . . 6 𝐴 = (Atoms‘𝐾)
106, 7, 8, 9exatleN 38579 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 𝑋𝑅 = 𝑃))
115, 10syld3an3 1408 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (𝑅 𝑋𝑅 = 𝑃))
1211necon3bbid 2977 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (¬ 𝑅 𝑋𝑅𝑃))
131, 12mpbird 257 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → ¬ 𝑅 𝑋)
14 simp1l 1196 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝐾 ∈ HL)
15 hlatl 38534 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1614, 15syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝐾 ∈ AtLat)
17 simp23 1207 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝑅𝐴)
18 simp1r 1197 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝑋𝐵)
19 cdlema2.m . . . 4 = (meet‘𝐾)
20 cdlema2.z . . . 4 0 = (0.‘𝐾)
216, 7, 19, 20, 9atnle 38491 . . 3 ((𝐾 ∈ AtLat ∧ 𝑅𝐴𝑋𝐵) → (¬ 𝑅 𝑋 ↔ (𝑅 𝑋) = 0 ))
2216, 17, 18, 21syl3anc 1370 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (¬ 𝑅 𝑋 ↔ (𝑅 𝑋) = 0 ))
2313, 22mpbid 231 1 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (𝑅 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939   class class class wbr 5148  cfv 6543  (class class class)co 7412  Basecbs 17149  lecple 17209  joincjn 18269  meetcmee 18270  0.cp0 18381  Atomscatm 38437  AtLatcal 38438  HLchlt 38524
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-lat 18390  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator