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Theorem cdlema2N 34579
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 1130 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝑅𝑃)
2 simp3rl 1132 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝑃 𝑋)
3 simp3rr 1133 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → ¬ 𝑄 𝑋)
4 simp3lr 1131 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝑅 (𝑃 𝑄))
52, 3, 43jca 1240 . . . . 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 34191 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 𝑋𝑅 = 𝑃))
115, 10syld3an3 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (𝑅 𝑋𝑅 = 𝑃))
1211necon3bbid 2827 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (¬ 𝑅 𝑋𝑅𝑃))
131, 12mpbird 247 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → ¬ 𝑅 𝑋)
14 simp1l 1083 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝐾 ∈ HL)
15 hlatl 34148 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1614, 15syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝐾 ∈ AtLat)
17 simp23 1094 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → 𝑅𝐴)
18 simp1r 1084 . . 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 34105 . . 3 ((𝐾 ∈ AtLat ∧ 𝑅𝐴𝑋𝐵) → (¬ 𝑅 𝑋 ↔ (𝑅 𝑋) = 0 ))
2216, 17, 18, 21syl3anc 1323 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (¬ 𝑅 𝑋 ↔ (𝑅 𝑋) = 0 ))
2313, 22mpbid 222 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4615  cfv 5849  (class class class)co 6607  Basecbs 15784  lecple 15872  joincjn 16868  meetcmee 16869  0.cp0 16961  Atomscatm 34051  AtLatcal 34052  HLchlt 34138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-preset 16852  df-poset 16870  df-plt 16882  df-lub 16898  df-glb 16899  df-join 16900  df-meet 16901  df-p0 16963  df-lat 16970  df-covers 34054  df-ats 34055  df-atl 34086  df-cvlat 34110  df-hlat 34139
This theorem is referenced by: (None)
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