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Theorem cdlemd3 36354
Description: Part of proof of Lemma D in [Crawley] p. 113. The 𝑅𝑃 requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemd3.l = (le‘𝐾)
cdlemd3.j = (join‘𝐾)
cdlemd3.a 𝐴 = (Atoms‘𝐾)
cdlemd3.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
cdlemd3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑅 (𝑃 𝑆))

Proof of Theorem cdlemd3
StepHypRef Expression
1 simp33 1225 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑆 (𝑃 𝑄))
2 simp1l 1211 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐾 ∈ HL)
3 simp31 1223 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅𝐴)
4 simp32 1224 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆𝐴)
5 simp21l 1346 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃𝐴)
6 simp233 1375 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅𝑃)
7 cdlemd3.l . . . . 5 = (le‘𝐾)
8 cdlemd3.j . . . . 5 = (join‘𝐾)
9 cdlemd3.a . . . . 5 𝐴 = (Atoms‘𝐾)
107, 8, 9hlatexch1 35549 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑃𝐴) ∧ 𝑅𝑃) → (𝑅 (𝑃 𝑆) → 𝑆 (𝑃 𝑅)))
112, 3, 4, 5, 6, 10syl131anc 1451 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑅 (𝑃 𝑆) → 𝑆 (𝑃 𝑅)))
12 simp22l 1348 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑄𝐴)
137, 8, 9hlatlej1 35529 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
142, 5, 12, 13syl3anc 1439 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃 (𝑃 𝑄))
15 simp232 1374 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
162hllatd 35518 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐾 ∈ Lat)
17 eqid 2778 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
1817, 9atbase 35443 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
195, 18syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃 ∈ (Base‘𝐾))
2017, 9atbase 35443 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
213, 20syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅 ∈ (Base‘𝐾))
2217, 9atbase 35443 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
2312, 22syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑄 ∈ (Base‘𝐾))
2417, 8latjcl 17437 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
2516, 19, 23, 24syl3anc 1439 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑃 𝑄) ∈ (Base‘𝐾))
2617, 7, 8latjle12 17448 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ↔ (𝑃 𝑅) (𝑃 𝑄)))
2716, 19, 21, 25, 26syl13anc 1440 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝑃 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)) ↔ (𝑃 𝑅) (𝑃 𝑄)))
2814, 15, 27mpbi2and 702 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑃 𝑅) (𝑃 𝑄))
2917, 9atbase 35443 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
304, 29syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆 ∈ (Base‘𝐾))
3117, 8latjcl 17437 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑅) ∈ (Base‘𝐾))
3216, 19, 21, 31syl3anc 1439 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑃 𝑅) ∈ (Base‘𝐾))
3317, 7lattr 17442 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑆 (𝑃 𝑅) ∧ (𝑃 𝑅) (𝑃 𝑄)) → 𝑆 (𝑃 𝑄)))
3416, 30, 32, 25, 33syl13anc 1440 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝑆 (𝑃 𝑅) ∧ (𝑃 𝑅) (𝑃 𝑄)) → 𝑆 (𝑃 𝑄)))
3528, 34mpan2d 684 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑆 (𝑃 𝑅) → 𝑆 (𝑃 𝑄)))
3611, 35syld 47 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑅 (𝑃 𝑆) → 𝑆 (𝑃 𝑄)))
371, 36mtod 190 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑅 (𝑃 𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969   class class class wbr 4886  cfv 6135  (class class class)co 6922  Basecbs 16255  lecple 16345  joincjn 17330  Latclat 17431  Atomscatm 35417  HLchlt 35504  LHypclh 36138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-proset 17314  df-poset 17332  df-plt 17344  df-lub 17360  df-glb 17361  df-join 17362  df-meet 17363  df-p0 17425  df-lat 17432  df-covers 35420  df-ats 35421  df-atl 35452  df-cvlat 35476  df-hlat 35505
This theorem is referenced by:  cdlemd4  36355
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