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Theorem cdlemd3 36010
 Description: Part of proof of Lemma D in [Crawley] p. 113. The 𝑅 ≠ 𝑃 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 1253 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑆 (𝑃 𝑄))
2 simp1l 1239 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐾 ∈ HL)
3 simp31 1251 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅𝐴)
4 simp32 1252 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆𝐴)
5 simp21l 1374 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃𝐴)
6 simp233 1403 . . . 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 35204 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑃𝐴) ∧ 𝑅𝑃) → (𝑅 (𝑃 𝑆) → 𝑆 (𝑃 𝑅)))
112, 3, 4, 5, 6, 10syl131anc 1489 . . 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 1376 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑄𝐴)
137, 8, 9hlatlej1 35184 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
142, 5, 12, 13syl3anc 1476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃 (𝑃 𝑄))
15 simp232 1402 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
162hllatd 35173 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐾 ∈ Lat)
17 eqid 2771 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
1817, 9atbase 35098 . . . . . . 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 35098 . . . . . . 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 35098 . . . . . . . 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 17259 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
2516, 19, 23, 24syl3anc 1476 . . . . . 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 17270 . . . . . 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 1478 . . . . 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 691 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑃 𝑅) (𝑃 𝑄))
2917, 9atbase 35098 . . . . . 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 17259 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑅) ∈ (Base‘𝐾))
3216, 19, 21, 31syl3anc 1476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑃 𝑅) ∈ (Base‘𝐾))
3317, 7lattr 17264 . . . . 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 1478 . . . 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 674 . . 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 189 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 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943   class class class wbr 4787  ‘cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  Latclat 17253  Atomscatm 35072  HLchlt 35159  LHypclh 35793 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160 This theorem is referenced by:  cdlemd4  36011
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