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Theorem cdlemd1 37493
 Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemd1.l = (le‘𝐾)
cdlemd1.j = (join‘𝐾)
cdlemd1.m = (meet‘𝐾)
cdlemd1.a 𝐴 = (Atoms‘𝐾)
cdlemd1.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
cdlemd1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑅 = ((𝑃 ((𝑃 𝑅) 𝑊)) (𝑄 ((𝑄 𝑅) 𝑊))))

Proof of Theorem cdlemd1
StepHypRef Expression
1 simpll 766 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝐾 ∈ HL)
2 simpr1l 1227 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑃𝐴)
3 simpr2l 1229 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑄𝐴)
4 simpr31 1260 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑅𝐴)
5 simpr32 1261 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑃𝑄)
6 simpr33 1262 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → ¬ 𝑅 (𝑃 𝑄))
7 cdlemd1.l . . . 4 = (le‘𝐾)
8 cdlemd1.j . . . 4 = (join‘𝐾)
9 cdlemd1.m . . . 4 = (meet‘𝐾)
10 cdlemd1.a . . . 4 𝐴 = (Atoms‘𝐾)
117, 8, 9, 102llnma2 37084 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑃) (𝑅 𝑄)) = 𝑅)
121, 2, 3, 4, 5, 6, 11syl132anc 1385 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → ((𝑅 𝑃) (𝑅 𝑄)) = 𝑅)
13 hllat 36658 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1413ad2antrr 725 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝐾 ∈ Lat)
15 eqid 2801 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1615, 10atbase 36584 . . . . . 6 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
174, 16syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑅 ∈ (Base‘𝐾))
1815, 10atbase 36584 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
192, 18syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑃 ∈ (Base‘𝐾))
2015, 8latjcom 17665 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑅 𝑃) = (𝑃 𝑅))
2114, 17, 19, 20syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑅 𝑃) = (𝑃 𝑅))
22 simpl 486 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
23 simpr1 1191 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
24 cdlemd1.h . . . . . 6 𝐻 = (LHyp‘𝐾)
2515, 7, 8, 9, 10, 24cdlemc1 37486 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 ((𝑃 𝑅) 𝑊)) = (𝑃 𝑅))
2622, 17, 23, 25syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑃 ((𝑃 𝑅) 𝑊)) = (𝑃 𝑅))
2721, 26eqtr4d 2839 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑅 𝑃) = (𝑃 ((𝑃 𝑅) 𝑊)))
2815, 10atbase 36584 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
293, 28syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑄 ∈ (Base‘𝐾))
3015, 8latjcom 17665 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑅 𝑄) = (𝑄 𝑅))
3114, 17, 29, 30syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑅 𝑄) = (𝑄 𝑅))
32 simpr2 1192 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
3315, 7, 8, 9, 10, 24cdlemc1 37486 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 ((𝑄 𝑅) 𝑊)) = (𝑄 𝑅))
3422, 17, 32, 33syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑄 ((𝑄 𝑅) 𝑊)) = (𝑄 𝑅))
3531, 34eqtr4d 2839 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑅 𝑄) = (𝑄 ((𝑄 𝑅) 𝑊)))
3627, 35oveq12d 7157 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → ((𝑅 𝑃) (𝑅 𝑄)) = ((𝑃 ((𝑃 𝑅) 𝑊)) (𝑄 ((𝑄 𝑅) 𝑊))))
3712, 36eqtr3d 2838 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑅 = ((𝑃 ((𝑃 𝑅) 𝑊)) (𝑄 ((𝑄 𝑅) 𝑊))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  Basecbs 16479  lecple 16568  joincjn 17550  meetcmee 17551  Latclat 17651  Atomscatm 36558  HLchlt 36645  LHypclh 37279 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-proset 17534  df-poset 17552  df-plt 17564  df-lub 17580  df-glb 17581  df-join 17582  df-meet 17583  df-p0 17645  df-p1 17646  df-lat 17652  df-clat 17714  df-oposet 36471  df-ol 36473  df-oml 36474  df-covers 36561  df-ats 36562  df-atl 36593  df-cvlat 36617  df-hlat 36646  df-psubsp 36798  df-pmap 36799  df-padd 37091  df-lhyp 37283 This theorem is referenced by:  cdlemd2  37494
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