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Theorem cdlemd1 36006
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 750 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝐾 ∈ HL)
2 simpr1l 1290 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑃𝐴)
3 simpr2l 1294 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑄𝐴)
4 simpr31 1344 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑅𝐴)
5 simpr32 1346 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑃𝑄)
6 simpr33 1348 . . 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 35596 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑃) (𝑅 𝑄)) = 𝑅)
121, 2, 3, 4, 5, 6, 11syl132anc 1494 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → ((𝑅 𝑃) (𝑅 𝑄)) = 𝑅)
13 hllat 35170 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1413ad2antrr 705 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝐾 ∈ Lat)
15 eqid 2771 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1615, 10atbase 35096 . . . . . 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 35096 . . . . . 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 17267 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑅 𝑃) = (𝑃 𝑅))
2114, 17, 19, 20syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑅 𝑃) = (𝑃 𝑅))
22 simpl 468 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
23 simpr1 1233 . . . . 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 35999 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 ((𝑃 𝑅) 𝑊)) = (𝑃 𝑅))
2622, 17, 23, 25syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑃 ((𝑃 𝑅) 𝑊)) = (𝑃 𝑅))
2721, 26eqtr4d 2808 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑅 𝑃) = (𝑃 ((𝑃 𝑅) 𝑊)))
2815, 10atbase 35096 . . . . . 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 17267 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑅 𝑄) = (𝑄 𝑅))
3114, 17, 29, 30syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑅 𝑄) = (𝑄 𝑅))
32 simpr2 1235 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
3315, 7, 8, 9, 10, 24cdlemc1 35999 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 ((𝑄 𝑅) 𝑊)) = (𝑄 𝑅))
3422, 17, 32, 33syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑄 ((𝑄 𝑅) 𝑊)) = (𝑄 𝑅))
3531, 34eqtr4d 2808 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → (𝑅 𝑄) = (𝑄 ((𝑄 𝑅) 𝑊)))
3627, 35oveq12d 6814 . 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 2807 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 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  meetcmee 17153  Latclat 17253  Atomscatm 35070  HLchlt 35157  LHypclh 35791
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-iin 4658  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-mpt2 6801  df-1st 7319  df-2nd 7320  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-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34983  df-ol 34985  df-oml 34986  df-covers 35073  df-ats 35074  df-atl 35105  df-cvlat 35129  df-hlat 35158  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795
This theorem is referenced by:  cdlemd2  36007
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