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Theorem dalawlem2 34176
 Description: Lemma for dalaw 34190. Utility lemma that breaks ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) into a join of two pieces. (Contributed by NM, 6-Oct-2012.)
Hypotheses
Ref Expression
dalawlem.l = (le‘𝐾)
dalawlem.j = (join‘𝐾)
dalawlem.m = (meet‘𝐾)
dalawlem.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
dalawlem2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))

Proof of Theorem dalawlem2
StepHypRef Expression
1 simp1 1054 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ HL)
2 hllat 33668 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
31, 2syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ Lat)
4 simp2l 1080 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑃𝐴)
5 simp2r 1081 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑄𝐴)
6 eqid 2610 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
7 dalawlem.j . . . . . . 7 = (join‘𝐾)
8 dalawlem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatjcl 33671 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
101, 4, 5, 9syl3anc 1318 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
11 simp3r 1083 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑇𝐴)
126, 8atbase 33594 . . . . . 6 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
1311, 12syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑇 ∈ (Base‘𝐾))
14 dalawlem.l . . . . . 6 = (le‘𝐾)
156, 14, 7latlej1 16883 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑇))
163, 10, 13, 15syl3anc 1318 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑇))
17 simp3l 1082 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆𝐴)
186, 8atbase 33594 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆 ∈ (Base‘𝐾))
206, 14, 7latlej1 16883 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
213, 10, 19, 20syl3anc 1318 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
226, 7latjcl 16874 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾))
233, 10, 13, 22syl3anc 1318 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾))
246, 7latjcl 16874 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
253, 10, 19, 24syl3anc 1318 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
26 dalawlem.m . . . . . 6 = (meet‘𝐾)
276, 14, 26latlem12 16901 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))) → (((𝑃 𝑄) ((𝑃 𝑄) 𝑇) ∧ (𝑃 𝑄) ((𝑃 𝑄) 𝑆)) ↔ (𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆))))
283, 10, 23, 25, 27syl13anc 1320 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) ((𝑃 𝑄) 𝑇) ∧ (𝑃 𝑄) ((𝑃 𝑄) 𝑆)) ↔ (𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆))))
2916, 21, 28mpbi2and 958 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)))
306, 26latmcl 16875 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) ∈ (Base‘𝐾))
313, 23, 25, 30syl3anc 1318 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) ∈ (Base‘𝐾))
326, 7, 8hlatjcl 33671 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
331, 17, 11, 32syl3anc 1318 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑆 𝑇) ∈ (Base‘𝐾))
346, 14, 26latmlem1 16904 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾))) → ((𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇))))
353, 10, 31, 33, 34syl13anc 1320 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇))))
3629, 35mpd 15 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)))
376, 14, 7latlej2 16884 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ((𝑃 𝑄) 𝑆))
383, 10, 19, 37syl3anc 1318 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆 ((𝑃 𝑄) 𝑆))
396, 14, 7, 26, 8atmod3i1 34168 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆𝐴 ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) ∧ 𝑆 ((𝑃 𝑄) 𝑆)) → (𝑆 (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑆) (𝑆 𝑇)))
401, 17, 25, 13, 38, 39syl131anc 1331 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑆 (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑆) (𝑆 𝑇)))
4140oveq2d 6565 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑇) (𝑆 (((𝑃 𝑄) 𝑆) 𝑇))) = (((𝑃 𝑄) 𝑇) (((𝑃 𝑄) 𝑆) (𝑆 𝑇))))
426, 26latmcl 16875 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
433, 25, 13, 42syl3anc 1318 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
446, 14, 7, 26latmlej22 16916 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑇 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑃 𝑄) 𝑇))
453, 13, 25, 10, 44syl13anc 1320 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑃 𝑄) 𝑇))
466, 14, 7, 26, 8atmod2i2 34166 . . . 4 ((𝐾 ∈ HL ∧ (𝑆𝐴 ∧ ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾) ∧ (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾)) ∧ (((𝑃 𝑄) 𝑆) 𝑇) ((𝑃 𝑄) 𝑇)) → ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑇) (𝑆 (((𝑃 𝑄) 𝑆) 𝑇))))
471, 17, 23, 43, 45, 46syl131anc 1331 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑇) (𝑆 (((𝑃 𝑄) 𝑆) 𝑇))))
48 hlol 33666 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
491, 48syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ OL)
506, 26latmassOLD 33534 . . . 4 ((𝐾 ∈ OL ∧ (((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾))) → ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)) = (((𝑃 𝑄) 𝑇) (((𝑃 𝑄) 𝑆) (𝑆 𝑇))))
5149, 23, 25, 33, 50syl13anc 1320 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)) = (((𝑃 𝑄) 𝑇) (((𝑃 𝑄) 𝑆) (𝑆 𝑇))))
5241, 47, 513eqtr4rd 2655 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)) = ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))
5336, 52breqtrd 4609 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868  OLcol 33479  Atomscatm 33568  HLchlt 33655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-psubsp 33807  df-pmap 33808  df-padd 34100 This theorem is referenced by:  dalawlem5  34179  dalawlem8  34182
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