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Theorem dalawlem2 35679
Description: Lemma for dalaw 35693. 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 1131 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ HL)
2 hllat 35171 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
31, 2syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ Lat)
4 simp2l 1242 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑃𝐴)
5 simp2r 1243 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑄𝐴)
6 eqid 2760 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
7 dalawlem.j . . . . . . 7 = (join‘𝐾)
8 dalawlem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatjcl 35174 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
101, 4, 5, 9syl3anc 1477 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
11 simp3r 1245 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑇𝐴)
126, 8atbase 35097 . . . . . 6 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
1311, 12syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑇 ∈ (Base‘𝐾))
14 dalawlem.l . . . . . 6 = (le‘𝐾)
156, 14, 7latlej1 17281 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑇))
163, 10, 13, 15syl3anc 1477 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑇))
17 simp3l 1244 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆𝐴)
186, 8atbase 35097 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆 ∈ (Base‘𝐾))
206, 14, 7latlej1 17281 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
213, 10, 19, 20syl3anc 1477 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
226, 7latjcl 17272 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾))
233, 10, 13, 22syl3anc 1477 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾))
246, 7latjcl 17272 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
253, 10, 19, 24syl3anc 1477 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
26 dalawlem.m . . . . . 6 = (meet‘𝐾)
276, 14, 26latlem12 17299 . . . . 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 1479 . . . 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 994 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)))
306, 26latmcl 17273 . . . . 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 1477 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) ∈ (Base‘𝐾))
326, 7, 8hlatjcl 35174 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
331, 17, 11, 32syl3anc 1477 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑆 𝑇) ∈ (Base‘𝐾))
346, 14, 26latmlem1 17302 . . . 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 1479 . . 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 17282 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ((𝑃 𝑄) 𝑆))
383, 10, 19, 37syl3anc 1477 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆 ((𝑃 𝑄) 𝑆))
396, 14, 7, 26, 8atmod3i1 35671 . . . . 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 1490 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑆 (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑆) (𝑆 𝑇)))
4140oveq2d 6830 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑇) (𝑆 (((𝑃 𝑄) 𝑆) 𝑇))) = (((𝑃 𝑄) 𝑇) (((𝑃 𝑄) 𝑆) (𝑆 𝑇))))
426, 26latmcl 17273 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
433, 25, 13, 42syl3anc 1477 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
446, 14, 7, 26latmlej22 17314 . . . . 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 1479 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑃 𝑄) 𝑇))
466, 14, 7, 26, 8atmod2i2 35669 . . . 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 1490 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑇) (𝑆 (((𝑃 𝑄) 𝑆) 𝑇))))
48 hlol 35169 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
491, 48syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ OL)
506, 26latmassOLD 35037 . . . 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 1479 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)) = (((𝑃 𝑄) 𝑇) (((𝑃 𝑄) 𝑆) (𝑆 𝑇))))
5241, 47, 513eqtr4rd 2805 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)) = ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))
5336, 52breqtrd 4830 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139   class class class wbr 4804  cfv 6049  (class class class)co 6814  Basecbs 16079  lecple 16170  joincjn 17165  meetcmee 17166  Latclat 17266  OLcol 34982  Atomscatm 35071  HLchlt 35158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-psubsp 35310  df-pmap 35311  df-padd 35603
This theorem is referenced by:  dalawlem5  35682  dalawlem8  35685
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