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Theorem dalawlem2 37080
Description: Lemma for dalaw 37094. 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 1133 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ HL)
21hllatd 36572 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ Lat)
3 simp2l 1196 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑃𝐴)
4 simp2r 1197 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑄𝐴)
5 eqid 2824 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
6 dalawlem.j . . . . . . 7 = (join‘𝐾)
7 dalawlem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
85, 6, 7hlatjcl 36575 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
91, 3, 4, 8syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
10 simp3r 1199 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑇𝐴)
115, 7atbase 36497 . . . . . 6 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
1210, 11syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑇 ∈ (Base‘𝐾))
13 dalawlem.l . . . . . 6 = (le‘𝐾)
145, 13, 6latlej1 17668 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑇))
152, 9, 12, 14syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑇))
16 simp3l 1198 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆𝐴)
175, 7atbase 36497 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1816, 17syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆 ∈ (Base‘𝐾))
195, 13, 6latlej1 17668 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
202, 9, 18, 19syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
215, 6latjcl 17659 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾))
222, 9, 12, 21syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾))
235, 6latjcl 17659 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
242, 9, 18, 23syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
25 dalawlem.m . . . . . 6 = (meet‘𝐾)
265, 13, 25latlem12 17686 . . . . 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))))
272, 9, 22, 24, 26syl13anc 1369 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) ((𝑃 𝑄) 𝑇) ∧ (𝑃 𝑄) ((𝑃 𝑄) 𝑆)) ↔ (𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆))))
2815, 20, 27mpbi2and 711 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑃 𝑄) (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)))
295, 25latmcl 17660 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) ∈ (Base‘𝐾))
302, 22, 24, 29syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) ∈ (Base‘𝐾))
315, 6, 7hlatjcl 36575 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
321, 16, 10, 31syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑆 𝑇) ∈ (Base‘𝐾))
335, 13, 25latmlem1 17689 . . . 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))))
342, 9, 30, 32, 33syl13anc 1369 . . 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))))
3528, 34mpd 15 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)))
365, 13, 6latlej2 17669 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ((𝑃 𝑄) 𝑆))
372, 9, 18, 36syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝑆 ((𝑃 𝑄) 𝑆))
385, 13, 6, 25, 7atmod3i1 37072 . . . . 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)))
391, 16, 24, 12, 37, 38syl131anc 1380 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (𝑆 (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑆) (𝑆 𝑇)))
4039oveq2d 7162 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑇) (𝑆 (((𝑃 𝑄) 𝑆) 𝑇))) = (((𝑃 𝑄) 𝑇) (((𝑃 𝑄) 𝑆) (𝑆 𝑇))))
415, 25latmcl 17660 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
422, 24, 12, 41syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
435, 13, 6, 25latmlej22 17701 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑇 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑃 𝑄) 𝑇))
442, 12, 24, 9, 43syl13anc 1369 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑃 𝑄) 𝑇))
455, 13, 6, 25, 7atmod2i2 37070 . . . 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))))
461, 16, 22, 42, 44, 45syl131anc 1380 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) = (((𝑃 𝑄) 𝑇) (𝑆 (((𝑃 𝑄) 𝑆) 𝑇))))
47 hlol 36569 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
481, 47syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → 𝐾 ∈ OL)
495, 25latmassOLD 36437 . . . 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))))
5048, 22, 24, 32, 49syl13anc 1369 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)) = (((𝑃 𝑄) 𝑇) (((𝑃 𝑄) 𝑆) (𝑆 𝑇))))
5140, 46, 503eqtr4rd 2870 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((((𝑃 𝑄) 𝑇) ((𝑃 𝑄) 𝑆)) (𝑆 𝑇)) = ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))
5235, 51breqtrd 5079 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115   class class class wbr 5053  cfv 6344  (class class class)co 7146  Basecbs 16481  lecple 16570  joincjn 17552  meetcmee 17553  Latclat 17653  OLcol 36382  Atomscatm 36471  HLchlt 36558
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-iin 4909  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-proset 17536  df-poset 17554  df-plt 17566  df-lub 17582  df-glb 17583  df-join 17584  df-meet 17585  df-p0 17647  df-lat 17654  df-clat 17716  df-oposet 36384  df-ol 36386  df-oml 36387  df-covers 36474  df-ats 36475  df-atl 36506  df-cvlat 36530  df-hlat 36559  df-psubsp 36711  df-pmap 36712  df-padd 37004
This theorem is referenced by:  dalawlem5  37083  dalawlem8  37086
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