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Theorem dalawlem10 39503
Description: Lemma for dalaw 39509. Combine dalawlem5 39498, dalawlem8 39501, and dalawlem9 . (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
dalawlem10 (((𝐾 ∈ HL ∧ ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Proof of Theorem dalawlem10
StepHypRef Expression
1 simp11 1200 . 2 (((𝐾 ∈ HL ∧ ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
2 simp12 1201 . . 3 (((𝐾 ∈ HL ∧ ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)))
3 3oran 1106 . . 3 ((((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∨ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∨ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ↔ ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)))
42, 3sylibr 233 . 2 (((𝐾 ∈ HL ∧ ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∨ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∨ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)))
5 simp13 1202 . 2 (((𝐾 ∈ HL ∧ ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))
6 simp2 1134 . 2 (((𝐾 ∈ HL ∧ ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃𝐴𝑄𝐴𝑅𝐴))
7 simp3 1135 . 2 (((𝐾 ∈ HL ∧ ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆𝐴𝑇𝐴𝑈𝐴))
8 dalawlem.l . . . . . . . 8 = (le‘𝐾)
9 dalawlem.j . . . . . . . 8 = (join‘𝐾)
10 dalawlem.m . . . . . . . 8 = (meet‘𝐾)
11 dalawlem.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
128, 9, 10, 11dalawlem5 39498 . . . . . . 7 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
13123expib 1119 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
14133exp 1116 . . . . 5 (𝐾 ∈ HL → (((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
158, 9, 10, 11dalawlem8 39501 . . . . . . 7 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
16153expib 1119 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
17163exp 1116 . . . . 5 (𝐾 ∈ HL → (((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
188, 9, 10, 11dalawlem9 39502 . . . . . . 7 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
19183expib 1119 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
20193exp 1116 . . . . 5 (𝐾 ∈ HL → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
2114, 17, 203jaod 1425 . . . 4 (𝐾 ∈ HL → ((((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∨ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∨ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
22213imp 1108 . . 3 ((𝐾 ∈ HL ∧ (((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∨ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∨ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
23223impib 1113 . 2 (((𝐾 ∈ HL ∧ (((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∨ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∨ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
241, 4, 5, 6, 7, 23syl311anc 1381 1 (((𝐾 ∈ HL ∧ ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3o 1083  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5149  cfv 6549  (class class class)co 7419  lecple 17259  joincjn 18322  meetcmee 18323  Atomscatm 38885  HLchlt 38972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-proset 18306  df-poset 18324  df-plt 18341  df-lub 18357  df-glb 18358  df-join 18359  df-meet 18360  df-p0 18436  df-lat 18443  df-clat 18510  df-oposet 38798  df-ol 38800  df-oml 38801  df-covers 38888  df-ats 38889  df-atl 38920  df-cvlat 38944  df-hlat 38973  df-psubsp 39126  df-pmap 39127  df-padd 39419
This theorem is referenced by:  dalawlem14  39507
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