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Theorem dalawlem13 37172
 Description: Lemma for dalaw 37175. Special case to eliminate the requirement ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 in dalawlem1 37160. (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)
dalawlem2.o 𝑂 = (LPlanes‘𝐾)
Assertion
Ref Expression
dalawlem13 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Proof of Theorem dalawlem13
StepHypRef Expression
1 simp11 1200 . 2 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
2 simp12 1201 . . 3 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂)
3 simp22 1204 . . . . . . 7 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄𝐴)
4 simp23 1205 . . . . . . 7 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑅𝐴)
5 simp21 1203 . . . . . . 7 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃𝐴)
6 dalawlem.l . . . . . . . 8 = (le‘𝐾)
7 dalawlem.j . . . . . . . 8 = (join‘𝐾)
8 dalawlem.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
9 dalawlem2.o . . . . . . . 8 𝑂 = (LPlanes‘𝐾)
106, 7, 8, 9islpln2a 36837 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑃𝐴)) → (((𝑄 𝑅) 𝑃) ∈ 𝑂 ↔ (𝑄𝑅 ∧ ¬ 𝑃 (𝑄 𝑅))))
111, 3, 4, 5, 10syl13anc 1369 . . . . . 6 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑅) 𝑃) ∈ 𝑂 ↔ (𝑄𝑅 ∧ ¬ 𝑃 (𝑄 𝑅))))
12 df-ne 2991 . . . . . . . 8 (𝑄𝑅 ↔ ¬ 𝑄 = 𝑅)
1312anbi1i 626 . . . . . . 7 ((𝑄𝑅 ∧ ¬ 𝑃 (𝑄 𝑅)) ↔ (¬ 𝑄 = 𝑅 ∧ ¬ 𝑃 (𝑄 𝑅)))
14 pm4.56 986 . . . . . . 7 ((¬ 𝑄 = 𝑅 ∧ ¬ 𝑃 (𝑄 𝑅)) ↔ ¬ (𝑄 = 𝑅𝑃 (𝑄 𝑅)))
1513, 14bitri 278 . . . . . 6 ((𝑄𝑅 ∧ ¬ 𝑃 (𝑄 𝑅)) ↔ ¬ (𝑄 = 𝑅𝑃 (𝑄 𝑅)))
1611, 15syl6rbb 291 . . . . 5 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (¬ (𝑄 = 𝑅𝑃 (𝑄 𝑅)) ↔ ((𝑄 𝑅) 𝑃) ∈ 𝑂))
177, 8hlatjrot 36662 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑃𝐴)) → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
181, 3, 4, 5, 17syl13anc 1369 . . . . . 6 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
1918eleq1d 2877 . . . . 5 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑅) 𝑃) ∈ 𝑂 ↔ ((𝑃 𝑄) 𝑅) ∈ 𝑂))
2016, 19bitrd 282 . . . 4 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (¬ (𝑄 = 𝑅𝑃 (𝑄 𝑅)) ↔ ((𝑃 𝑄) 𝑅) ∈ 𝑂))
2120con1bid 359 . . 3 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ↔ (𝑄 = 𝑅𝑃 (𝑄 𝑅))))
222, 21mpbid 235 . 2 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 = 𝑅𝑃 (𝑄 𝑅)))
23 simp13 1202 . 2 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))
24 simp2 1134 . 2 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃𝐴𝑄𝐴𝑅𝐴))
25 simp3 1135 . 2 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆𝐴𝑇𝐴𝑈𝐴))
26 dalawlem.m . . . . . . . 8 = (meet‘𝐾)
276, 7, 26, 8dalawlem12 37171 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
28273expib 1119 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
29283exp 1116 . . . . 5 (𝐾 ∈ HL → (𝑄 = 𝑅 → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
306, 7, 26, 8dalawlem11 37170 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 (𝑄 𝑅) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
31303expib 1119 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 (𝑄 𝑅) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
32313exp 1116 . . . . 5 (𝐾 ∈ HL → (𝑃 (𝑄 𝑅) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
3329, 32jaod 856 . . . 4 (𝐾 ∈ HL → ((𝑄 = 𝑅𝑃 (𝑄 𝑅)) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))))
34333imp 1108 . . 3 ((𝐾 ∈ HL ∧ (𝑄 = 𝑅𝑃 (𝑄 𝑅)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) → (((𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
35343impib 1113 . 2 (((𝐾 ∈ HL ∧ (𝑄 = 𝑅𝑃 (𝑄 𝑅)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
361, 22, 23, 24, 25, 35syl311anc 1381 1 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((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   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  lecple 16567  joincjn 17549  meetcmee 17550  Atomscatm 36552  HLchlt 36639  LPlanesclpl 36781 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-lat 17651  df-clat 17713  df-oposet 36465  df-ol 36467  df-oml 36468  df-covers 36555  df-ats 36556  df-atl 36587  df-cvlat 36611  df-hlat 36640  df-llines 36787  df-lplanes 36788  df-psubsp 36792  df-pmap 36793  df-padd 37085 This theorem is referenced by:  dalawlem14  37173
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