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Theorem dalawlem13 37893
Description: Lemma for dalaw 37896. Special case to eliminate the requirement ((𝑃 𝑄) 𝑅) ∈ 𝑂 in dalawlem1 37881. (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 1202 . 2 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
2 simp12 1203 . . 3 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂)
3 simp22 1206 . . . . . . 7 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄𝐴)
4 simp23 1207 . . . . . . 7 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑅𝐴)
5 simp21 1205 . . . . . . 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 37558 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑃𝐴)) → (((𝑄 𝑅) 𝑃) ∈ 𝑂 ↔ (𝑄𝑅 ∧ ¬ 𝑃 (𝑄 𝑅))))
111, 3, 4, 5, 10syl13anc 1371 . . . . . 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 2946 . . . . . . . 8 (𝑄𝑅 ↔ ¬ 𝑄 = 𝑅)
1312anbi1i 624 . . . . . . 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 274 . . . . . 6 ((𝑄𝑅 ∧ ¬ 𝑃 (𝑄 𝑅)) ↔ ¬ (𝑄 = 𝑅𝑃 (𝑄 𝑅)))
1611, 15bitr2di 288 . . . . 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 37383 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑃𝐴)) → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
181, 3, 4, 5, 17syl13anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
1918eleq1d 2825 . . . . 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 278 . . . 4 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (¬ (𝑄 = 𝑅𝑃 (𝑄 𝑅)) ↔ ((𝑃 𝑄) 𝑅) ∈ 𝑂))
2120con1bid 356 . . 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 231 . 2 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 = 𝑅𝑃 (𝑄 𝑅)))
23 simp13 1204 . 2 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))
24 simp2 1136 . 2 (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃𝐴𝑄𝐴𝑅𝐴))
25 simp3 1137 . 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 37892 . . . . . . 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 1121 . . . . . 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 1118 . . . . 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 37891 . . . . . . 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 1121 . . . . . 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 1118 . . . . 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 1110 . . 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 1115 . 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 1383 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 205  wa 396  wo 844  w3a 1086   = wceq 1542  wcel 2110  wne 2945   class class class wbr 5079  cfv 6432  (class class class)co 7271  lecple 16967  joincjn 18027  meetcmee 18028  Atomscatm 37273  HLchlt 37360  LPlanesclpl 37502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-proset 18011  df-poset 18029  df-plt 18046  df-lub 18062  df-glb 18063  df-join 18064  df-meet 18065  df-p0 18141  df-lat 18148  df-clat 18215  df-oposet 37186  df-ol 37188  df-oml 37189  df-covers 37276  df-ats 37277  df-atl 37308  df-cvlat 37332  df-hlat 37361  df-llines 37508  df-lplanes 37509  df-psubsp 37513  df-pmap 37514  df-padd 37806
This theorem is referenced by:  dalawlem14  37894
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