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Theorem 3dimlem3OLDN 36154
Description: Lemma for 3dim1 36159. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
3dim0.j = (join‘𝐾)
3dim0.l = (le‘𝐾)
3dim0.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
3dimlem3OLDN ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑇 ((𝑃 𝑄) 𝑅)))

Proof of Theorem 3dimlem3OLDN
StepHypRef Expression
1 simpr1 1187 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑃𝑄)
2 simpr2 1188 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → ¬ 𝑃 (𝑄 𝑅))
3 simpl11 1241 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝐾 ∈ HL)
4 simpl2l 1219 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑅𝐴)
5 simpl12 1242 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑃𝐴)
6 simpl13 1243 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑄𝐴)
7 simpl3l 1221 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑄𝑅)
87necomd 3039 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑅𝑄)
9 3dim0.l . . . . . 6 = (le‘𝐾)
10 3dim0.j . . . . . 6 = (join‘𝐾)
11 3dim0.a . . . . . 6 𝐴 = (Atoms‘𝐾)
129, 10, 11hlatexch2 36088 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑃𝐴𝑄𝐴) ∧ 𝑅𝑄) → (𝑅 (𝑃 𝑄) → 𝑃 (𝑅 𝑄)))
133, 4, 5, 6, 8, 12syl131anc 1376 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑅 (𝑃 𝑄) → 𝑃 (𝑅 𝑄)))
1410, 11hlatjcom 36060 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) = (𝑅 𝑄))
153, 6, 4, 14syl3anc 1364 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑄 𝑅) = (𝑅 𝑄))
1615breq2d 4978 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑃 (𝑄 𝑅) ↔ 𝑃 (𝑅 𝑄)))
1713, 16sylibrd 260 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑅 (𝑃 𝑄) → 𝑃 (𝑄 𝑅)))
182, 17mtod 199 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → ¬ 𝑅 (𝑃 𝑄))
19 simpl3r 1222 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → ¬ 𝑇 ((𝑄 𝑅) 𝑆))
20 hllat 36055 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213, 20syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝐾 ∈ Lat)
22 eqid 2795 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2322, 11atbase 35981 . . . . . . 7 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
246, 23syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑄 ∈ (Base‘𝐾))
2522, 11atbase 35981 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
264, 25syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑅 ∈ (Base‘𝐾))
2722, 11atbase 35981 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
285, 27syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑃 ∈ (Base‘𝐾))
2922, 10latjrot 17544 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
3021, 24, 26, 28, 29syl13anc 1365 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
31 simpr3 1189 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑃 ((𝑄 𝑅) 𝑆))
32 simpl2r 1220 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → 𝑆𝐴)
3322, 10, 11hlatjcl 36059 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
343, 6, 4, 33syl3anc 1364 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑄 𝑅) ∈ (Base‘𝐾))
3522, 9, 10, 11hlexchb1 36076 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑆𝐴 ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) ∧ ¬ 𝑃 (𝑄 𝑅)) → (𝑃 ((𝑄 𝑅) 𝑆) ↔ ((𝑄 𝑅) 𝑃) = ((𝑄 𝑅) 𝑆)))
363, 5, 32, 34, 2, 35syl131anc 1376 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑃 ((𝑄 𝑅) 𝑆) ↔ ((𝑄 𝑅) 𝑃) = ((𝑄 𝑅) 𝑆)))
3731, 36mpbid 233 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → ((𝑄 𝑅) 𝑃) = ((𝑄 𝑅) 𝑆))
3830, 37eqtr3d 2833 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → ((𝑃 𝑄) 𝑅) = ((𝑄 𝑅) 𝑆))
3938breq2d 4978 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑇 ((𝑃 𝑄) 𝑅) ↔ 𝑇 ((𝑄 𝑅) 𝑆)))
4019, 39mtbird 326 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → ¬ 𝑇 ((𝑃 𝑄) 𝑅))
411, 18, 403jca 1121 1 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))) ∧ (𝑃𝑄 ∧ ¬ 𝑃 (𝑄 𝑅) ∧ 𝑃 ((𝑄 𝑅) 𝑆))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑇 ((𝑃 𝑄) 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984   class class class wbr 4966  cfv 6230  (class class class)co 7021  Basecbs 16317  lecple 16406  joincjn 17388  Latclat 17489  Atomscatm 35955  HLchlt 36042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-proset 17372  df-poset 17390  df-plt 17402  df-lub 17418  df-glb 17419  df-join 17420  df-meet 17421  df-p0 17483  df-lat 17490  df-covers 35958  df-ats 35959  df-atl 35990  df-cvlat 36014  df-hlat 36043
This theorem is referenced by: (None)
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