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Theorem trlval2 35768
Description: The value of the trace of a lattice translation, given any atom 𝑃 not under the fiducial co-atom 𝑊. Note: this requires only the weaker assumption 𝐾 ∈ Lat; we use 𝐾 ∈ HL for convenience. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
trlval2.l = (le‘𝐾)
trlval2.j = (join‘𝐾)
trlval2.m = (meet‘𝐾)
trlval2.a 𝐴 = (Atoms‘𝐾)
trlval2.h 𝐻 = (LHyp‘𝐾)
trlval2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlval2.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
trlval2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))

Proof of Theorem trlval2
Dummy variables 𝑥 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hllat 34968 . . 3 (𝐾 ∈ HL → 𝐾 ∈ Lat)
21anim1i 591 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐾 ∈ Lat ∧ 𝑊𝐻))
3 eqid 2651 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
4 trlval2.l . . . . 5 = (le‘𝐾)
5 trlval2.j . . . . 5 = (join‘𝐾)
6 trlval2.m . . . . 5 = (meet‘𝐾)
7 trlval2.a . . . . 5 𝐴 = (Atoms‘𝐾)
8 trlval2.h . . . . 5 𝐻 = (LHyp‘𝐾)
9 trlval2.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
10 trlval2.r . . . . 5 𝑅 = ((trL‘𝐾)‘𝑊)
113, 4, 5, 6, 7, 8, 9, 10trlval 35767 . . . 4 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑥 ∈ (Base‘𝐾)∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
12113adant3 1101 . . 3 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = (𝑥 ∈ (Base‘𝐾)∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
13 simp1l 1105 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐾 ∈ Lat)
14 simp3l 1109 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃𝐴)
153, 7atbase 34894 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1614, 15syl 17 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ∈ (Base‘𝐾))
173, 8, 9ltrncl 35729 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
1816, 17syld3an3 1411 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) ∈ (Base‘𝐾))
193, 5latjcl 17098 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
2013, 16, 18, 19syl3anc 1366 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
21 simp1r 1106 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊𝐻)
223, 8lhpbase 35602 . . . . . 6 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2321, 22syl 17 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑊 ∈ (Base‘𝐾))
243, 6latmcl 17099 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) 𝑊) ∈ (Base‘𝐾))
2513, 20, 23, 24syl3anc 1366 . . . 4 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑃 (𝐹𝑃)) 𝑊) ∈ (Base‘𝐾))
26 simpl3l 1136 . . . . . 6 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑃𝐴)
27 simpl3r 1137 . . . . . 6 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → ¬ 𝑃 𝑊)
28 breq1 4688 . . . . . . . . . 10 (𝑞 = 𝑃 → (𝑞 𝑊𝑃 𝑊))
2928notbid 307 . . . . . . . . 9 (𝑞 = 𝑃 → (¬ 𝑞 𝑊 ↔ ¬ 𝑃 𝑊))
30 id 22 . . . . . . . . . . . 12 (𝑞 = 𝑃𝑞 = 𝑃)
31 fveq2 6229 . . . . . . . . . . . 12 (𝑞 = 𝑃 → (𝐹𝑞) = (𝐹𝑃))
3230, 31oveq12d 6708 . . . . . . . . . . 11 (𝑞 = 𝑃 → (𝑞 (𝐹𝑞)) = (𝑃 (𝐹𝑃)))
3332oveq1d 6705 . . . . . . . . . 10 (𝑞 = 𝑃 → ((𝑞 (𝐹𝑞)) 𝑊) = ((𝑃 (𝐹𝑃)) 𝑊))
3433eqeq2d 2661 . . . . . . . . 9 (𝑞 = 𝑃 → (𝑥 = ((𝑞 (𝐹𝑞)) 𝑊) ↔ 𝑥 = ((𝑃 (𝐹𝑃)) 𝑊)))
3529, 34imbi12d 333 . . . . . . . 8 (𝑞 = 𝑃 → ((¬ 𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ (¬ 𝑃 𝑊𝑥 = ((𝑃 (𝐹𝑃)) 𝑊))))
3635rspcv 3336 . . . . . . 7 (𝑃𝐴 → (∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) → (¬ 𝑃 𝑊𝑥 = ((𝑃 (𝐹𝑃)) 𝑊))))
3736com23 86 . . . . . 6 (𝑃𝐴 → (¬ 𝑃 𝑊 → (∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) → 𝑥 = ((𝑃 (𝐹𝑃)) 𝑊))))
3826, 27, 37sylc 65 . . . . 5 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) → 𝑥 = ((𝑃 (𝐹𝑃)) 𝑊)))
39 simp11 1111 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → (𝐾 ∈ Lat ∧ 𝑊𝐻))
40 simp12 1112 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → 𝐹𝑇)
41 simp13l 1196 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → 𝑃𝐴)
42 simp13r 1197 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → ¬ 𝑃 𝑊)
43 simp3 1083 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → 𝑞𝐴)
44 simp2 1082 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → ¬ 𝑞 𝑊)
454, 5, 6, 7, 8, 9ltrnu 35725 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
4639, 40, 41, 42, 43, 44, 45syl222anc 1382 . . . . . . . . . 10 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
47 eqeq2 2662 . . . . . . . . . . 11 (((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) ↔ 𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))
4847biimpd 219 . . . . . . . . . 10 (((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → 𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))
4946, 48syl 17 . . . . . . . . 9 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ¬ 𝑞 𝑊𝑞𝐴) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → 𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))
50493exp 1283 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (¬ 𝑞 𝑊 → (𝑞𝐴 → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → 𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))))
5150com24 95 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → (𝑞𝐴 → (¬ 𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)))))
5251ralrimdv 2997 . . . . . 6 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → ∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
5352adantr 480 . . . . 5 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = ((𝑃 (𝐹𝑃)) 𝑊) → ∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))))
5438, 53impbid 202 . . . 4 ((((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ 𝑥 = ((𝑃 (𝐹𝑃)) 𝑊)))
5525, 54riota5 6677 . . 3 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑥 ∈ (Base‘𝐾)∀𝑞𝐴𝑞 𝑊𝑥 = ((𝑞 (𝐹𝑞)) 𝑊))) = ((𝑃 (𝐹𝑃)) 𝑊))
5612, 55eqtrd 2685 . 2 (((𝐾 ∈ Lat ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
572, 56syl3an1 1399 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941   class class class wbr 4685  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Latclat 17092  Atomscatm 34868  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  trLctrl 35763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-lat 17093  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764
This theorem is referenced by:  trlcl  35769  trlcnv  35770  trljat1  35771  trljat2  35772  trlat  35774  trl0  35775  trlle  35789  trlval3  35792  trlval5  35794  cdlemd6  35808  cdlemf  36168  cdlemg4a  36213  cdlemg4b1  36214  cdlemg4b2  36215  cdlemg4  36222  cdlemg11b  36247  cdlemg13a  36256  cdlemg13  36257  cdlemg17a  36266  cdlemg17dN  36268  cdlemg17e  36270  cdlemg17f  36271  trlcoabs2N  36327  trlcolem  36331  cdlemg42  36334  cdlemg43  36335  cdlemi1  36423  cdlemk4  36439  cdlemk39  36521  dia2dimlem1  36670  dia2dimlem2  36671  dia2dimlem3  36672  cdlemm10N  36724  cdlemn2  36801  cdlemn10  36812  dihjatcclem3  37026
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