Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trlval Structured version   Visualization version   GIF version

Theorem trlval 39630
Description: The value of the trace of a lattice translation. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
trlset.b 𝐵 = (Base‘𝐾)
trlset.l = (le‘𝐾)
trlset.j = (join‘𝐾)
trlset.m = (meet‘𝐾)
trlset.a 𝐴 = (Atoms‘𝐾)
trlset.h 𝐻 = (LHyp‘𝐾)
trlset.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlset.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
trlval (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑥,𝑝,𝐾   𝑊,𝑝,𝑥   𝐹,𝑝,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑝)   𝑅(𝑥,𝑝)   𝑇(𝑥,𝑝)   𝐻(𝑥,𝑝)   (𝑥,𝑝)   (𝑥,𝑝)   (𝑥,𝑝)   𝑉(𝑥,𝑝)

Proof of Theorem trlval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 trlset.b . . . 4 𝐵 = (Base‘𝐾)
2 trlset.l . . . 4 = (le‘𝐾)
3 trlset.j . . . 4 = (join‘𝐾)
4 trlset.m . . . 4 = (meet‘𝐾)
5 trlset.a . . . 4 𝐴 = (Atoms‘𝐾)
6 trlset.h . . . 4 𝐻 = (LHyp‘𝐾)
7 trlset.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
8 trlset.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
91, 2, 3, 4, 5, 6, 7, 8trlset 39629 . . 3 ((𝐾𝑉𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
109fveq1d 6894 . 2 ((𝐾𝑉𝑊𝐻) → (𝑅𝐹) = ((𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))‘𝐹))
11 fveq1 6891 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑝) = (𝐹𝑝))
1211oveq2d 7431 . . . . . . . 8 (𝑓 = 𝐹 → (𝑝 (𝑓𝑝)) = (𝑝 (𝐹𝑝)))
1312oveq1d 7430 . . . . . . 7 (𝑓 = 𝐹 → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑝 (𝐹𝑝)) 𝑊))
1413eqeq2d 2739 . . . . . 6 (𝑓 = 𝐹 → (𝑥 = ((𝑝 (𝑓𝑝)) 𝑊) ↔ 𝑥 = ((𝑝 (𝐹𝑝)) 𝑊)))
1514imbi2d 340 . . . . 5 (𝑓 = 𝐹 → ((¬ 𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)) ↔ (¬ 𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
1615ralbidv 3173 . . . 4 (𝑓 = 𝐹 → (∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)) ↔ ∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
1716riotabidv 7373 . . 3 (𝑓 = 𝐹 → (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
18 eqid 2728 . . 3 (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
19 riotaex 7375 . . 3 (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))) ∈ V
2017, 18, 19fvmpt 7000 . 2 (𝐹𝑇 → ((𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))‘𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
2110, 20sylan9eq 2788 1 (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3057   class class class wbr 5143  cmpt 5226  cfv 6543  crio 7370  (class class class)co 7415  Basecbs 17174  lecple 17234  joincjn 18297  meetcmee 18298  Atomscatm 38730  LHypclh 39452  LTrncltrn 39569  trLctrl 39626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7371  df-ov 7418  df-trl 39627
This theorem is referenced by:  trlval2  39631
  Copyright terms: Public domain W3C validator