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Theorem trlval 34467
 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 34466 . . 3 ((𝐾𝑉𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
109fveq1d 6105 . 2 ((𝐾𝑉𝑊𝐻) → (𝑅𝐹) = ((𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))‘𝐹))
11 fveq1 6102 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑝) = (𝐹𝑝))
1211oveq2d 6565 . . . . . . . 8 (𝑓 = 𝐹 → (𝑝 (𝑓𝑝)) = (𝑝 (𝐹𝑝)))
1312oveq1d 6564 . . . . . . 7 (𝑓 = 𝐹 → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑝 (𝐹𝑝)) 𝑊))
1413eqeq2d 2620 . . . . . 6 (𝑓 = 𝐹 → (𝑥 = ((𝑝 (𝑓𝑝)) 𝑊) ↔ 𝑥 = ((𝑝 (𝐹𝑝)) 𝑊)))
1514imbi2d 329 . . . . 5 (𝑓 = 𝐹 → ((¬ 𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)) ↔ (¬ 𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
1615ralbidv 2969 . . . 4 (𝑓 = 𝐹 → (∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)) ↔ ∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
1716riotabidv 6513 . . 3 (𝑓 = 𝐹 → (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
18 eqid 2610 . . 3 (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
19 riotaex 6515 . . 3 (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))) ∈ V
2017, 18, 19fvmpt 6191 . 2 (𝐹𝑇 → ((𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))‘𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
2110, 20sylan9eq 2664 1 (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  LHypclh 34288  LTrncltrn 34405  trLctrl 34463 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-trl 34464 This theorem is referenced by:  trlval2  34468
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