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Theorem trlset 40144
Description: The set of traces of lattice translations for a fiducial co-atom 𝑊. (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
trlset ((𝐾𝐶𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑓,𝑝,𝑥,𝐾   𝑇,𝑓   𝑓,𝑊,𝑝,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐵(𝑓,𝑝)   𝐶(𝑥,𝑓,𝑝)   𝑅(𝑥,𝑓,𝑝)   𝑇(𝑥,𝑝)   𝐻(𝑥,𝑓,𝑝)   (𝑥,𝑓,𝑝)   (𝑥,𝑓,𝑝)   (𝑥,𝑓,𝑝)

Proof of Theorem trlset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 trlset.r . . 3 𝑅 = ((trL‘𝐾)‘𝑊)
2 trlset.b . . . . 5 𝐵 = (Base‘𝐾)
3 trlset.l . . . . 5 = (le‘𝐾)
4 trlset.j . . . . 5 = (join‘𝐾)
5 trlset.m . . . . 5 = (meet‘𝐾)
6 trlset.a . . . . 5 𝐴 = (Atoms‘𝐾)
7 trlset.h . . . . 5 𝐻 = (LHyp‘𝐾)
82, 3, 4, 5, 6, 7trlfset 40143 . . . 4 (𝐾𝐶 → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
98fveq1d 6909 . . 3 (𝐾𝐶 → ((trL‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊))
101, 9eqtrid 2787 . 2 (𝐾𝐶𝑅 = ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊))
11 fveq2 6907 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
12 breq2 5152 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑝 𝑤𝑝 𝑊))
1312notbid 318 . . . . . . . 8 (𝑤 = 𝑊 → (¬ 𝑝 𝑤 ↔ ¬ 𝑝 𝑊))
14 oveq2 7439 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑝 (𝑓𝑝)) 𝑊))
1514eqeq2d 2746 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 = ((𝑝 (𝑓𝑝)) 𝑤) ↔ 𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))
1613, 15imbi12d 344 . . . . . . 7 (𝑤 = 𝑊 → ((¬ 𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)) ↔ (¬ 𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
1716ralbidv 3176 . . . . . 6 (𝑤 = 𝑊 → (∀𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)) ↔ ∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
1817riotabidv 7390 . . . . 5 (𝑤 = 𝑊 → (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
1911, 18mpteq12dv 5239 . . . 4 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
20 eqid 2735 . . . 4 (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))
21 fvex 6920 . . . . 5 ((LTrn‘𝐾)‘𝑊) ∈ V
2221mptex 7243 . . . 4 (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) ∈ V
2319, 20, 22fvmpt 7016 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
24 trlset.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
2524mpteq1i 5244 . . 3 (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
2623, 25eqtr4di 2793 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊) = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
2710, 26sylan9eq 2795 1 ((𝐾𝐶𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  cmpt 5231  cfv 6563  crio 7387  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Atomscatm 39245  LHypclh 39967  LTrncltrn 40084  trLctrl 40141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-trl 40142
This theorem is referenced by:  trlval  40145
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