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Theorem trlset 40163
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 40162 . . . 4 (𝐾𝐶 → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
98fveq1d 6908 . . 3 (𝐾𝐶 → ((trL‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊))
101, 9eqtrid 2789 . 2 (𝐾𝐶𝑅 = ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊))
11 fveq2 6906 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
12 breq2 5147 . . . . . . . . 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 2748 . . . . . . . 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 3178 . . . . . 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 5233 . . . 4 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
20 eqid 2737 . . . 4 (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))
21 fvex 6919 . . . . 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 5238 . . 3 (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
2623, 25eqtr4di 2795 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊) = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
2710, 26sylan9eq 2797 1 ((𝐾𝐶𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061   class class class wbr 5143  cmpt 5225  cfv 6561  crio 7387  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  Atomscatm 39264  LHypclh 39986  LTrncltrn 40103  trLctrl 40160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-trl 40161
This theorem is referenced by:  trlval  40164
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