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Theorem trlset 37369
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 37368 . . . 4 (𝐾𝐶 → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
98fveq1d 6661 . . 3 (𝐾𝐶 → ((trL‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊))
101, 9syl5eq 2871 . 2 (𝐾𝐶𝑅 = ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊))
11 fveq2 6659 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
12 breq2 5057 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑝 𝑤𝑝 𝑊))
1312notbid 321 . . . . . . . 8 (𝑤 = 𝑊 → (¬ 𝑝 𝑤 ↔ ¬ 𝑝 𝑊))
14 oveq2 7154 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑝 (𝑓𝑝)) 𝑊))
1514eqeq2d 2835 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 = ((𝑝 (𝑓𝑝)) 𝑤) ↔ 𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))
1613, 15imbi12d 348 . . . . . . 7 (𝑤 = 𝑊 → ((¬ 𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)) ↔ (¬ 𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
1716ralbidv 3192 . . . . . 6 (𝑤 = 𝑊 → (∀𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)) ↔ ∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
1817riotabidv 7106 . . . . 5 (𝑤 = 𝑊 → (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
1911, 18mpteq12dv 5138 . . . 4 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
20 eqid 2824 . . . 4 (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))
21 fvex 6672 . . . . 5 ((LTrn‘𝐾)‘𝑊) ∈ V
2221mptex 6975 . . . 4 (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) ∈ V
2319, 20, 22fvmpt 6757 . . 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 5143 . . 3 (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
2623, 25syl6eqr 2877 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊) = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
2710, 26sylan9eq 2879 1 ((𝐾𝐶𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133   class class class wbr 5053  cmpt 5133  cfv 6344  crio 7103  (class class class)co 7146  Basecbs 16481  lecple 16570  joincjn 17552  meetcmee 17553  Atomscatm 36471  LHypclh 37192  LTrncltrn 37309  trLctrl 37366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7104  df-ov 7149  df-trl 37367
This theorem is referenced by:  trlval  37370
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