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Theorem trlfset 37820
Description: The set of all traces of lattice translations for a lattice 𝐾. (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)
Assertion
Ref Expression
trlfset (𝐾𝐶 → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑤,𝐻   𝑓,𝑝,𝑤,𝑥,𝐾
Allowed substitution hints:   𝐴(𝑥,𝑤,𝑓)   𝐵(𝑤,𝑓,𝑝)   𝐶(𝑥,𝑤,𝑓,𝑝)   𝐻(𝑥,𝑓,𝑝)   (𝑥,𝑤,𝑓,𝑝)   (𝑥,𝑤,𝑓,𝑝)   (𝑥,𝑤,𝑓,𝑝)

Proof of Theorem trlfset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3417 . 2 (𝐾𝐶𝐾 ∈ V)
2 fveq2 6677 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 trlset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2792 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6677 . . . . . 6 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
65fveq1d 6679 . . . . 5 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7 fveq2 6677 . . . . . . 7 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
8 trlset.b . . . . . . 7 𝐵 = (Base‘𝐾)
97, 8eqtr4di 2792 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
10 fveq2 6677 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
11 trlset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
1210, 11eqtr4di 2792 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
13 fveq2 6677 . . . . . . . . . . 11 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
14 trlset.l . . . . . . . . . . 11 = (le‘𝐾)
1513, 14eqtr4di 2792 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = )
1615breqd 5042 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑤𝑝 𝑤))
1716notbid 321 . . . . . . . 8 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑤 ↔ ¬ 𝑝 𝑤))
18 fveq2 6677 . . . . . . . . . . 11 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
19 trlset.m . . . . . . . . . . 11 = (meet‘𝐾)
2018, 19eqtr4di 2792 . . . . . . . . . 10 (𝑘 = 𝐾 → (meet‘𝑘) = )
21 fveq2 6677 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
22 trlset.j . . . . . . . . . . . 12 = (join‘𝐾)
2321, 22eqtr4di 2792 . . . . . . . . . . 11 (𝑘 = 𝐾 → (join‘𝑘) = )
2423oveqd 7190 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(join‘𝑘)(𝑓𝑝)) = (𝑝 (𝑓𝑝)))
25 eqidd 2740 . . . . . . . . . 10 (𝑘 = 𝐾𝑤 = 𝑤)
2620, 24, 25oveq123d 7194 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) = ((𝑝 (𝑓𝑝)) 𝑤))
2726eqeq2d 2750 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤) ↔ 𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))
2817, 27imbi12d 348 . . . . . . 7 (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤)) ↔ (¬ 𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))
2912, 28raleqbidv 3305 . . . . . 6 (𝑘 = 𝐾 → (∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤)) ↔ ∀𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))
309, 29riotaeqbidv 7133 . . . . 5 (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤))) = (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))
316, 30mpteq12dv 5116 . . . 4 (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))
324, 31mpteq12dv 5116 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤))))) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
33 df-trl 37819 . . 3 trL = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤))))))
3432, 33, 3mptfvmpt 7004 . 2 (𝐾 ∈ V → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
351, 34syl 17 1 (𝐾𝐶 → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2114  wral 3054  Vcvv 3399   class class class wbr 5031  cmpt 5111  cfv 6340  crio 7129  (class class class)co 7173  Basecbs 16589  lecple 16678  joincjn 17673  meetcmee 17674  Atomscatm 36923  LHypclh 37644  LTrncltrn 37761  trLctrl 37818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7130  df-ov 7176  df-trl 37819
This theorem is referenced by:  trlset  37821
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