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Theorem trlfset 39026
Description: The set of all traces of lattice translations for a lattice 𝐾. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
trlset.b 𝐡 = (Baseβ€˜πΎ)
trlset.l ≀ = (leβ€˜πΎ)
trlset.j ∨ = (joinβ€˜πΎ)
trlset.m ∧ = (meetβ€˜πΎ)
trlset.a 𝐴 = (Atomsβ€˜πΎ)
trlset.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
trlfset (𝐾 ∈ 𝐢 β†’ (trLβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀))))))
Distinct variable groups:   𝐴,𝑝   π‘₯,𝐡   𝑀,𝐻   𝑓,𝑝,𝑀,π‘₯,𝐾
Allowed substitution hints:   𝐴(π‘₯,𝑀,𝑓)   𝐡(𝑀,𝑓,𝑝)   𝐢(π‘₯,𝑀,𝑓,𝑝)   𝐻(π‘₯,𝑓,𝑝)   ∨ (π‘₯,𝑀,𝑓,𝑝)   ≀ (π‘₯,𝑀,𝑓,𝑝)   ∧ (π‘₯,𝑀,𝑓,𝑝)

Proof of Theorem trlfset
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝐢 β†’ 𝐾 ∈ V)
2 fveq2 6891 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 trlset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2790 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6891 . . . . . 6 (π‘˜ = 𝐾 β†’ (LTrnβ€˜π‘˜) = (LTrnβ€˜πΎ))
65fveq1d 6893 . . . . 5 (π‘˜ = 𝐾 β†’ ((LTrnβ€˜π‘˜)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘€))
7 fveq2 6891 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
8 trlset.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
97, 8eqtr4di 2790 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
10 fveq2 6891 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
11 trlset.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
1210, 11eqtr4di 2790 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
13 fveq2 6891 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = (leβ€˜πΎ))
14 trlset.l . . . . . . . . . . 11 ≀ = (leβ€˜πΎ)
1513, 14eqtr4di 2790 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = ≀ )
1615breqd 5159 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (𝑝(leβ€˜π‘˜)𝑀 ↔ 𝑝 ≀ 𝑀))
1716notbid 317 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (Β¬ 𝑝(leβ€˜π‘˜)𝑀 ↔ Β¬ 𝑝 ≀ 𝑀))
18 fveq2 6891 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (meetβ€˜π‘˜) = (meetβ€˜πΎ))
19 trlset.m . . . . . . . . . . 11 ∧ = (meetβ€˜πΎ)
2018, 19eqtr4di 2790 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (meetβ€˜π‘˜) = ∧ )
21 fveq2 6891 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = (joinβ€˜πΎ))
22 trlset.j . . . . . . . . . . . 12 ∨ = (joinβ€˜πΎ)
2321, 22eqtr4di 2790 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = ∨ )
2423oveqd 7425 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (𝑝(joinβ€˜π‘˜)(π‘“β€˜π‘)) = (𝑝 ∨ (π‘“β€˜π‘)))
25 eqidd 2733 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ 𝑀 = 𝑀)
2620, 24, 25oveq123d 7429 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ ((𝑝(joinβ€˜π‘˜)(π‘“β€˜π‘))(meetβ€˜π‘˜)𝑀) = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀))
2726eqeq2d 2743 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (π‘₯ = ((𝑝(joinβ€˜π‘˜)(π‘“β€˜π‘))(meetβ€˜π‘˜)𝑀) ↔ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀)))
2817, 27imbi12d 344 . . . . . . 7 (π‘˜ = 𝐾 β†’ ((Β¬ 𝑝(leβ€˜π‘˜)𝑀 β†’ π‘₯ = ((𝑝(joinβ€˜π‘˜)(π‘“β€˜π‘))(meetβ€˜π‘˜)𝑀)) ↔ (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀))))
2912, 28raleqbidv 3342 . . . . . 6 (π‘˜ = 𝐾 β†’ (βˆ€π‘ ∈ (Atomsβ€˜π‘˜)(Β¬ 𝑝(leβ€˜π‘˜)𝑀 β†’ π‘₯ = ((𝑝(joinβ€˜π‘˜)(π‘“β€˜π‘))(meetβ€˜π‘˜)𝑀)) ↔ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀))))
309, 29riotaeqbidv 7367 . . . . 5 (π‘˜ = 𝐾 β†’ (β„©π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘ ∈ (Atomsβ€˜π‘˜)(Β¬ 𝑝(leβ€˜π‘˜)𝑀 β†’ π‘₯ = ((𝑝(joinβ€˜π‘˜)(π‘“β€˜π‘))(meetβ€˜π‘˜)𝑀))) = (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀))))
316, 30mpteq12dv 5239 . . . 4 (π‘˜ = 𝐾 β†’ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (β„©π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘ ∈ (Atomsβ€˜π‘˜)(Β¬ 𝑝(leβ€˜π‘˜)𝑀 β†’ π‘₯ = ((𝑝(joinβ€˜π‘˜)(π‘“β€˜π‘))(meetβ€˜π‘˜)𝑀)))) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀)))))
324, 31mpteq12dv 5239 . . 3 (π‘˜ = 𝐾 β†’ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (β„©π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘ ∈ (Atomsβ€˜π‘˜)(Β¬ 𝑝(leβ€˜π‘˜)𝑀 β†’ π‘₯ = ((𝑝(joinβ€˜π‘˜)(π‘“β€˜π‘))(meetβ€˜π‘˜)𝑀))))) = (𝑀 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀))))))
33 df-trl 39025 . . 3 trL = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (β„©π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘ ∈ (Atomsβ€˜π‘˜)(Β¬ 𝑝(leβ€˜π‘˜)𝑀 β†’ π‘₯ = ((𝑝(joinβ€˜π‘˜)(π‘“β€˜π‘))(meetβ€˜π‘˜)𝑀))))))
3432, 33, 3mptfvmpt 7229 . 2 (𝐾 ∈ V β†’ (trLβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀))))))
351, 34syl 17 1 (𝐾 ∈ 𝐢 β†’ (trLβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀))))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   class class class wbr 5148   ↦ cmpt 5231  β€˜cfv 6543  β„©crio 7363  (class class class)co 7408  Basecbs 17143  lecple 17203  joincjn 18263  meetcmee 18264  Atomscatm 38128  LHypclh 38850  LTrncltrn 38967  trLctrl 39024
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-trl 39025
This theorem is referenced by:  trlset  39027
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