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Theorem trlset 39496
Description: The set of traces of lattice translations for a fiducial co-atom π‘Š. (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)
trlset.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
trlset.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
trlset ((𝐾 ∈ 𝐢 ∧ π‘Š ∈ 𝐻) β†’ 𝑅 = (𝑓 ∈ 𝑇 ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))))
Distinct variable groups:   𝐴,𝑝   π‘₯,𝐡   𝑓,𝑝,π‘₯,𝐾   𝑇,𝑓   𝑓,π‘Š,𝑝,π‘₯
Allowed substitution hints:   𝐴(π‘₯,𝑓)   𝐡(𝑓,𝑝)   𝐢(π‘₯,𝑓,𝑝)   𝑅(π‘₯,𝑓,𝑝)   𝑇(π‘₯,𝑝)   𝐻(π‘₯,𝑓,𝑝)   ∨ (π‘₯,𝑓,𝑝)   ≀ (π‘₯,𝑓,𝑝)   ∧ (π‘₯,𝑓,𝑝)

Proof of Theorem trlset
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 trlset.r . . 3 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
2 trlset.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
3 trlset.l . . . . 5 ≀ = (leβ€˜πΎ)
4 trlset.j . . . . 5 ∨ = (joinβ€˜πΎ)
5 trlset.m . . . . 5 ∧ = (meetβ€˜πΎ)
6 trlset.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
7 trlset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
82, 3, 4, 5, 6, 7trlfset 39495 . . . 4 (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))))))
98fveq1d 6893 . . 3 (𝐾 ∈ 𝐢 β†’ ((trLβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀)))))β€˜π‘Š))
101, 9eqtrid 2783 . 2 (𝐾 ∈ 𝐢 β†’ 𝑅 = ((𝑀 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀)))))β€˜π‘Š))
11 fveq2 6891 . . . . 5 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
12 breq2 5152 . . . . . . . . 9 (𝑀 = π‘Š β†’ (𝑝 ≀ 𝑀 ↔ 𝑝 ≀ π‘Š))
1312notbid 318 . . . . . . . 8 (𝑀 = π‘Š β†’ (Β¬ 𝑝 ≀ 𝑀 ↔ Β¬ 𝑝 ≀ π‘Š))
14 oveq2 7420 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀) = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š))
1514eqeq2d 2742 . . . . . . . 8 (𝑀 = π‘Š β†’ (π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀) ↔ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))
1613, 15imbi12d 344 . . . . . . 7 (𝑀 = π‘Š β†’ ((Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀)) ↔ (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š))))
1716ralbidv 3176 . . . . . 6 (𝑀 = π‘Š β†’ (βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀)) ↔ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š))))
1817riotabidv 7370 . . . . 5 (𝑀 = π‘Š β†’ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀))) = (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š))))
1911, 18mpteq12dv 5239 . . . 4 (𝑀 = π‘Š β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀)))) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))))
20 eqid 2731 . . . 4 (𝑀 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜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)))))
21 fvex 6904 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V
2221mptex 7227 . . . 4 (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))) ∈ V
2319, 20, 22fvmpt 6998 . . 3 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀)))))β€˜π‘Š) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))))
24 trlset.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
2524mpteq1i 5244 . . 3 (𝑓 ∈ 𝑇 ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š))))
2623, 25eqtr4di 2789 . 2 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀)))))β€˜π‘Š) = (𝑓 ∈ 𝑇 ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))))
2710, 26sylan9eq 2791 1 ((𝐾 ∈ 𝐢 ∧ π‘Š ∈ 𝐻) β†’ 𝑅 = (𝑓 ∈ 𝑇 ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060   class class class wbr 5148   ↦ cmpt 5231  β€˜cfv 6543  β„©crio 7367  (class class class)co 7412  Basecbs 17151  lecple 17211  joincjn 18274  meetcmee 18275  Atomscatm 38597  LHypclh 39319  LTrncltrn 39436  trLctrl 39493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-trl 39494
This theorem is referenced by:  trlval  39497
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