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Theorem trlset 39032
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 39031 . . . 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 6894 . . 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 2785 . 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 6892 . . . . 5 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
12 breq2 5153 . . . . . . . . 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 7417 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀) = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š))
1514eqeq2d 2744 . . . . . . . 8 (𝑀 = π‘Š β†’ (π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀) ↔ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))
1613, 15imbi12d 345 . . . . . . 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 3178 . . . . . 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 7367 . . . . 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 5240 . . . 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 2733 . . . 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 6905 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V
2221mptex 7225 . . . 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 6999 . . 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 5245 . . 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 2791 . 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 2793 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   class class class wbr 5149   ↦ cmpt 5232  β€˜cfv 6544  β„©crio 7364  (class class class)co 7409  Basecbs 17144  lecple 17204  joincjn 18264  meetcmee 18265  Atomscatm 38133  LHypclh 38855  LTrncltrn 38972  trLctrl 39029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-trl 39030
This theorem is referenced by:  trlval  39033
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