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Theorem trlval 39635
Description: The value of the trace of a lattice translation. (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
trlval (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜πΉ) = (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))))
Distinct variable groups:   𝐴,𝑝   π‘₯,𝐡   π‘₯,𝑝,𝐾   π‘Š,𝑝,π‘₯   𝐹,𝑝,π‘₯
Allowed substitution hints:   𝐴(π‘₯)   𝐡(𝑝)   𝑅(π‘₯,𝑝)   𝑇(π‘₯,𝑝)   𝐻(π‘₯,𝑝)   ∨ (π‘₯,𝑝)   ≀ (π‘₯,𝑝)   ∧ (π‘₯,𝑝)   𝑉(π‘₯,𝑝)

Proof of Theorem trlval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 trlset.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 trlset.l . . . 4 ≀ = (leβ€˜πΎ)
3 trlset.j . . . 4 ∨ = (joinβ€˜πΎ)
4 trlset.m . . . 4 ∧ = (meetβ€˜πΎ)
5 trlset.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
6 trlset.h . . . 4 𝐻 = (LHypβ€˜πΎ)
7 trlset.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
8 trlset.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
91, 2, 3, 4, 5, 6, 7, 8trlset 39634 . . 3 ((𝐾 ∈ 𝑉 ∧ 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)))))
109fveq1d 6899 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (π‘…β€˜πΉ) = ((𝑓 ∈ 𝑇 ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š))))β€˜πΉ))
11 fveq1 6896 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘) = (πΉβ€˜π‘))
1211oveq2d 7436 . . . . . . . 8 (𝑓 = 𝐹 β†’ (𝑝 ∨ (π‘“β€˜π‘)) = (𝑝 ∨ (πΉβ€˜π‘)))
1312oveq1d 7435 . . . . . . 7 (𝑓 = 𝐹 β†’ ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š) = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))
1413eqeq2d 2739 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š) ↔ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š)))
1514imbi2d 340 . . . . 5 (𝑓 = 𝐹 β†’ ((Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)) ↔ (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))))
1615ralbidv 3174 . . . 4 (𝑓 = 𝐹 β†’ (βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)) ↔ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))))
1716riotabidv 7378 . . 3 (𝑓 = 𝐹 β†’ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š))) = (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))))
18 eqid 2728 . . 3 (𝑓 ∈ 𝑇 ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))) = (𝑓 ∈ 𝑇 ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š))))
19 riotaex 7380 . . 3 (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))) ∈ V
2017, 18, 19fvmpt 7005 . 2 (𝐹 ∈ 𝑇 β†’ ((𝑓 ∈ 𝑇 ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š))))β€˜πΉ) = (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))))
2110, 20sylan9eq 2788 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜πΉ) = (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058   class class class wbr 5148   ↦ cmpt 5231  β€˜cfv 6548  β„©crio 7375  (class class class)co 7420  Basecbs 17180  lecple 17240  joincjn 18303  meetcmee 18304  Atomscatm 38735  LHypclh 39457  LTrncltrn 39574  trLctrl 39631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-trl 39632
This theorem is referenced by:  trlval2  39636
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