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Theorem trlval 38628
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 38627 . . 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 6845 . 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 6842 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘) = (πΉβ€˜π‘))
1211oveq2d 7374 . . . . . . . 8 (𝑓 = 𝐹 β†’ (𝑝 ∨ (π‘“β€˜π‘)) = (𝑝 ∨ (πΉβ€˜π‘)))
1312oveq1d 7373 . . . . . . 7 (𝑓 = 𝐹 β†’ ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š) = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))
1413eqeq2d 2748 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š) ↔ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š)))
1514imbi2d 341 . . . . 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 3175 . . . 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 7316 . . 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 2737 . . 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 7318 . . 3 (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))) ∈ V
2017, 18, 19fvmpt 6949 . 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 2797 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065   class class class wbr 5106   ↦ cmpt 5189  β€˜cfv 6497  β„©crio 7313  (class class class)co 7358  Basecbs 17084  lecple 17141  joincjn 18201  meetcmee 18202  Atomscatm 37728  LHypclh 38450  LTrncltrn 38567  trLctrl 38624
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-trl 38625
This theorem is referenced by:  trlval2  38629
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