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Theorem trlval 36237
Description: The value of the trace of a lattice translation. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
trlset.b 𝐵 = (Base‘𝐾)
trlset.l = (le‘𝐾)
trlset.j = (join‘𝐾)
trlset.m = (meet‘𝐾)
trlset.a 𝐴 = (Atoms‘𝐾)
trlset.h 𝐻 = (LHyp‘𝐾)
trlset.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlset.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
trlval (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑥,𝑝,𝐾   𝑊,𝑝,𝑥   𝐹,𝑝,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑝)   𝑅(𝑥,𝑝)   𝑇(𝑥,𝑝)   𝐻(𝑥,𝑝)   (𝑥,𝑝)   (𝑥,𝑝)   (𝑥,𝑝)   𝑉(𝑥,𝑝)

Proof of Theorem trlval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 trlset.b . . . 4 𝐵 = (Base‘𝐾)
2 trlset.l . . . 4 = (le‘𝐾)
3 trlset.j . . . 4 = (join‘𝐾)
4 trlset.m . . . 4 = (meet‘𝐾)
5 trlset.a . . . 4 𝐴 = (Atoms‘𝐾)
6 trlset.h . . . 4 𝐻 = (LHyp‘𝐾)
7 trlset.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
8 trlset.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
91, 2, 3, 4, 5, 6, 7, 8trlset 36236 . . 3 ((𝐾𝑉𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
109fveq1d 6435 . 2 ((𝐾𝑉𝑊𝐻) → (𝑅𝐹) = ((𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))‘𝐹))
11 fveq1 6432 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑝) = (𝐹𝑝))
1211oveq2d 6921 . . . . . . . 8 (𝑓 = 𝐹 → (𝑝 (𝑓𝑝)) = (𝑝 (𝐹𝑝)))
1312oveq1d 6920 . . . . . . 7 (𝑓 = 𝐹 → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑝 (𝐹𝑝)) 𝑊))
1413eqeq2d 2835 . . . . . 6 (𝑓 = 𝐹 → (𝑥 = ((𝑝 (𝑓𝑝)) 𝑊) ↔ 𝑥 = ((𝑝 (𝐹𝑝)) 𝑊)))
1514imbi2d 332 . . . . 5 (𝑓 = 𝐹 → ((¬ 𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)) ↔ (¬ 𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
1615ralbidv 3195 . . . 4 (𝑓 = 𝐹 → (∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)) ↔ ∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
1716riotabidv 6868 . . 3 (𝑓 = 𝐹 → (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
18 eqid 2825 . . 3 (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
19 riotaex 6870 . . 3 (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))) ∈ V
2017, 18, 19fvmpt 6529 . 2 (𝐹𝑇 → ((𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))‘𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
2110, 20sylan9eq 2881 1 (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1658  wcel 2166  wral 3117   class class class wbr 4873  cmpt 4952  cfv 6123  crio 6865  (class class class)co 6905  Basecbs 16222  lecple 16312  joincjn 17297  meetcmee 17298  Atomscatm 35338  LHypclh 36059  LTrncltrn 36176  trLctrl 36233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-trl 36234
This theorem is referenced by:  trlval2  36238
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