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Theorem trlval 37417
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 37416 . . 3 ((𝐾𝑉𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
109fveq1d 6654 . 2 ((𝐾𝑉𝑊𝐻) → (𝑅𝐹) = ((𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))‘𝐹))
11 fveq1 6651 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑝) = (𝐹𝑝))
1211oveq2d 7156 . . . . . . . 8 (𝑓 = 𝐹 → (𝑝 (𝑓𝑝)) = (𝑝 (𝐹𝑝)))
1312oveq1d 7155 . . . . . . 7 (𝑓 = 𝐹 → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑝 (𝐹𝑝)) 𝑊))
1413eqeq2d 2833 . . . . . 6 (𝑓 = 𝐹 → (𝑥 = ((𝑝 (𝑓𝑝)) 𝑊) ↔ 𝑥 = ((𝑝 (𝐹𝑝)) 𝑊)))
1514imbi2d 344 . . . . 5 (𝑓 = 𝐹 → ((¬ 𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)) ↔ (¬ 𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
1615ralbidv 3187 . . . 4 (𝑓 = 𝐹 → (∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)) ↔ ∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
1716riotabidv 7100 . . 3 (𝑓 = 𝐹 → (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
18 eqid 2822 . . 3 (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
19 riotaex 7102 . . 3 (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))) ∈ V
2017, 18, 19fvmpt 6750 . 2 (𝐹𝑇 → ((𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))‘𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
2110, 20sylan9eq 2877 1 (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130   class class class wbr 5042  cmpt 5122  cfv 6334  crio 7097  (class class class)co 7140  Basecbs 16474  lecple 16563  joincjn 17545  meetcmee 17546  Atomscatm 36518  LHypclh 37239  LTrncltrn 37356  trLctrl 37413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-trl 37414
This theorem is referenced by:  trlval2  37418
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