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Theorem trlset 36182
Description: The set of traces of lattice translations for a fiducial co-atom 𝑊. (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
trlset ((𝐾𝐶𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑓,𝑝,𝑥,𝐾   𝑇,𝑓   𝑓,𝑊,𝑝,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐵(𝑓,𝑝)   𝐶(𝑥,𝑓,𝑝)   𝑅(𝑥,𝑓,𝑝)   𝑇(𝑥,𝑝)   𝐻(𝑥,𝑓,𝑝)   (𝑥,𝑓,𝑝)   (𝑥,𝑓,𝑝)   (𝑥,𝑓,𝑝)

Proof of Theorem trlset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 trlset.r . . 3 𝑅 = ((trL‘𝐾)‘𝑊)
2 trlset.b . . . . 5 𝐵 = (Base‘𝐾)
3 trlset.l . . . . 5 = (le‘𝐾)
4 trlset.j . . . . 5 = (join‘𝐾)
5 trlset.m . . . . 5 = (meet‘𝐾)
6 trlset.a . . . . 5 𝐴 = (Atoms‘𝐾)
7 trlset.h . . . . 5 𝐻 = (LHyp‘𝐾)
82, 3, 4, 5, 6, 7trlfset 36181 . . . 4 (𝐾𝐶 → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))
98fveq1d 6413 . . 3 (𝐾𝐶 → ((trL‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊))
101, 9syl5eq 2845 . 2 (𝐾𝐶𝑅 = ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊))
11 fveq2 6411 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
12 breq2 4847 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑝 𝑤𝑝 𝑊))
1312notbid 310 . . . . . . . 8 (𝑤 = 𝑊 → (¬ 𝑝 𝑤 ↔ ¬ 𝑝 𝑊))
14 oveq2 6886 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑝 (𝑓𝑝)) 𝑊))
1514eqeq2d 2809 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 = ((𝑝 (𝑓𝑝)) 𝑤) ↔ 𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))
1613, 15imbi12d 336 . . . . . . 7 (𝑤 = 𝑊 → ((¬ 𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)) ↔ (¬ 𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
1716ralbidv 3167 . . . . . 6 (𝑤 = 𝑊 → (∀𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)) ↔ ∀𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
1817riotabidv 6841 . . . . 5 (𝑤 = 𝑊 → (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
1911, 18mpteq12dv 4926 . . . 4 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
20 eqid 2799 . . . 4 (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))
21 fvex 6424 . . . . 5 ((LTrn‘𝐾)‘𝑊) ∈ V
2221mptex 6715 . . . 4 (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) ∈ V
2319, 20, 22fvmpt 6507 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
24 trlset.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
25 eqid 2799 . . . 4 (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))
2624, 25mpteq12i 4935 . . 3 (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊))))
2723, 26syl6eqr 2851 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤)))))‘𝑊) = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
2810, 27sylan9eq 2853 1 ((𝐾𝐶𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089   class class class wbr 4843  cmpt 4922  cfv 6101  crio 6838  (class class class)co 6878  Basecbs 16184  lecple 16274  joincjn 17259  meetcmee 17260  Atomscatm 35284  LHypclh 36005  LTrncltrn 36122  trLctrl 36179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-trl 36180
This theorem is referenced by:  trlval  36183
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