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Theorem tendofset 38772
Description: The set of all trace-preserving endomorphisms on the set of translations for a lattice 𝐾. (Contributed by NM, 8-Jun-2013.)
Hypotheses
Ref Expression
tendoset.l = (le‘𝐾)
tendoset.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
tendofset (𝐾𝑉 → (TEndo‘𝐾) = (𝑤𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓))}))
Distinct variable groups:   𝑤,𝐻   𝑤,𝑠,𝑓,𝑔,𝐾
Allowed substitution hints:   𝐻(𝑓,𝑔,𝑠)   (𝑤,𝑓,𝑔,𝑠)   𝑉(𝑤,𝑓,𝑔,𝑠)

Proof of Theorem tendofset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3450 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6774 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 tendoset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2796 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6774 . . . . . . . 8 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
65fveq1d 6776 . . . . . . 7 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
76, 6feq23d 6595 . . . . . 6 (𝑘 = 𝐾 → (𝑠:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ↔ 𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤)))
86raleqdv 3348 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔))))
96, 8raleqbidv 3336 . . . . . 6 (𝑘 = 𝐾 → (∀𝑓 ∈ ((LTrn‘𝑘)‘𝑤)∀𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔))))
10 fveq2 6774 . . . . . . . . . 10 (𝑘 = 𝐾 → (trL‘𝑘) = (trL‘𝐾))
1110fveq1d 6776 . . . . . . . . 9 (𝑘 = 𝐾 → ((trL‘𝑘)‘𝑤) = ((trL‘𝐾)‘𝑤))
1211fveq1d 6776 . . . . . . . 8 (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘(𝑠𝑓)) = (((trL‘𝐾)‘𝑤)‘(𝑠𝑓)))
13 fveq2 6774 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
14 tendoset.l . . . . . . . . 9 = (le‘𝐾)
1513, 14eqtr4di 2796 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1611fveq1d 6776 . . . . . . . 8 (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘𝑓) = (((trL‘𝐾)‘𝑤)‘𝑓))
1712, 15, 16breq123d 5088 . . . . . . 7 (𝑘 = 𝐾 → ((((trL‘𝑘)‘𝑤)‘(𝑠𝑓))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑓) ↔ (((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓)))
186, 17raleqbidv 3336 . . . . . 6 (𝑘 = 𝐾 → (∀𝑓 ∈ ((LTrn‘𝑘)‘𝑤)(((trL‘𝑘)‘𝑤)‘(𝑠𝑓))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑓) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓)))
197, 9, 183anbi123d 1435 . . . . 5 (𝑘 = 𝐾 → ((𝑠:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝑘)‘𝑤)∀𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝑘)‘𝑤)(((trL‘𝑘)‘𝑤)‘(𝑠𝑓))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑓)) ↔ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓))))
2019abbidv 2807 . . . 4 (𝑘 = 𝐾 → {𝑠 ∣ (𝑠:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝑘)‘𝑤)∀𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝑘)‘𝑤)(((trL‘𝑘)‘𝑤)‘(𝑠𝑓))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑓))} = {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓))})
214, 20mpteq12dv 5165 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑠 ∣ (𝑠:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝑘)‘𝑤)∀𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝑘)‘𝑤)(((trL‘𝑘)‘𝑤)‘(𝑠𝑓))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑓))}) = (𝑤𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓))}))
22 df-tendo 38769 . . 3 TEndo = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑠 ∣ (𝑠:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝑘)‘𝑤)∀𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝑘)‘𝑤)(((trL‘𝑘)‘𝑤)‘(𝑠𝑓))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑓))}))
2321, 22, 3mptfvmpt 7104 . 2 (𝐾 ∈ V → (TEndo‘𝐾) = (𝑤𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓))}))
241, 23syl 17 1 (𝐾𝑉 → (TEndo‘𝐾) = (𝑤𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2106  {cab 2715  wral 3064  Vcvv 3432   class class class wbr 5074  cmpt 5157  ccom 5593  wf 6429  cfv 6433  lecple 16969  LHypclh 37998  LTrncltrn 38115  trLctrl 38172  TEndoctendo 38766
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-tendo 38769
This theorem is referenced by:  tendoset  38773
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