Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendofset Structured version   Visualization version   GIF version

Theorem tendofset 37888
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 3512 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6664 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 tendoset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2874 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6664 . . . . . . . 8 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
65fveq1d 6666 . . . . . . 7 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
76, 6feq23d 6503 . . . . . 6 (𝑘 = 𝐾 → (𝑠:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ↔ 𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤)))
86raleqdv 3415 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔))))
96, 8raleqbidv 3401 . . . . . 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 6664 . . . . . . . . . 10 (𝑘 = 𝐾 → (trL‘𝑘) = (trL‘𝐾))
1110fveq1d 6666 . . . . . . . . 9 (𝑘 = 𝐾 → ((trL‘𝑘)‘𝑤) = ((trL‘𝐾)‘𝑤))
1211fveq1d 6666 . . . . . . . 8 (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘(𝑠𝑓)) = (((trL‘𝐾)‘𝑤)‘(𝑠𝑓)))
13 fveq2 6664 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
14 tendoset.l . . . . . . . . 9 = (le‘𝐾)
1513, 14syl6eqr 2874 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1611fveq1d 6666 . . . . . . . 8 (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘𝑓) = (((trL‘𝐾)‘𝑤)‘𝑓))
1712, 15, 16breq123d 5072 . . . . . . 7 (𝑘 = 𝐾 → ((((trL‘𝑘)‘𝑤)‘(𝑠𝑓))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑓) ↔ (((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓)))
186, 17raleqbidv 3401 . . . . . 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 1432 . . . . 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 2885 . . . 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 5143 . . 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 37885 . . 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 6984 . 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 1083   = wceq 1533  wcel 2110  {cab 2799  wral 3138  Vcvv 3494   class class class wbr 5058  cmpt 5138  ccom 5553  wf 6345  cfv 6349  lecple 16566  LHypclh 37114  LTrncltrn 37231  trLctrl 37288  TEndoctendo 37882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-tendo 37885
This theorem is referenced by:  tendoset  37889
  Copyright terms: Public domain W3C validator