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Theorem tendoset 40761
Description: The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom 𝑊. (Contributed by NM, 8-Jun-2013.)
Hypotheses
Ref Expression
tendoset.l = (le‘𝐾)
tendoset.h 𝐻 = (LHyp‘𝐾)
tendoset.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendoset.r 𝑅 = ((trL‘𝐾)‘𝑊)
tendoset.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
Assertion
Ref Expression
tendoset ((𝐾𝑉𝑊𝐻) → 𝐸 = {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))})
Distinct variable groups:   𝑓,𝑠,𝑔,𝐾   𝑇,𝑓,𝑔,𝑠   𝑊,𝑠,𝑓,𝑔
Allowed substitution hints:   𝑅(𝑓,𝑔,𝑠)   𝐸(𝑓,𝑔,𝑠)   𝐻(𝑓,𝑔,𝑠)   (𝑓,𝑔,𝑠)   𝑉(𝑓,𝑔,𝑠)

Proof of Theorem tendoset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 tendoset.e . 2 𝐸 = ((TEndo‘𝐾)‘𝑊)
2 tendoset.l . . . . 5 = (le‘𝐾)
3 tendoset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3tendofset 40760 . . . 4 (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))}))
54fveq1d 6908 . . 3 (𝐾𝑉 → ((TEndo‘𝐾)‘𝑊) = ((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))})‘𝑊))
6 fveq2 6906 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
76, 6feq23d 6731 . . . . . . 7 (𝑤 = 𝑊 → (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ↔ 𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)))
86raleqdv 3326 . . . . . . . 8 (𝑤 = 𝑊 → (∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔))))
96, 8raleqbidv 3346 . . . . . . 7 (𝑤 = 𝑊 → (∀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 6906 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = ((trL‘𝐾)‘𝑊))
11 tendoset.r . . . . . . . . . . 11 𝑅 = ((trL‘𝐾)‘𝑊)
1210, 11eqtr4di 2795 . . . . . . . . . 10 (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = 𝑅)
1312fveq1d 6908 . . . . . . . . 9 (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) = (𝑅‘(𝑠𝑓)))
1412fveq1d 6908 . . . . . . . . 9 (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘𝑓) = (𝑅𝑓))
1513, 14breq12d 5156 . . . . . . . 8 (𝑤 = 𝑊 → ((((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓) ↔ (𝑅‘(𝑠𝑓)) (𝑅𝑓)))
166, 15raleqbidv 3346 . . . . . . 7 (𝑤 = 𝑊 → (∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓)))
177, 9, 163anbi123d 1438 . . . . . 6 (𝑤 = 𝑊 → ((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)) ↔ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓))))
1817abbidv 2808 . . . . 5 (𝑤 = 𝑊 → {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))} = {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓))})
19 eqid 2737 . . . . 5 (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))}) = (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))})
20 fvex 6919 . . . . . . . 8 ((LTrn‘𝐾)‘𝑊) ∈ V
2120, 20mapval 8878 . . . . . . 7 (((LTrn‘𝐾)‘𝑊) ↑m ((LTrn‘𝐾)‘𝑊)) = {𝑠𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)}
22 ovex 7464 . . . . . . 7 (((LTrn‘𝐾)‘𝑊) ↑m ((LTrn‘𝐾)‘𝑊)) ∈ V
2321, 22eqeltrri 2838 . . . . . 6 {𝑠𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)} ∈ V
24 simp1 1137 . . . . . . 7 ((𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓)) → 𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
2524ss2abi 4067 . . . . . 6 {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓))} ⊆ {𝑠𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)}
2623, 25ssexi 5322 . . . . 5 {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓))} ∈ V
2718, 19, 26fvmpt 7016 . . . 4 (𝑊𝐻 → ((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))})‘𝑊) = {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓))})
28 tendoset.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
2928, 28feq23i 6730 . . . . . 6 (𝑠:𝑇𝑇𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
3028raleqi 3324 . . . . . . 7 (∀𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)))
3128, 30raleqbii 3344 . . . . . 6 (∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)))
3228raleqi 3324 . . . . . 6 (∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓))
3329, 31, 323anbi123i 1156 . . . . 5 ((𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓)) ↔ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓)))
3433abbii 2809 . . . 4 {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))} = {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓))}
3527, 34eqtr4di 2795 . . 3 (𝑊𝐻 → ((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))})‘𝑊) = {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))})
365, 35sylan9eq 2797 . 2 ((𝐾𝑉𝑊𝐻) → ((TEndo‘𝐾)‘𝑊) = {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))})
371, 36eqtrid 2789 1 ((𝐾𝑉𝑊𝐻) → 𝐸 = {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wral 3061  Vcvv 3480   class class class wbr 5143  cmpt 5225  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  lecple 17304  LHypclh 39986  LTrncltrn 40103  trLctrl 40160  TEndoctendo 40754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-tendo 40757
This theorem is referenced by:  istendo  40762
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