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Theorem tendoset 38013
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 38012 . . . 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 6654 . . 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 6652 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
76, 6feq23d 6489 . . . . . . 7 (𝑤 = 𝑊 → (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ↔ 𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)))
86raleqdv 3392 . . . . . . . 8 (𝑤 = 𝑊 → (∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔))))
96, 8raleqbidv 3382 . . . . . . 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 6652 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = ((trL‘𝐾)‘𝑊))
11 tendoset.r . . . . . . . . . . 11 𝑅 = ((trL‘𝐾)‘𝑊)
1210, 11eqtr4di 2875 . . . . . . . . . 10 (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = 𝑅)
1312fveq1d 6654 . . . . . . . . 9 (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) = (𝑅‘(𝑠𝑓)))
1412fveq1d 6654 . . . . . . . . 9 (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘𝑓) = (𝑅𝑓))
1513, 14breq12d 5055 . . . . . . . 8 (𝑤 = 𝑊 → ((((trL‘𝐾)‘𝑤)‘(𝑠𝑓)) (((trL‘𝐾)‘𝑤)‘𝑓) ↔ (𝑅‘(𝑠𝑓)) (𝑅𝑓)))
166, 15raleqbidv 3382 . . . . . . 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 1433 . . . . . 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 2886 . . . . 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 2822 . . . . 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 6665 . . . . . . . 8 ((LTrn‘𝐾)‘𝑊) ∈ V
2120, 20mapval 8405 . . . . . . 7 (((LTrn‘𝐾)‘𝑊) ↑m ((LTrn‘𝐾)‘𝑊)) = {𝑠𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)}
22 ovex 7173 . . . . . . 7 (((LTrn‘𝐾)‘𝑊) ↑m ((LTrn‘𝐾)‘𝑊)) ∈ V
2321, 22eqeltrri 2911 . . . . . 6 {𝑠𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)} ∈ V
24 simp1 1133 . . . . . . 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 4018 . . . . . 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 5202 . . . . 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 6750 . . . 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 6488 . . . . . 6 (𝑠:𝑇𝑇𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
3028raleqi 3390 . . . . . . 7 (∀𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)))
3128, 30raleqbii 3222 . . . . . 6 (∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)))
3228raleqi 3390 . . . . . 6 (∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠𝑓)) (𝑅𝑓))
3329, 31, 323anbi123i 1152 . . . . 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 2887 . . . 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 2875 . . 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 2877 . 2 ((𝐾𝑉𝑊𝐻) → ((TEndo‘𝐾)‘𝑊) = {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))})
371, 36syl5eq 2869 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 399  w3a 1084   = wceq 1538  wcel 2114  {cab 2800  wral 3130  Vcvv 3469   class class class wbr 5042  cmpt 5122  ccom 5536  wf 6330  cfv 6334  (class class class)co 7140  m cmap 8393  lecple 16563  LHypclh 37238  LTrncltrn 37355  trLctrl 37412  TEndoctendo 38006
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-pow 5243  ax-pr 5307  ax-un 7446
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-pw 4513  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-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8395  df-tendo 38009
This theorem is referenced by:  istendo  38014
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