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Theorem istendo 40419
Description: The predicate "is a trace-preserving endomorphism". Similar to definition of trace-preserving endomorphism in [Crawley] p. 117, penultimate line. (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
istendo ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
Distinct variable groups:   𝑓,𝑔,𝐾   𝑇,𝑓,𝑔   𝑓,𝑊,𝑔   𝑆,𝑓,𝑔
Allowed substitution hints:   𝑅(𝑓,𝑔)   𝐸(𝑓,𝑔)   𝐻(𝑓,𝑔)   (𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem istendo
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 tendoset.l . . . 4 = (le‘𝐾)
2 tendoset.h . . . 4 𝐻 = (LHyp‘𝐾)
3 tendoset.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 tendoset.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
5 tendoset.e . . . 4 𝐸 = ((TEndo‘𝐾)‘𝑊)
61, 2, 3, 4, 5tendoset 40418 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐸 = {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))})
76eleq2d 2811 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸𝑆 ∈ {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))}))
83fvexi 6914 . . . . 5 𝑇 ∈ V
9 fex 7242 . . . . 5 ((𝑆:𝑇𝑇𝑇 ∈ V) → 𝑆 ∈ V)
108, 9mpan2 689 . . . 4 (𝑆:𝑇𝑇𝑆 ∈ V)
11103ad2ant1 1130 . . 3 ((𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)) → 𝑆 ∈ V)
12 feq1 6708 . . . 4 (𝑠 = 𝑆 → (𝑠:𝑇𝑇𝑆:𝑇𝑇))
13 fveq1 6899 . . . . . 6 (𝑠 = 𝑆 → (𝑠‘(𝑓𝑔)) = (𝑆‘(𝑓𝑔)))
14 fveq1 6899 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑓) = (𝑆𝑓))
15 fveq1 6899 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑔) = (𝑆𝑔))
1614, 15coeq12d 5870 . . . . . 6 (𝑠 = 𝑆 → ((𝑠𝑓) ∘ (𝑠𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)))
1713, 16eqeq12d 2741 . . . . 5 (𝑠 = 𝑆 → ((𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔))))
18172ralbidv 3208 . . . 4 (𝑠 = 𝑆 → (∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔))))
1914fveq2d 6904 . . . . . 6 (𝑠 = 𝑆 → (𝑅‘(𝑠𝑓)) = (𝑅‘(𝑆𝑓)))
2019breq1d 5162 . . . . 5 (𝑠 = 𝑆 → ((𝑅‘(𝑠𝑓)) (𝑅𝑓) ↔ (𝑅‘(𝑆𝑓)) (𝑅𝑓)))
2120ralbidv 3167 . . . 4 (𝑠 = 𝑆 → (∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓) ↔ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)))
2212, 18, 213anbi123d 1432 . . 3 (𝑠 = 𝑆 → ((𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓)) ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
2311, 22elab3 3673 . 2 (𝑆 ∈ {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))} ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)))
247, 23bitrdi 286 1 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  {cab 2702  wral 3050  Vcvv 3461   class class class wbr 5152  ccom 5685  wf 6549  cfv 6553  lecple 17268  LHypclh 39643  LTrncltrn 39760  trLctrl 39817  TEndoctendo 40411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8856  df-tendo 40414
This theorem is referenced by:  tendotp  40420  istendod  40421  tendof  40422  tendovalco  40424
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