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Theorem istendo 36828
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 36827 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐸 = {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))})
76eleq2d 2892 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸𝑆 ∈ {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))}))
83fvexi 6447 . . . . 5 𝑇 ∈ V
9 fex 6745 . . . . 5 ((𝑆:𝑇𝑇𝑇 ∈ V) → 𝑆 ∈ V)
108, 9mpan2 682 . . . 4 (𝑆:𝑇𝑇𝑆 ∈ V)
11103ad2ant1 1167 . . 3 ((𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)) → 𝑆 ∈ V)
12 feq1 6259 . . . 4 (𝑠 = 𝑆 → (𝑠:𝑇𝑇𝑆:𝑇𝑇))
13 fveq1 6432 . . . . . 6 (𝑠 = 𝑆 → (𝑠‘(𝑓𝑔)) = (𝑆‘(𝑓𝑔)))
14 fveq1 6432 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑓) = (𝑆𝑓))
15 fveq1 6432 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑔) = (𝑆𝑔))
1614, 15coeq12d 5519 . . . . . 6 (𝑠 = 𝑆 → ((𝑠𝑓) ∘ (𝑠𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)))
1713, 16eqeq12d 2840 . . . . 5 (𝑠 = 𝑆 → ((𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔))))
18172ralbidv 3198 . . . 4 (𝑠 = 𝑆 → (∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔))))
1914fveq2d 6437 . . . . . 6 (𝑠 = 𝑆 → (𝑅‘(𝑠𝑓)) = (𝑅‘(𝑆𝑓)))
2019breq1d 4883 . . . . 5 (𝑠 = 𝑆 → ((𝑅‘(𝑠𝑓)) (𝑅𝑓) ↔ (𝑅‘(𝑆𝑓)) (𝑅𝑓)))
2120ralbidv 3195 . . . 4 (𝑠 = 𝑆 → (∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓) ↔ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)))
2212, 18, 213anbi123d 1564 . . 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 3579 . 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, 23syl6bb 279 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 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  {cab 2811  wral 3117  Vcvv 3414   class class class wbr 4873  ccom 5346  wf 6119  cfv 6123  lecple 16312  LHypclh 36052  LTrncltrn 36169  trLctrl 36226  TEndoctendo 36820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-tendo 36823
This theorem is referenced by:  tendotp  36829  istendod  36830  tendof  36831  tendovalco  36833
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