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Theorem tendotp 40266
Description: Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-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
tendotp (((𝐾𝑉𝑊𝐻) ∧ 𝑆𝐸𝐹𝑇) → (𝑅‘(𝑆𝐹)) (𝑅𝐹))

Proof of Theorem tendotp
Dummy variables 𝑓 𝑔 are mutually distinct and 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, 5istendo 40265 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
7 2fveq3 6907 . . . . . 6 (𝑓 = 𝐹 → (𝑅‘(𝑆𝑓)) = (𝑅‘(𝑆𝐹)))
8 fveq2 6902 . . . . . 6 (𝑓 = 𝐹 → (𝑅𝑓) = (𝑅𝐹))
97, 8breq12d 5165 . . . . 5 (𝑓 = 𝐹 → ((𝑅‘(𝑆𝑓)) (𝑅𝑓) ↔ (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
109rspccv 3608 . . . 4 (∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓) → (𝐹𝑇 → (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
11103ad2ant3 1132 . . 3 ((𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)) → (𝐹𝑇 → (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
126, 11biimtrdi 252 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 → (𝐹𝑇 → (𝑅‘(𝑆𝐹)) (𝑅𝐹))))
13123imp 1108 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑆𝐸𝐹𝑇) → (𝑅‘(𝑆𝐹)) (𝑅𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058   class class class wbr 5152  ccom 5686  wf 6549  cfv 6553  lecple 17247  LHypclh 39489  LTrncltrn 39606  trLctrl 39663  TEndoctendo 40257
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  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 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  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 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-tendo 40260
This theorem is referenced by:  tendococl  40277  tendoid  40278  tendopltp  40285  tendoicl  40301  cdlemi1  40323  tendotr  40335  cdleml1N  40481  dva1dim  40490  dialss  40551  diblss  40675
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