Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendotp Structured version   Visualization version   GIF version

Theorem tendotp 36543
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 36542 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
7 fveq2 6344 . . . . . . 7 (𝑓 = 𝐹 → (𝑆𝑓) = (𝑆𝐹))
87fveq2d 6348 . . . . . 6 (𝑓 = 𝐹 → (𝑅‘(𝑆𝑓)) = (𝑅‘(𝑆𝐹)))
9 fveq2 6344 . . . . . 6 (𝑓 = 𝐹 → (𝑅𝑓) = (𝑅𝐹))
108, 9breq12d 4809 . . . . 5 (𝑓 = 𝐹 → ((𝑅‘(𝑆𝑓)) (𝑅𝑓) ↔ (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
1110rspccv 3438 . . . 4 (∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓) → (𝐹𝑇 → (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
12113ad2ant3 1129 . . 3 ((𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)) → (𝐹𝑇 → (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
136, 12syl6bi 243 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 → (𝐹𝑇 → (𝑅‘(𝑆𝐹)) (𝑅𝐹))))
14133imp 1101 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑆𝐸𝐹𝑇) → (𝑅‘(𝑆𝐹)) (𝑅𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1624  wcel 2131  wral 3042   class class class wbr 4796  ccom 5262  wf 6037  cfv 6041  lecple 16142  LHypclh 35765  LTrncltrn 35882  trLctrl 35940  TEndoctendo 36534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-map 8017  df-tendo 36537
This theorem is referenced by:  tendococl  36554  tendoid  36555  tendopltp  36562  tendoicl  36578  cdlemi1  36600  tendotr  36612  cdleml1N  36758  dva1dim  36767  dialss  36829  diblss  36953
  Copyright terms: Public domain W3C validator