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Theorem tendotp 35515
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 35514 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
7 fveq2 6150 . . . . . . 7 (𝑓 = 𝐹 → (𝑆𝑓) = (𝑆𝐹))
87fveq2d 6154 . . . . . 6 (𝑓 = 𝐹 → (𝑅‘(𝑆𝑓)) = (𝑅‘(𝑆𝐹)))
9 fveq2 6150 . . . . . 6 (𝑓 = 𝐹 → (𝑅𝑓) = (𝑅𝐹))
108, 9breq12d 4631 . . . . 5 (𝑓 = 𝐹 → ((𝑅‘(𝑆𝑓)) (𝑅𝑓) ↔ (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
1110rspccv 3297 . . . 4 (∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓) → (𝐹𝑇 → (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
12113ad2ant3 1082 . . 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 1254 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑆𝐸𝐹𝑇) → (𝑅‘(𝑆𝐹)) (𝑅𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912   class class class wbr 4618  ccom 5083  wf 5846  cfv 5850  lecple 15864  LHypclh 34736  LTrncltrn 34853  trLctrl 34911  TEndoctendo 35506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-tendo 35509
This theorem is referenced by:  tendococl  35526  tendoid  35527  tendopltp  35534  tendoicl  35550  cdlemi1  35572  tendotr  35584  cdleml1N  35730  dva1dim  35739  dialss  35801  diblss  35925
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