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Theorem tendotp 38337
 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 38336 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
7 2fveq3 6663 . . . . . 6 (𝑓 = 𝐹 → (𝑅‘(𝑆𝑓)) = (𝑅‘(𝑆𝐹)))
8 fveq2 6658 . . . . . 6 (𝑓 = 𝐹 → (𝑅𝑓) = (𝑅𝐹))
97, 8breq12d 5045 . . . . 5 (𝑓 = 𝐹 → ((𝑅‘(𝑆𝑓)) (𝑅𝑓) ↔ (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
109rspccv 3538 . . . 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, 11syl6bi 256 . 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070   class class class wbr 5032   ∘ ccom 5528  ⟶wf 6331  ‘cfv 6335  lecple 16630  LHypclh 37560  LTrncltrn 37677  trLctrl 37734  TEndoctendo 38328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8418  df-tendo 38331 This theorem is referenced by:  tendococl  38348  tendoid  38349  tendopltp  38356  tendoicl  38372  cdlemi1  38394  tendotr  38406  cdleml1N  38552  dva1dim  38561  dialss  38622  diblss  38746
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