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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 ≀ = (leβ€˜πΎ)
tendoset.h 𝐻 = (LHypβ€˜πΎ)
tendoset.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
tendoset.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
tendoset.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
tendotp (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜(π‘†β€˜πΉ)) ≀ (π‘…β€˜πΉ))
Proof of Theorem tendotp
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tendoset.l . . . 4 ≀ = (leβ€˜πΎ)
2 tendoset.h . . . 4 𝐻 = (LHypβ€˜πΎ)
3 tendoset.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
4 tendoset.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
5 tendoset.e . . . 4 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5istendo 40265 . . 3 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑆 ∈ 𝐸 ↔ (𝑆:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“))))
7 2fveq3 6907 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘…β€˜(π‘†β€˜π‘“)) = (π‘…β€˜(π‘†β€˜πΉ)))
8 fveq2 6902 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘…β€˜π‘“) = (π‘…β€˜πΉ))
97, 8breq12d 5165 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“) ↔ (π‘…β€˜(π‘†β€˜πΉ)) ≀ (π‘…β€˜πΉ)))
109rspccv 3608 . . . 4 (βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“) β†’ (𝐹 ∈ 𝑇 β†’ (π‘…β€˜(π‘†β€˜πΉ)) ≀ (π‘…β€˜πΉ)))
11103ad2ant3 1132 . . 3 ((𝑆:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“)) β†’ (𝐹 ∈ 𝑇 β†’ (π‘…β€˜(π‘†β€˜πΉ)) ≀ (π‘…β€˜πΉ)))
126, 11biimtrdi 252 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑆 ∈ 𝐸 β†’ (𝐹 ∈ 𝑇 β†’ (π‘…β€˜(π‘†β€˜πΉ)) ≀ (π‘…β€˜πΉ))))
13123imp 1108 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜(π‘†β€˜πΉ)) ≀ (π‘…β€˜πΉ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058   class class class wbr 5152   ∘ ccom 5686  βŸΆwf 6549  β€˜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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