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Theorem tendotp 40144
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 40143 . . 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 6889 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘…β€˜(π‘†β€˜π‘“)) = (π‘…β€˜(π‘†β€˜πΉ)))
8 fveq2 6884 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘…β€˜π‘“) = (π‘…β€˜πΉ))
97, 8breq12d 5154 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“) ↔ (π‘…β€˜(π‘†β€˜πΉ)) ≀ (π‘…β€˜πΉ)))
109rspccv 3603 . . . 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, 11syl6bi 253 . 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   class class class wbr 5141   ∘ ccom 5673  βŸΆwf 6532  β€˜cfv 6536  lecple 17210  LHypclh 39367  LTrncltrn 39484  trLctrl 39541  TEndoctendo 40135
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-tendo 40138
This theorem is referenced by:  tendococl  40155  tendoid  40156  tendopltp  40163  tendoicl  40179  cdlemi1  40201  tendotr  40213  cdleml1N  40359  dva1dim  40368  dialss  40429  diblss  40553
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