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Theorem tendotp 31459
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  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  S  e.  E  /\  F  e.  T
)  ->  ( R `  ( S `  F
) )  .<_  ( R `
 F ) )

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 31458 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( S  e.  E  <->  ( S : T --> T  /\  A. f  e.  T  A. g  e.  T  ( S `  ( f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g )
)  /\  A. f  e.  T  ( R `  ( S `  f
) )  .<_  ( R `
 f ) ) ) )
7 fveq2 5720 . . . . . . 7  |-  ( f  =  F  ->  ( S `  f )  =  ( S `  F ) )
87fveq2d 5724 . . . . . 6  |-  ( f  =  F  ->  ( R `  ( S `  f ) )  =  ( R `  ( S `  F )
) )
9 fveq2 5720 . . . . . 6  |-  ( f  =  F  ->  ( R `  f )  =  ( R `  F ) )
108, 9breq12d 4217 . . . . 5  |-  ( f  =  F  ->  (
( R `  ( S `  f )
)  .<_  ( R `  f )  <->  ( R `  ( S `  F
) )  .<_  ( R `
 F ) ) )
1110rspccv 3041 . . . 4  |-  ( A. f  e.  T  ( R `  ( S `  f ) )  .<_  ( R `  f )  ->  ( F  e.  T  ->  ( R `  ( S `  F
) )  .<_  ( R `
 F ) ) )
12113ad2ant3 980 . . 3  |-  ( ( S : T --> T  /\  A. f  e.  T  A. g  e.  T  ( S `  ( f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g )
)  /\  A. f  e.  T  ( R `  ( S `  f
) )  .<_  ( R `
 f ) )  ->  ( F  e.  T  ->  ( R `  ( S `  F
) )  .<_  ( R `
 F ) ) )
136, 12syl6bi 220 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( S  e.  E  ->  ( F  e.  T  ->  ( R `  ( S `  F )
)  .<_  ( R `  F ) ) ) )
14133imp 1147 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  S  e.  E  /\  F  e.  T
)  ->  ( R `  ( S `  F
) )  .<_  ( R `
 F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204    o. ccom 4874   -->wf 5442   ` cfv 5446   lecple 13526   LHypclh 30682   LTrncltrn 30799   trLctrl 30856   TEndoctendo 31450
This theorem is referenced by:  tendococl  31470  tendoid  31471  tendopltp  31478  tendoicl  31494  cdlemi1  31516  tendotr  31528  cdleml1N  31674  dva1dim  31683  dialss  31745  diblss  31869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-tendo 31453
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