Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendotp Unicode version

Theorem tendotp 30950
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 30949 . . 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 5525 . . . . . . 7  |-  ( f  =  F  ->  ( S `  f )  =  ( S `  F ) )
87fveq2d 5529 . . . . . 6  |-  ( f  =  F  ->  ( R `  ( S `  f ) )  =  ( R `  ( S `  F )
) )
9 fveq2 5525 . . . . . 6  |-  ( f  =  F  ->  ( R `  f )  =  ( R `  F ) )
108, 9breq12d 4036 . . . . 5  |-  ( f  =  F  ->  (
( R `  ( S `  f )
)  .<_  ( R `  f )  <->  ( R `  ( S `  F
) )  .<_  ( R `
 F ) ) )
1110rspccv 2881 . . . 4  |-  ( A. f  e.  T  ( R `  ( S `  f ) )  .<_  ( R `  f )  ->  ( F  e.  T  ->  ( R `  ( S `  F
) )  .<_  ( R `
 F ) ) )
12113ad2ant3 978 . . 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 219 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( S  e.  E  ->  ( F  e.  T  ->  ( R `  ( S `  F )
)  .<_  ( R `  F ) ) ) )
14133imp 1145 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    o. ccom 4693   -->wf 5251   ` cfv 5255   lecple 13215   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   TEndoctendo 30941
This theorem is referenced by:  tendococl  30961  tendoid  30962  tendopltp  30969  tendoicl  30985  cdlemi1  31007  tendotr  31019  cdleml1N  31165  dva1dim  31174  dialss  31236  diblss  31360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-tendo 30944
  Copyright terms: Public domain W3C validator