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Theorem tendotp 30875
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 30874 . . 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 5668 . . . . . . 7  |-  ( f  =  F  ->  ( S `  f )  =  ( S `  F ) )
87fveq2d 5672 . . . . . 6  |-  ( f  =  F  ->  ( R `  ( S `  f ) )  =  ( R `  ( S `  F )
) )
9 fveq2 5668 . . . . . 6  |-  ( f  =  F  ->  ( R `  f )  =  ( R `  F ) )
108, 9breq12d 4166 . . . . 5  |-  ( f  =  F  ->  (
( R `  ( S `  f )
)  .<_  ( R `  f )  <->  ( R `  ( S `  F
) )  .<_  ( R `
 F ) ) )
1110rspccv 2992 . . . 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 1649    e. wcel 1717   A.wral 2649   class class class wbr 4153    o. ccom 4822   -->wf 5390   ` cfv 5394   lecple 13463   LHypclh 30098   LTrncltrn 30215   trLctrl 30272   TEndoctendo 30866
This theorem is referenced by:  tendococl  30886  tendoid  30887  tendopltp  30894  tendoicl  30910  cdlemi1  30932  tendotr  30944  cdleml1N  31090  dva1dim  31099  dialss  31161  diblss  31285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-map 6956  df-tendo 30869
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