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Theorem tendof 30879
Description: Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
Hypotheses
Ref Expression
tendof.h  |-  H  =  ( LHyp `  K
)
tendof.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendof.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
tendof  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  S  e.  E )  ->  S : T --> T )

Proof of Theorem tendof
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2389 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . . . 4  |-  H  =  ( LHyp `  K
)
3 tendof.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2389 . . . 4  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendof.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
61, 2, 3, 4, 5istendo 30876 . . 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  ( (
( trL `  K
) `  W ) `  ( S `  f
) ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  f ) ) ) )
7 simp1 957 . . 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  ( (
( trL `  K
) `  W ) `  ( S `  f
) ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  f ) )  ->  S : T --> T )
86, 7syl6bi 220 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( S  e.  E  ->  S : T --> T ) )
98imp 419 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  S  e.  E )  ->  S : T --> T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   class class class wbr 4155    o. ccom 4824   -->wf 5392   ` cfv 5396   lecple 13465   LHypclh 30100   LTrncltrn 30217   trLctrl 30274   TEndoctendo 30868
This theorem is referenced by:  tendoeq1  30880  tendocoval  30882  tendocl  30883  tendo1mul  30886  tendo1mulr  30887  tendococl  30888  tendoconid  30945  tendospass  31136  dvhlveclem  31225  dicvscacl  31308
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-map 6958  df-tendo 30871
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