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Theorem tendovalco 34403
Description: Value of composition of translations in 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
tendovalco  |-  ( ( ( K  e.  V  /\  W  e.  H  /\  S  e.  E
)  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( S `  ( F  o.  G )
)  =  ( ( S `  F )  o.  ( S `  G ) ) )

Proof of Theorem tendovalco
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2471 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 tendof.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2471 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendof.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
61, 2, 3, 4, 5istendo 34398 . . . 4  |-  ( ( 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 coeq1 4997 . . . . . . . . 9  |-  ( f  =  F  ->  (
f  o.  g )  =  ( F  o.  g ) )
87fveq2d 5883 . . . . . . . 8  |-  ( f  =  F  ->  ( S `  ( f  o.  g ) )  =  ( S `  ( F  o.  g )
) )
9 fveq2 5879 . . . . . . . . 9  |-  ( f  =  F  ->  ( S `  f )  =  ( S `  F ) )
109coeq1d 5001 . . . . . . . 8  |-  ( f  =  F  ->  (
( S `  f
)  o.  ( S `
 g ) )  =  ( ( S `
 F )  o.  ( S `  g
) ) )
118, 10eqeq12d 2486 . . . . . . 7  |-  ( f  =  F  ->  (
( S `  (
f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g ) )  <->  ( S `  ( F  o.  g
) )  =  ( ( S `  F
)  o.  ( S `
 g ) ) ) )
12 coeq2 4998 . . . . . . . . 9  |-  ( g  =  G  ->  ( F  o.  g )  =  ( F  o.  G ) )
1312fveq2d 5883 . . . . . . . 8  |-  ( g  =  G  ->  ( S `  ( F  o.  g ) )  =  ( S `  ( F  o.  G )
) )
14 fveq2 5879 . . . . . . . . 9  |-  ( g  =  G  ->  ( S `  g )  =  ( S `  G ) )
1514coeq2d 5002 . . . . . . . 8  |-  ( g  =  G  ->  (
( S `  F
)  o.  ( S `
 g ) )  =  ( ( S `
 F )  o.  ( S `  G
) ) )
1613, 15eqeq12d 2486 . . . . . . 7  |-  ( g  =  G  ->  (
( S `  ( F  o.  g )
)  =  ( ( S `  F )  o.  ( S `  g ) )  <->  ( S `  ( F  o.  G
) )  =  ( ( S `  F
)  o.  ( S `
 G ) ) ) )
1711, 16rspc2v 3147 . . . . . 6  |-  ( ( F  e.  T  /\  G  e.  T )  ->  ( A. f  e.  T  A. g  e.  T  ( S `  ( f  o.  g
) )  =  ( ( S `  f
)  o.  ( S `
 g ) )  ->  ( S `  ( F  o.  G
) )  =  ( ( S `  F
)  o.  ( S `
 G ) ) ) )
1817com12 31 . . . . 5  |-  ( A. f  e.  T  A. g  e.  T  ( S `  ( f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g )
)  ->  ( ( F  e.  T  /\  G  e.  T )  ->  ( S `  ( F  o.  G )
)  =  ( ( S `  F )  o.  ( S `  G ) ) ) )
19183ad2ant2 1052 . . . 4  |-  ( ( 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 ) )  -> 
( ( F  e.  T  /\  G  e.  T )  ->  ( S `  ( F  o.  G ) )  =  ( ( S `  F )  o.  ( S `  G )
) ) )
206, 19syl6bi 236 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( S  e.  E  ->  ( ( F  e.  T  /\  G  e.  T )  ->  ( S `  ( F  o.  G ) )  =  ( ( S `  F )  o.  ( S `  G )
) ) ) )
21203impia 1228 . 2  |-  ( ( K  e.  V  /\  W  e.  H  /\  S  e.  E )  ->  ( ( F  e.  T  /\  G  e.  T )  ->  ( S `  ( F  o.  G ) )  =  ( ( S `  F )  o.  ( S `  G )
) ) )
2221imp 436 1  |-  ( ( ( K  e.  V  /\  W  e.  H  /\  S  e.  E
)  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( S `  ( F  o.  G )
)  =  ( ( S `  F )  o.  ( S `  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   class class class wbr 4395    o. ccom 4843   -->wf 5585   ` cfv 5589   lecple 15275   LHypclh 33620   LTrncltrn 33737   trLctrl 33795   TEndoctendo 34390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-map 7492  df-tendo 34393
This theorem is referenced by:  tendoco2  34406  tendococl  34410  tendodi1  34422  tendoicl  34434  cdlemi2  34457  tendospdi1  34659
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