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Theorem tendovalco 31576
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 2296 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 tendof.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2296 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendof.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
61, 2, 3, 4, 5istendo 31571 . . . 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 4857 . . . . . . . . 9  |-  ( f  =  F  ->  (
f  o.  g )  =  ( F  o.  g ) )
87fveq2d 5545 . . . . . . . 8  |-  ( f  =  F  ->  ( S `  ( f  o.  g ) )  =  ( S `  ( F  o.  g )
) )
9 fveq2 5541 . . . . . . . . 9  |-  ( f  =  F  ->  ( S `  f )  =  ( S `  F ) )
109coeq1d 4861 . . . . . . . 8  |-  ( f  =  F  ->  (
( S `  f
)  o.  ( S `
 g ) )  =  ( ( S `
 F )  o.  ( S `  g
) ) )
118, 10eqeq12d 2310 . . . . . . 7  |-  ( f  =  F  ->  (
( S `  (
f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g ) )  <->  ( S `  ( F  o.  g
) )  =  ( ( S `  F
)  o.  ( S `
 g ) ) ) )
12 coeq2 4858 . . . . . . . . 9  |-  ( g  =  G  ->  ( F  o.  g )  =  ( F  o.  G ) )
1312fveq2d 5545 . . . . . . . 8  |-  ( g  =  G  ->  ( S `  ( F  o.  g ) )  =  ( S `  ( F  o.  G )
) )
14 fveq2 5541 . . . . . . . . 9  |-  ( g  =  G  ->  ( S `  g )  =  ( S `  G ) )
1514coeq2d 4862 . . . . . . . 8  |-  ( g  =  G  ->  (
( S `  F
)  o.  ( S `
 g ) )  =  ( ( S `
 F )  o.  ( S `  G
) ) )
1613, 15eqeq12d 2310 . . . . . . 7  |-  ( g  =  G  ->  (
( S `  ( F  o.  g )
)  =  ( ( S `  F )  o.  ( S `  g ) )  <->  ( S `  ( F  o.  G
) )  =  ( ( S `  F
)  o.  ( S `
 G ) ) ) )
1711, 16rspc2v 2903 . . . . . 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 27 . . . . 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 977 . . . 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 219 . . 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 1148 . 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 418 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039    o. ccom 4709   -->wf 5267   ` cfv 5271   lecple 13231   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563
This theorem is referenced by:  tendoco2  31579  tendococl  31583  tendodi1  31595  tendoicl  31607  cdlemi2  31630  tendospdi1  31832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-tendo 31566
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