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Theorem tendovalco 35778
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 2467 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 tendof.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2467 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendof.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
61, 2, 3, 4, 5istendo 35773 . . . 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 5160 . . . . . . . . 9  |-  ( f  =  F  ->  (
f  o.  g )  =  ( F  o.  g ) )
87fveq2d 5870 . . . . . . . 8  |-  ( f  =  F  ->  ( S `  ( f  o.  g ) )  =  ( S `  ( F  o.  g )
) )
9 fveq2 5866 . . . . . . . . 9  |-  ( f  =  F  ->  ( S `  f )  =  ( S `  F ) )
109coeq1d 5164 . . . . . . . 8  |-  ( f  =  F  ->  (
( S `  f
)  o.  ( S `
 g ) )  =  ( ( S `
 F )  o.  ( S `  g
) ) )
118, 10eqeq12d 2489 . . . . . . 7  |-  ( f  =  F  ->  (
( S `  (
f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g ) )  <->  ( S `  ( F  o.  g
) )  =  ( ( S `  F
)  o.  ( S `
 g ) ) ) )
12 coeq2 5161 . . . . . . . . 9  |-  ( g  =  G  ->  ( F  o.  g )  =  ( F  o.  G ) )
1312fveq2d 5870 . . . . . . . 8  |-  ( g  =  G  ->  ( S `  ( F  o.  g ) )  =  ( S `  ( F  o.  G )
) )
14 fveq2 5866 . . . . . . . . 9  |-  ( g  =  G  ->  ( S `  g )  =  ( S `  G ) )
1514coeq2d 5165 . . . . . . . 8  |-  ( g  =  G  ->  (
( S `  F
)  o.  ( S `
 g ) )  =  ( ( S `
 F )  o.  ( S `  G
) ) )
1613, 15eqeq12d 2489 . . . . . . 7  |-  ( g  =  G  ->  (
( S `  ( F  o.  g )
)  =  ( ( S `  F )  o.  ( S `  g ) )  <->  ( S `  ( F  o.  G
) )  =  ( ( S `  F
)  o.  ( S `
 G ) ) ) )
1711, 16rspc2v 3223 . . . . . 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 1018 . . . 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 228 . . 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 1193 . 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 429 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   class class class wbr 4447    o. ccom 5003   -->wf 5584   ` cfv 5588   lecple 14565   LHypclh 34997   LTrncltrn 35114   trLctrl 35171   TEndoctendo 35765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-map 7423  df-tendo 35768
This theorem is referenced by:  tendoco2  35781  tendococl  35785  tendodi1  35797  tendoicl  35809  cdlemi2  35832  tendospdi1  36034
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