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Theorem tendovalco 31624
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 2438 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 tendof.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2438 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendof.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
61, 2, 3, 4, 5istendo 31619 . . . 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 5032 . . . . . . . . 9  |-  ( f  =  F  ->  (
f  o.  g )  =  ( F  o.  g ) )
87fveq2d 5734 . . . . . . . 8  |-  ( f  =  F  ->  ( S `  ( f  o.  g ) )  =  ( S `  ( F  o.  g )
) )
9 fveq2 5730 . . . . . . . . 9  |-  ( f  =  F  ->  ( S `  f )  =  ( S `  F ) )
109coeq1d 5036 . . . . . . . 8  |-  ( f  =  F  ->  (
( S `  f
)  o.  ( S `
 g ) )  =  ( ( S `
 F )  o.  ( S `  g
) ) )
118, 10eqeq12d 2452 . . . . . . 7  |-  ( f  =  F  ->  (
( S `  (
f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g ) )  <->  ( S `  ( F  o.  g
) )  =  ( ( S `  F
)  o.  ( S `
 g ) ) ) )
12 coeq2 5033 . . . . . . . . 9  |-  ( g  =  G  ->  ( F  o.  g )  =  ( F  o.  G ) )
1312fveq2d 5734 . . . . . . . 8  |-  ( g  =  G  ->  ( S `  ( F  o.  g ) )  =  ( S `  ( F  o.  G )
) )
14 fveq2 5730 . . . . . . . . 9  |-  ( g  =  G  ->  ( S `  g )  =  ( S `  G ) )
1514coeq2d 5037 . . . . . . . 8  |-  ( g  =  G  ->  (
( S `  F
)  o.  ( S `
 g ) )  =  ( ( S `
 F )  o.  ( S `  G
) ) )
1613, 15eqeq12d 2452 . . . . . . 7  |-  ( g  =  G  ->  (
( S `  ( F  o.  g )
)  =  ( ( S `  F )  o.  ( S `  g ) )  <->  ( S `  ( F  o.  G
) )  =  ( ( S `  F
)  o.  ( S `
 G ) ) ) )
1711, 16rspc2v 3060 . . . . . 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 30 . . . . 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 980 . . . 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 221 . . 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 1151 . 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 420 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   class class class wbr 4214    o. ccom 4884   -->wf 5452   ` cfv 5456   lecple 13538   LHypclh 30843   LTrncltrn 30960   trLctrl 31017   TEndoctendo 31611
This theorem is referenced by:  tendoco2  31627  tendococl  31631  tendodi1  31643  tendoicl  31655  cdlemi2  31678  tendospdi1  31880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-tendo 31614
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