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Theorem tendovalco 31401
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 2435 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 tendof.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2435 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendof.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
61, 2, 3, 4, 5istendo 31396 . . . 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 5021 . . . . . . . . 9  |-  ( f  =  F  ->  (
f  o.  g )  =  ( F  o.  g ) )
87fveq2d 5723 . . . . . . . 8  |-  ( f  =  F  ->  ( S `  ( f  o.  g ) )  =  ( S `  ( F  o.  g )
) )
9 fveq2 5719 . . . . . . . . 9  |-  ( f  =  F  ->  ( S `  f )  =  ( S `  F ) )
109coeq1d 5025 . . . . . . . 8  |-  ( f  =  F  ->  (
( S `  f
)  o.  ( S `
 g ) )  =  ( ( S `
 F )  o.  ( S `  g
) ) )
118, 10eqeq12d 2449 . . . . . . 7  |-  ( f  =  F  ->  (
( S `  (
f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g ) )  <->  ( S `  ( F  o.  g
) )  =  ( ( S `  F
)  o.  ( S `
 g ) ) ) )
12 coeq2 5022 . . . . . . . . 9  |-  ( g  =  G  ->  ( F  o.  g )  =  ( F  o.  G ) )
1312fveq2d 5723 . . . . . . . 8  |-  ( g  =  G  ->  ( S `  ( F  o.  g ) )  =  ( S `  ( F  o.  G )
) )
14 fveq2 5719 . . . . . . . . 9  |-  ( g  =  G  ->  ( S `  g )  =  ( S `  G ) )
1514coeq2d 5026 . . . . . . . 8  |-  ( g  =  G  ->  (
( S `  F
)  o.  ( S `
 g ) )  =  ( ( S `
 F )  o.  ( S `  G
) ) )
1613, 15eqeq12d 2449 . . . . . . 7  |-  ( g  =  G  ->  (
( S `  ( F  o.  g )
)  =  ( ( S `  F )  o.  ( S `  g ) )  <->  ( S `  ( F  o.  G
) )  =  ( ( S `  F
)  o.  ( S `
 G ) ) ) )
1711, 16rspc2v 3050 . . . . . 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 29 . . . . 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 979 . . . 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 220 . . 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 1150 . 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 419 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204    o. ccom 4873   -->wf 5441   ` cfv 5445   lecple 13524   LHypclh 30620   LTrncltrn 30737   trLctrl 30794   TEndoctendo 31388
This theorem is referenced by:  tendoco2  31404  tendococl  31408  tendodi1  31420  tendoicl  31432  cdlemi2  31455  tendospdi1  31657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-map 7011  df-tendo 31391
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