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Theorem grplactval 14814
Description: The value of the left group action of element  A of group  G at  B. (Contributed by Paul Chapman, 18-Mar-2008.)
Hypotheses
Ref Expression
grplact.1  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
grplact.2  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
grplactval  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( F `  A ) `  B
)  =  ( A 
.+  B ) )
Distinct variable groups:    g, a, A    G, a, g    .+ , a,
g    X, a, g    B, a
Allowed substitution hints:    B( g)    F( g, a)

Proof of Theorem grplactval
StepHypRef Expression
1 grplact.1 . . . 4  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
2 grplact.2 . . . 4  |-  X  =  ( Base `  G
)
31, 2grplactfval 14813 . . 3  |-  ( A  e.  X  ->  ( F `  A )  =  ( a  e.  X  |->  ( A  .+  a ) ) )
43fveq1d 5671 . 2  |-  ( A  e.  X  ->  (
( F `  A
) `  B )  =  ( ( a  e.  X  |->  ( A 
.+  a ) ) `
 B ) )
5 oveq2 6029 . . 3  |-  ( a  =  B  ->  ( A  .+  a )  =  ( A  .+  B
) )
6 eqid 2388 . . 3  |-  ( a  e.  X  |->  ( A 
.+  a ) )  =  ( a  e.  X  |->  ( A  .+  a ) )
7 ovex 6046 . . 3  |-  ( A 
.+  B )  e. 
_V
85, 6, 7fvmpt 5746 . 2  |-  ( B  e.  X  ->  (
( a  e.  X  |->  ( A  .+  a
) ) `  B
)  =  ( A 
.+  B ) )
94, 8sylan9eq 2440 1  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( F `  A ) `  B
)  =  ( A 
.+  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4208   ` cfv 5395  (class class class)co 6021   Basecbs 13397
This theorem is referenced by:  cayleylem2  15039  dchrsum2  20920  sumdchr2  20922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024
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