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Theorem grplactval 14878
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 14877 . . 3  |-  ( A  e.  X  ->  ( F `  A )  =  ( a  e.  X  |->  ( A  .+  a ) ) )
43fveq1d 5722 . 2  |-  ( A  e.  X  ->  (
( F `  A
) `  B )  =  ( ( a  e.  X  |->  ( A 
.+  a ) ) `
 B ) )
5 oveq2 6081 . . 3  |-  ( a  =  B  ->  ( A  .+  a )  =  ( A  .+  B
) )
6 eqid 2435 . . 3  |-  ( a  e.  X  |->  ( A 
.+  a ) )  =  ( a  e.  X  |->  ( A  .+  a ) )
7 ovex 6098 . . 3  |-  ( A 
.+  B )  e. 
_V
85, 6, 7fvmpt 5798 . 2  |-  ( B  e.  X  ->  (
( a  e.  X  |->  ( A  .+  a
) ) `  B
)  =  ( A 
.+  B ) )
94, 8sylan9eq 2487 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 1652    e. wcel 1725    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   Basecbs 13461
This theorem is referenced by:  cayleylem2  15103  dchrsum2  21044  sumdchr2  21046
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-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-pr 4395
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-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 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076
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