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Theorem grplactval 14563
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 14562 . . 3  |-  ( A  e.  X  ->  ( F `  A )  =  ( a  e.  X  |->  ( A  .+  a ) ) )
43fveq1d 5527 . 2  |-  ( A  e.  X  ->  (
( F `  A
) `  B )  =  ( ( a  e.  X  |->  ( A 
.+  a ) ) `
 B ) )
5 oveq2 5866 . . 3  |-  ( a  =  B  ->  ( A  .+  a )  =  ( A  .+  B
) )
6 eqid 2283 . . 3  |-  ( a  e.  X  |->  ( A 
.+  a ) )  =  ( a  e.  X  |->  ( A  .+  a ) )
7 ovex 5883 . . 3  |-  ( A 
.+  B )  e. 
_V
85, 6, 7fvmpt 5602 . 2  |-  ( B  e.  X  ->  (
( a  e.  X  |->  ( A  .+  a
) ) `  B
)  =  ( A 
.+  B ) )
94, 8sylan9eq 2335 1  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( F `  A ) `  B
)  =  ( A 
.+  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148
This theorem is referenced by:  cayleylem2  14788  dchrsum2  20507  sumdchr2  20509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861
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