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Definition df-ga 14744
Description: Define the class of all group actions. A group  G acts on a set  S if a permutation on  S is associated with every element of  G in such a way that the identity permutation on  S is associated with the neutral element of 
G, and the composition of the permutations associated with two elements of  G is identical with the permutation associated to the composition of these two elements (in the same order) in the group  G. (Contributed by Jeff Hankins, 10-Aug-2009.)
Assertion
Ref Expression
df-ga  |-  GrpAct  =  ( g  e.  Grp , 
s  e.  _V  |->  [_ ( Base `  g )  /  b ]_ {
m  e.  ( s  ^m  ( b  X.  s ) )  | 
A. x  e.  s  ( ( ( 0g
`  g ) m x )  =  x  /\  A. y  e.  b  A. z  e.  b  ( ( y ( +g  `  g
) z ) m x )  =  ( y m ( z m x ) ) ) } )
Distinct variable group:    g, b, m, s, x, y, z

Detailed syntax breakdown of Definition df-ga
StepHypRef Expression
1 cga 14743 . 2  class  GrpAct
2 vg . . 3  set  g
3 vs . . 3  set  s
4 cgrp 14362 . . 3  class  Grp
5 cvv 2788 . . 3  class  _V
6 vb . . . 4  set  b
72cv 1622 . . . . 5  class  g
8 cbs 13148 . . . . 5  class  Base
97, 8cfv 5255 . . . 4  class  ( Base `  g )
10 c0g 13400 . . . . . . . . . 10  class  0g
117, 10cfv 5255 . . . . . . . . 9  class  ( 0g
`  g )
12 vx . . . . . . . . . 10  set  x
1312cv 1622 . . . . . . . . 9  class  x
14 vm . . . . . . . . . 10  set  m
1514cv 1622 . . . . . . . . 9  class  m
1611, 13, 15co 5858 . . . . . . . 8  class  ( ( 0g `  g ) m x )
1716, 13wceq 1623 . . . . . . 7  wff  ( ( 0g `  g ) m x )  =  x
18 vy . . . . . . . . . . . . 13  set  y
1918cv 1622 . . . . . . . . . . . 12  class  y
20 vz . . . . . . . . . . . . 13  set  z
2120cv 1622 . . . . . . . . . . . 12  class  z
22 cplusg 13208 . . . . . . . . . . . . 13  class  +g
237, 22cfv 5255 . . . . . . . . . . . 12  class  ( +g  `  g )
2419, 21, 23co 5858 . . . . . . . . . . 11  class  ( y ( +g  `  g
) z )
2524, 13, 15co 5858 . . . . . . . . . 10  class  ( ( y ( +g  `  g
) z ) m x )
2621, 13, 15co 5858 . . . . . . . . . . 11  class  ( z m x )
2719, 26, 15co 5858 . . . . . . . . . 10  class  ( y m ( z m x ) )
2825, 27wceq 1623 . . . . . . . . 9  wff  ( ( y ( +g  `  g
) z ) m x )  =  ( y m ( z m x ) )
296cv 1622 . . . . . . . . 9  class  b
3028, 20, 29wral 2543 . . . . . . . 8  wff  A. z  e.  b  ( (
y ( +g  `  g
) z ) m x )  =  ( y m ( z m x ) )
3130, 18, 29wral 2543 . . . . . . 7  wff  A. y  e.  b  A. z  e.  b  ( (
y ( +g  `  g
) z ) m x )  =  ( y m ( z m x ) )
3217, 31wa 358 . . . . . 6  wff  ( ( ( 0g `  g
) m x )  =  x  /\  A. y  e.  b  A. z  e.  b  (
( y ( +g  `  g ) z ) m x )  =  ( y m ( z m x ) ) )
333cv 1622 . . . . . 6  class  s
3432, 12, 33wral 2543 . . . . 5  wff  A. x  e.  s  ( (
( 0g `  g
) m x )  =  x  /\  A. y  e.  b  A. z  e.  b  (
( y ( +g  `  g ) z ) m x )  =  ( y m ( z m x ) ) )
3529, 33cxp 4687 . . . . . 6  class  ( b  X.  s )
36 cmap 6772 . . . . . 6  class  ^m
3733, 35, 36co 5858 . . . . 5  class  ( s  ^m  ( b  X.  s ) )
3834, 14, 37crab 2547 . . . 4  class  { m  e.  ( s  ^m  (
b  X.  s ) )  |  A. x  e.  s  ( (
( 0g `  g
) m x )  =  x  /\  A. y  e.  b  A. z  e.  b  (
( y ( +g  `  g ) z ) m x )  =  ( y m ( z m x ) ) ) }
396, 9, 38csb 3081 . . 3  class  [_ ( Base `  g )  / 
b ]_ { m  e.  ( s  ^m  (
b  X.  s ) )  |  A. x  e.  s  ( (
( 0g `  g
) m x )  =  x  /\  A. y  e.  b  A. z  e.  b  (
( y ( +g  `  g ) z ) m x )  =  ( y m ( z m x ) ) ) }
402, 3, 4, 5, 39cmpt2 5860 . 2  class  ( g  e.  Grp ,  s  e.  _V  |->  [_ ( Base `  g )  / 
b ]_ { m  e.  ( s  ^m  (
b  X.  s ) )  |  A. x  e.  s  ( (
( 0g `  g
) m x )  =  x  /\  A. y  e.  b  A. z  e.  b  (
( y ( +g  `  g ) z ) m x )  =  ( y m ( z m x ) ) ) } )
411, 40wceq 1623 1  wff  GrpAct  =  ( g  e.  Grp , 
s  e.  _V  |->  [_ ( Base `  g )  /  b ]_ {
m  e.  ( s  ^m  ( b  X.  s ) )  | 
A. x  e.  s  ( ( ( 0g
`  g ) m x )  =  x  /\  A. y  e.  b  A. z  e.  b  ( ( y ( +g  `  g
) z ) m x )  =  ( y m ( z m x ) ) ) } )
Colors of variables: wff set class
This definition is referenced by:  isga  14745
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