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Definition df-0g 13420
Description: Define group identity element. (Contributed by NM, 20-Aug-2011.)
Assertion
Ref Expression
df-0g  |-  0g  =  ( g  e.  _V  |->  ( iota e ( e  e.  ( Base `  g
)  /\  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) ) ) )
Distinct variable group:    e, g, x

Detailed syntax breakdown of Definition df-0g
StepHypRef Expression
1 c0g 13416 . 2  class  0g
2 vg . . 3  set  g
3 cvv 2801 . . 3  class  _V
4 ve . . . . . . 7  set  e
54cv 1631 . . . . . 6  class  e
62cv 1631 . . . . . . 7  class  g
7 cbs 13164 . . . . . . 7  class  Base
86, 7cfv 5271 . . . . . 6  class  ( Base `  g )
95, 8wcel 1696 . . . . 5  wff  e  e.  ( Base `  g
)
10 vx . . . . . . . . . 10  set  x
1110cv 1631 . . . . . . . . 9  class  x
12 cplusg 13224 . . . . . . . . . 10  class  +g
136, 12cfv 5271 . . . . . . . . 9  class  ( +g  `  g )
145, 11, 13co 5874 . . . . . . . 8  class  ( e ( +g  `  g
) x )
1514, 11wceq 1632 . . . . . . 7  wff  ( e ( +g  `  g
) x )  =  x
1611, 5, 13co 5874 . . . . . . . 8  class  ( x ( +g  `  g
) e )
1716, 11wceq 1632 . . . . . . 7  wff  ( x ( +g  `  g
) e )  =  x
1815, 17wa 358 . . . . . 6  wff  ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x )
1918, 10, 8wral 2556 . . . . 5  wff  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x )
209, 19wa 358 . . . 4  wff  ( e  e.  ( Base `  g
)  /\  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) )
2120, 4cio 5233 . . 3  class  ( iota e ( e  e.  ( Base `  g
)  /\  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) ) )
222, 3, 21cmpt 4093 . 2  class  ( g  e.  _V  |->  ( iota e ( e  e.  ( Base `  g
)  /\  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) ) ) )
231, 22wceq 1632 1  wff  0g  =  ( g  e.  _V  |->  ( iota e ( e  e.  ( Base `  g
)  /\  A. x  e.  ( Base `  g
) ( ( e ( +g  `  g
) x )  =  x  /\  ( x ( +g  `  g
) e )  =  x ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  grpidval  14400  fn0g  14401
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