Metamath Proof Explorer

Definition df-gex

Description: Define the exponent of a group. (Contributed by Mario Carneiro, 13-Jul-2014) (Revised by Stefan O'Rear, 4-Sep-2015) (Revised by AV, 26-Sep-2020)

Ref Expression
Assertion df-gex 
|- gEx = ( g e. _V |-> [_ { n e. NN | A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cgex 
 |-  gEx
1 vg 
 |-  g
2 cvv 
 |-  _V
3 vn 
 |-  n
4 cn 
 |-  NN
5 vx 
 |-  x
6 cbs 
 |-  Base
7 1 cv 
 |-  g
8 7 6 cfv 
 |-  ( Base ` g )
9 3 cv 
 |-  n
10 cmg 
 |-  .g
11 7 10 cfv 
 |-  ( .g ` g )
12 5 cv 
 |-  x
13 9 12 11 co 
 |-  ( n ( .g ` g ) x )
14 c0g 
 |-  0g
15 7 14 cfv 
 |-  ( 0g ` g )
16 13 15 wceq 
 |-  ( n ( .g ` g ) x ) = ( 0g ` g )
17 16 5 8 wral 
 |-  A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g )
18 17 3 4 crab 
 |-  { n e. NN | A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g ) }
19 vi 
 |-  i
20 19 cv 
 |-  i
21 c0 
 |-  (/)
22 20 21 wceq 
 |-  i = (/)
23 cc0 
 |-  0
24 cr 
 |-  RR
25 clt 
 |-  <
26 20 24 25 cinf 
 |-  inf ( i , RR , < )
27 22 23 26 cif 
 |-  if ( i = (/) , 0 , inf ( i , RR , < ) )
28 19 18 27 csb 
 |-  [_ { n e. NN | A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) )
29 1 2 28 cmpt 
 |-  ( g e. _V |-> [_ { n e. NN | A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) )
30 0 29 wceq 
 |-  gEx = ( g e. _V |-> [_ { n e. NN | A. x e. ( Base ` g ) ( n ( .g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) )