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Definition df-pgp 16039
Description: Define the set of p-groups, which are groups such that every element has a power of  p as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
Assertion
Ref Expression
df-pgp  |- pGrp  =  { <. p ,  g >.  |  ( ( p  e.  Prime  /\  g  e.  Grp )  /\  A. x  e.  ( Base `  g ) E. n  e.  NN0  ( ( od
`  g ) `  x )  =  ( p ^ n ) ) }
Distinct variable group:    g, n, p, x

Detailed syntax breakdown of Definition df-pgp
StepHypRef Expression
1 cpgp 16035 . 2  class pGrp
2 vp . . . . . . 7  setvar  p
32cv 1368 . . . . . 6  class  p
4 cprime 13768 . . . . . 6  class  Prime
53, 4wcel 1756 . . . . 5  wff  p  e. 
Prime
6 vg . . . . . . 7  setvar  g
76cv 1368 . . . . . 6  class  g
8 cgrp 15415 . . . . . 6  class  Grp
97, 8wcel 1756 . . . . 5  wff  g  e. 
Grp
105, 9wa 369 . . . 4  wff  ( p  e.  Prime  /\  g  e.  Grp )
11 vx . . . . . . . . 9  setvar  x
1211cv 1368 . . . . . . . 8  class  x
13 cod 16033 . . . . . . . . 9  class  od
147, 13cfv 5423 . . . . . . . 8  class  ( od
`  g )
1512, 14cfv 5423 . . . . . . 7  class  ( ( od `  g ) `
 x )
16 vn . . . . . . . . 9  setvar  n
1716cv 1368 . . . . . . . 8  class  n
18 cexp 11870 . . . . . . . 8  class  ^
193, 17, 18co 6096 . . . . . . 7  class  ( p ^ n )
2015, 19wceq 1369 . . . . . 6  wff  ( ( od `  g ) `
 x )  =  ( p ^ n
)
21 cn0 10584 . . . . . 6  class  NN0
2220, 16, 21wrex 2721 . . . . 5  wff  E. n  e.  NN0  ( ( od
`  g ) `  x )  =  ( p ^ n )
23 cbs 14179 . . . . . 6  class  Base
247, 23cfv 5423 . . . . 5  class  ( Base `  g )
2522, 11, 24wral 2720 . . . 4  wff  A. x  e.  ( Base `  g
) E. n  e. 
NN0  ( ( od
`  g ) `  x )  =  ( p ^ n )
2610, 25wa 369 . . 3  wff  ( ( p  e.  Prime  /\  g  e.  Grp )  /\  A. x  e.  ( Base `  g ) E. n  e.  NN0  ( ( od
`  g ) `  x )  =  ( p ^ n ) )
2726, 2, 6copab 4354 . 2  class  { <. p ,  g >.  |  ( ( p  e.  Prime  /\  g  e.  Grp )  /\  A. x  e.  (
Base `  g ) E. n  e.  NN0  ( ( od `  g ) `  x
)  =  ( p ^ n ) ) }
281, 27wceq 1369 1  wff pGrp  =  { <. p ,  g >.  |  ( ( p  e.  Prime  /\  g  e.  Grp )  /\  A. x  e.  ( Base `  g ) E. n  e.  NN0  ( ( od
`  g ) `  x )  =  ( p ^ n ) ) }
Colors of variables: wff setvar class
This definition is referenced by:  ispgp  16096
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