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Theorem ispgp 16485
Description: A group is a  P-group if every element has some power of  P as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
Hypotheses
Ref Expression
ispgp.1  |-  X  =  ( Base `  G
)
ispgp.2  |-  O  =  ( od `  G
)
Assertion
Ref Expression
ispgp  |-  ( P pGrp 
G  <->  ( P  e. 
Prime  /\  G  e.  Grp  /\ 
A. x  e.  X  E. n  e.  NN0  ( O `  x )  =  ( P ^
n ) ) )
Distinct variable groups:    x, n, G    P, n, x    x, X
Allowed substitution hints:    O( x, n)    X( n)

Proof of Theorem ispgp
Dummy variables  g  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  g  =  G )
21fveq2d 5876 . . . . 5  |-  ( ( p  =  P  /\  g  =  G )  ->  ( Base `  g
)  =  ( Base `  G ) )
3 ispgp.1 . . . . 5  |-  X  =  ( Base `  G
)
42, 3syl6eqr 2526 . . . 4  |-  ( ( p  =  P  /\  g  =  G )  ->  ( Base `  g
)  =  X )
51fveq2d 5876 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  ( od `  g
)  =  ( od
`  G ) )
6 ispgp.2 . . . . . . . 8  |-  O  =  ( od `  G
)
75, 6syl6eqr 2526 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  ( od `  g
)  =  O )
87fveq1d 5874 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( od `  g ) `  x
)  =  ( O `
 x ) )
9 simpl 457 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  p  =  P )
109oveq1d 6310 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  ( p ^ n
)  =  ( P ^ n ) )
118, 10eqeq12d 2489 . . . . 5  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( ( od
`  g ) `  x )  =  ( p ^ n )  <-> 
( O `  x
)  =  ( P ^ n ) ) )
1211rexbidv 2978 . . . 4  |-  ( ( p  =  P  /\  g  =  G )  ->  ( E. n  e. 
NN0  ( ( od
`  g ) `  x )  =  ( p ^ n )  <->  E. n  e.  NN0  ( O `  x )  =  ( P ^
n ) ) )
134, 12raleqbidv 3077 . . 3  |-  ( ( p  =  P  /\  g  =  G )  ->  ( A. x  e.  ( Base `  g
) E. n  e. 
NN0  ( ( od
`  g ) `  x )  =  ( p ^ n )  <->  A. x  e.  X  E. n  e.  NN0  ( O `  x )  =  ( P ^
n ) ) )
14 df-pgp 16428 . . 3  |- pGrp  =  { <. p ,  g >.  |  ( ( p  e.  Prime  /\  g  e.  Grp )  /\  A. x  e.  ( Base `  g ) E. n  e.  NN0  ( ( od
`  g ) `  x )  =  ( p ^ n ) ) }
1513, 14brab2ga 5081 . 2  |-  ( P pGrp 
G  <->  ( ( P  e.  Prime  /\  G  e. 
Grp )  /\  A. x  e.  X  E. n  e.  NN0  ( O `
 x )  =  ( P ^ n
) ) )
16 df-3an 975 . 2  |-  ( ( P  e.  Prime  /\  G  e.  Grp  /\  A. x  e.  X  E. n  e.  NN0  ( O `  x )  =  ( P ^ n ) )  <->  ( ( P  e.  Prime  /\  G  e. 
Grp )  /\  A. x  e.  X  E. n  e.  NN0  ( O `
 x )  =  ( P ^ n
) ) )
1715, 16bitr4i 252 1  |-  ( P pGrp 
G  <->  ( P  e. 
Prime  /\  G  e.  Grp  /\ 
A. x  e.  X  E. n  e.  NN0  ( O `  x )  =  ( P ^
n ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   NN0cn0 10807   ^cexp 12146   Primecprime 14093   Basecbs 14507   Grpcgrp 15925   odcod 16422   pGrp cpgp 16424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-iota 5557  df-fv 5602  df-ov 6298  df-pgp 16428
This theorem is referenced by:  pgpprm  16486  pgpgrp  16487  pgpfi1  16488  subgpgp  16490  pgpfi  16498
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