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Theorem ispgp 14903
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 447 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  g  =  G )
21fveq2d 5529 . . . . 5  |-  ( ( p  =  P  /\  g  =  G )  ->  ( Base `  g
)  =  ( Base `  G ) )
3 ispgp.1 . . . . 5  |-  X  =  ( Base `  G
)
42, 3syl6eqr 2333 . . . 4  |-  ( ( p  =  P  /\  g  =  G )  ->  ( Base `  g
)  =  X )
51fveq2d 5529 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  ( od `  g
)  =  ( od
`  G ) )
6 ispgp.2 . . . . . . . 8  |-  O  =  ( od `  G
)
75, 6syl6eqr 2333 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  ( od `  g
)  =  O )
87fveq1d 5527 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( od `  g ) `  x
)  =  ( O `
 x ) )
9 simpl 443 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  p  =  P )
109oveq1d 5873 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  ( p ^ n
)  =  ( P ^ n ) )
118, 10eqeq12d 2297 . . . . 5  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( ( od
`  g ) `  x )  =  ( p ^ n )  <-> 
( O `  x
)  =  ( P ^ n ) ) )
1211rexbidv 2564 . . . 4  |-  ( ( p  =  P  /\  g  =  G )  ->  ( E. n  e. 
NN0  ( ( od
`  g ) `  x )  =  ( p ^ n )  <->  E. n  e.  NN0  ( O `  x )  =  ( P ^
n ) ) )
134, 12raleqbidv 2748 . . 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 14846 . . 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 4763 . 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 936 . 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 243 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 set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   NN0cn0 9965   ^cexp 11104   Primecprime 12758   Basecbs 13148   Grpcgrp 14362   odcod 14840   pGrp cpgp 14842
This theorem is referenced by:  pgpprm  14904  pgpgrp  14905  pgpfi1  14906  subgpgp  14908  pgpfi  14916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-iota 5219  df-fv 5263  df-ov 5861  df-pgp 14846
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