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Theorem ispgp 14919
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 5545 . . . . 5  |-  ( ( p  =  P  /\  g  =  G )  ->  ( Base `  g
)  =  ( Base `  G ) )
3 ispgp.1 . . . . 5  |-  X  =  ( Base `  G
)
42, 3syl6eqr 2346 . . . 4  |-  ( ( p  =  P  /\  g  =  G )  ->  ( Base `  g
)  =  X )
51fveq2d 5545 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  ( od `  g
)  =  ( od
`  G ) )
6 ispgp.2 . . . . . . . 8  |-  O  =  ( od `  G
)
75, 6syl6eqr 2346 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  ( od `  g
)  =  O )
87fveq1d 5543 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( od `  g ) `  x
)  =  ( O `
 x ) )
9 simpl 443 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  p  =  P )
109oveq1d 5889 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  ( p ^ n
)  =  ( P ^ n ) )
118, 10eqeq12d 2310 . . . . 5  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( ( od
`  g ) `  x )  =  ( p ^ n )  <-> 
( O `  x
)  =  ( P ^ n ) ) )
1211rexbidv 2577 . . . 4  |-  ( ( p  =  P  /\  g  =  G )  ->  ( E. n  e. 
NN0  ( ( od
`  g ) `  x )  =  ( p ^ n )  <->  E. n  e.  NN0  ( O `  x )  =  ( P ^
n ) ) )
134, 12raleqbidv 2761 . . 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 14862 . . 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 4779 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   NN0cn0 9981   ^cexp 11120   Primecprime 12774   Basecbs 13164   Grpcgrp 14378   odcod 14856   pGrp cpgp 14858
This theorem is referenced by:  pgpprm  14920  pgpgrp  14921  pgpfi1  14922  subgpgp  14924  pgpfi  14932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-iota 5235  df-fv 5279  df-ov 5877  df-pgp 14862
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