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Theorem ispgp 15218
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 448 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  g  =  G )
21fveq2d 5724 . . . . 5  |-  ( ( p  =  P  /\  g  =  G )  ->  ( Base `  g
)  =  ( Base `  G ) )
3 ispgp.1 . . . . 5  |-  X  =  ( Base `  G
)
42, 3syl6eqr 2485 . . . 4  |-  ( ( p  =  P  /\  g  =  G )  ->  ( Base `  g
)  =  X )
51fveq2d 5724 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  ( od `  g
)  =  ( od
`  G ) )
6 ispgp.2 . . . . . . . 8  |-  O  =  ( od `  G
)
75, 6syl6eqr 2485 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  ( od `  g
)  =  O )
87fveq1d 5722 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( od `  g ) `  x
)  =  ( O `
 x ) )
9 simpl 444 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  p  =  P )
109oveq1d 6088 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  ( p ^ n
)  =  ( P ^ n ) )
118, 10eqeq12d 2449 . . . . 5  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( ( od
`  g ) `  x )  =  ( p ^ n )  <-> 
( O `  x
)  =  ( P ^ n ) ) )
1211rexbidv 2718 . . . 4  |-  ( ( p  =  P  /\  g  =  G )  ->  ( E. n  e. 
NN0  ( ( od
`  g ) `  x )  =  ( p ^ n )  <->  E. n  e.  NN0  ( O `  x )  =  ( P ^
n ) ) )
134, 12raleqbidv 2908 . . 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 15161 . . 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 4943 . 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 938 . 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 244 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   NN0cn0 10213   ^cexp 11374   Primecprime 13071   Basecbs 13461   Grpcgrp 14677   odcod 15155   pGrp cpgp 15157
This theorem is referenced by:  pgpprm  15219  pgpgrp  15220  pgpfi1  15221  subgpgp  15223  pgpfi  15231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-iota 5410  df-fv 5454  df-ov 6076  df-pgp 15161
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