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Theorem ispgp 16090
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 5694 . . . . 5  |-  ( ( p  =  P  /\  g  =  G )  ->  ( Base `  g
)  =  ( Base `  G ) )
3 ispgp.1 . . . . 5  |-  X  =  ( Base `  G
)
42, 3syl6eqr 2492 . . . 4  |-  ( ( p  =  P  /\  g  =  G )  ->  ( Base `  g
)  =  X )
51fveq2d 5694 . . . . . . . 8  |-  ( ( p  =  P  /\  g  =  G )  ->  ( od `  g
)  =  ( od
`  G ) )
6 ispgp.2 . . . . . . . 8  |-  O  =  ( od `  G
)
75, 6syl6eqr 2492 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  ( od `  g
)  =  O )
87fveq1d 5692 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( od `  g ) `  x
)  =  ( O `
 x ) )
9 simpl 457 . . . . . . 7  |-  ( ( p  =  P  /\  g  =  G )  ->  p  =  P )
109oveq1d 6105 . . . . . 6  |-  ( ( p  =  P  /\  g  =  G )  ->  ( p ^ n
)  =  ( P ^ n ) )
118, 10eqeq12d 2456 . . . . 5  |-  ( ( p  =  P  /\  g  =  G )  ->  ( ( ( od
`  g ) `  x )  =  ( p ^ n )  <-> 
( O `  x
)  =  ( P ^ n ) ) )
1211rexbidv 2735 . . . 4  |-  ( ( p  =  P  /\  g  =  G )  ->  ( E. n  e. 
NN0  ( ( od
`  g ) `  x )  =  ( p ^ n )  <->  E. n  e.  NN0  ( O `  x )  =  ( P ^
n ) ) )
134, 12raleqbidv 2930 . . 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 16033 . . 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 4911 . 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 967 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2714   E.wrex 2715   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   NN0cn0 10578   ^cexp 11864   Primecprime 13762   Basecbs 14173   Grpcgrp 15409   odcod 16027   pGrp cpgp 16029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-xp 4845  df-iota 5380  df-fv 5425  df-ov 6093  df-pgp 16033
This theorem is referenced by:  pgpprm  16091  pgpgrp  16092  pgpfi1  16093  subgpgp  16095  pgpfi  16103
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