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Theorem pgpprm 14920
Description: Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
Assertion
Ref Expression
pgpprm  |-  ( P pGrp 
G  ->  P  e.  Prime )

Proof of Theorem pgpprm
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2296 . . 3  |-  ( od
`  G )  =  ( od `  G
)
31, 2ispgp 14919 . 2  |-  ( P pGrp 
G  <->  ( P  e. 
Prime  /\  G  e.  Grp  /\ 
A. x  e.  (
Base `  G ) E. n  e.  NN0  ( ( od `  G ) `  x
)  =  ( P ^ n ) ) )
43simp1bi 970 1  |-  ( P pGrp 
G  ->  P  e.  Prime )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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:  subgpgp  14924  pgpssslw  14941  sylow2blem3  14949  pgpfac1lem2  15326  pgpfac1lem3a  15327  pgpfac1lem3  15328  pgpfac1lem4  15329  pgpfaclem1  15332
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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