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Theorem pgpprm 16419
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 2467 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2467 . . 3  |-  ( od
`  G )  =  ( od `  G
)
31, 2ispgp 16418 . 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 1011 1  |-  ( P pGrp 
G  ->  P  e.  Prime )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   NN0cn0 10795   ^cexp 12134   Primecprime 14076   Basecbs 14490   Grpcgrp 15727   odcod 16355   pGrp cpgp 16357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-iota 5551  df-fv 5596  df-ov 6287  df-pgp 16361
This theorem is referenced by:  subgpgp  16423  pgpssslw  16440  sylow2blem3  16448  pgpfac1lem2  16928  pgpfac1lem3a  16929  pgpfac1lem3  16930  pgpfac1lem4  16931  pgpfaclem1  16934
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