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Theorem pgpprm 17173
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 2420 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2420 . . 3  |-  ( od
`  G )  =  ( od `  G
)
31, 2ispgp 17172 . 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 1020 1  |-  ( P pGrp 
G  ->  P  e.  Prime )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   A.wral 2773   E.wrex 2774   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   NN0cn0 10858   ^cexp 12258   Primecprime 14582   Basecbs 15073   Grpcgrp 16613   odcod 17109   pGrp cpgp 17111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-xp 4851  df-iota 5556  df-fv 5600  df-ov 6299  df-pgp 17115
This theorem is referenced by:  subgpgp  17177  pgpssslw  17194  sylow2blem3  17202  pgpfac1lem2  17636  pgpfac1lem3a  17637  pgpfac1lem3  17638  pgpfac1lem4  17639  pgpfaclem1  17642
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