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Theorem pgpprm 16097
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 2443 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2443 . . 3  |-  ( od
`  G )  =  ( od `  G
)
31, 2ispgp 16096 . 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 1003 1  |-  ( P pGrp 
G  ->  P  e.  Prime )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   NN0cn0 10584   ^cexp 11870   Primecprime 13768   Basecbs 14179   Grpcgrp 15415   odcod 16033   pGrp cpgp 16035
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 4418  ax-nul 4426  ax-pr 4536
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-iota 5386  df-fv 5431  df-ov 6099  df-pgp 16039
This theorem is referenced by:  subgpgp  16101  pgpssslw  16118  sylow2blem3  16126  pgpfac1lem2  16581  pgpfac1lem3a  16582  pgpfac1lem3  16583  pgpfac1lem4  16584  pgpfaclem1  16587
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