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Theorem pgpgrp 15221
Description: Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
Assertion
Ref Expression
pgpgrp  |-  ( P pGrp 
G  ->  G  e.  Grp )

Proof of Theorem pgpgrp
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2436 . . 3  |-  ( od
`  G )  =  ( od `  G
)
31, 2ispgp 15219 . 2  |-  ( P pGrp 
G  <->  ( P  e. 
Prime  /\  G  e.  Grp  /\ 
A. x  e.  (
Base `  G ) E. n  e.  NN0  ( ( od `  G ) `  x
)  =  ( P ^ n ) ) )
43simp2bi 973 1  |-  ( P pGrp 
G  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2698   E.wrex 2699   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   NN0cn0 10214   ^cexp 11375   Primecprime 13072   Basecbs 13462   Grpcgrp 14678   odcod 15156   pGrp cpgp 15158
This theorem is referenced by:  pgphash  15234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-xp 4877  df-iota 5411  df-fv 5455  df-ov 6077  df-pgp 15162
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