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Theorem pgpgrp 15149
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 2381 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2381 . . 3  |-  ( od
`  G )  =  ( od `  G
)
31, 2ispgp 15147 . 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 1649    e. wcel 1717   A.wral 2643   E.wrex 2644   class class class wbr 4147   ` cfv 5388  (class class class)co 6014   NN0cn0 10147   ^cexp 11303   Primecprime 13000   Basecbs 13390   Grpcgrp 14606   odcod 15084   pGrp cpgp 15086
This theorem is referenced by:  pgphash  15162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pr 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-br 4148  df-opab 4202  df-xp 4818  df-iota 5352  df-fv 5396  df-ov 6017  df-pgp 15090
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