Theorem pgpgrp 18009

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 x n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` G ) = ( Base ` G )
2		eqid 2622	. . 3 |- ( od ` G ) = ( od ` G )
3	1, 2	ispgp 18007	. 2 |- ( P pGrp G <-> ( P e. Prime /\ G e. Grp /\ A. x e. ( Base ` G ) E. n e. NN0 ( ( od ` G ) ` x ) = ( P ^ n ) ) )
4	3	simp2bi 1077	1 |- ( P pGrp G -> G e. Grp )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   NN0cn0 11292   ^cexp 12860   Primecprime 15385   Basecbs 15857   Grpcgrp 17422   odcod 17944   pGrp cpgp 17946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-iota 5851  df-fv 5896  df-ov 6653  df-pgp 17950

This theorem is referenced by:  pgphash  18022