MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pgpprm Unicode version

Theorem pgpprm 15182
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 2404 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2404 . . 3  |-  ( od
`  G )  =  ( od `  G
)
31, 2ispgp 15181 . 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 972 1  |-  ( P pGrp 
G  ->  P  e.  Prime )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   NN0cn0 10177   ^cexp 11337   Primecprime 13034   Basecbs 13424   Grpcgrp 14640   odcod 15118   pGrp cpgp 15120
This theorem is referenced by:  subgpgp  15186  pgpssslw  15203  sylow2blem3  15211  pgpfac1lem2  15588  pgpfac1lem3a  15589  pgpfac1lem3  15590  pgpfac1lem4  15591  pgpfaclem1  15594
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-iota 5377  df-fv 5421  df-ov 6043  df-pgp 15124
  Copyright terms: Public domain W3C validator