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

Theorem pgpprm 15227
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 2436 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2436 . . 3  |-  ( od
`  G )  =  ( od `  G
)
31, 2ispgp 15226 . 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 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   NN0cn0 10221   ^cexp 11382   Primecprime 13079   Basecbs 13469   Grpcgrp 14685   odcod 15163   pGrp cpgp 15165
This theorem is referenced by:  subgpgp  15231  pgpssslw  15248  sylow2blem3  15256  pgpfac1lem2  15633  pgpfac1lem3a  15634  pgpfac1lem3  15635  pgpfac1lem4  15636  pgpfaclem1  15639
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 4330  ax-nul 4338  ax-pr 4403
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 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-iota 5418  df-fv 5462  df-ov 6084  df-pgp 15169
  Copyright terms: Public domain W3C validator