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

Theorem pgpprm 17173
 Description: Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
Assertion
Ref Expression
pgpprm pGrp

Proof of Theorem pgpprm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2420 . . 3
2 eqid 2420 . . 3
31, 2ispgp 17172 . 2 pGrp
43simp1bi 1020 1 pGrp
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1437   wcel 1867  wral 2773  wrex 2774   class class class wbr 4417  cfv 5592  (class class class)co 6296  cn0 10858  cexp 12258  cprime 14582  cbs 15073  cgrp 16613  cod 17109   pGrp cpgp 17111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-xp 4851  df-iota 5556  df-fv 5600  df-ov 6299  df-pgp 17115 This theorem is referenced by:  subgpgp  17177  pgpssslw  17194  sylow2blem3  17202  pgpfac1lem2  17636  pgpfac1lem3a  17637  pgpfac1lem3  17638  pgpfac1lem4  17639  pgpfaclem1  17642
 Copyright terms: Public domain W3C validator