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Theorem ispgp 15218
 Description: A group is a -group if every element has some power of as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
Hypotheses
Ref Expression
ispgp.1
ispgp.2
Assertion
Ref Expression
ispgp pGrp
Distinct variable groups:   ,,   ,,   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem ispgp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 448 . . . . . 6
21fveq2d 5724 . . . . 5
3 ispgp.1 . . . . 5
42, 3syl6eqr 2485 . . . 4
51fveq2d 5724 . . . . . . . 8
6 ispgp.2 . . . . . . . 8
75, 6syl6eqr 2485 . . . . . . 7
87fveq1d 5722 . . . . . 6
9 simpl 444 . . . . . . 7
109oveq1d 6088 . . . . . 6
118, 10eqeq12d 2449 . . . . 5
1211rexbidv 2718 . . . 4
134, 12raleqbidv 2908 . . 3
14 df-pgp 15161 . . 3 pGrp
1513, 14brab2ga 4943 . 2 pGrp
16 df-3an 938 . 2
1715, 16bitr4i 244 1 pGrp
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  wrex 2698   class class class wbr 4204  cfv 5446  (class class class)co 6073  cn0 10213  cexp 11374  cprime 13071  cbs 13461  cgrp 14677  cod 15155   pGrp cpgp 15157 This theorem is referenced by:  pgpprm  15219  pgpgrp  15220  pgpfi1  15221  subgpgp  15223  pgpfi  15231 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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-iota 5410  df-fv 5454  df-ov 6076  df-pgp 15161
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