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Theorem cyggrp 15501
 Description: A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
cyggrp CycGrp

Proof of Theorem cyggrp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3
2 eqid 2438 . . 3 .g .g
31, 2iscyg 15491 . 2 CycGrp .g
43simplbi 448 1 CycGrp
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  wrex 2708   cmpt 4268   crn 4881  cfv 5456  (class class class)co 6083  cz 10284  cbs 13471  cgrp 14687  .gcmg 14691  CycGrpccyg 15489 This theorem is referenced by:  cygznlem1  16849  cygznlem2a  16850  cygznlem3  16852 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891  df-iota 5420  df-fv 5464  df-ov 6086  df-cyg 15490
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