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Theorem cyggrp 17459
 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 2429 . . 3
2 eqid 2429 . . 3 .g .g
31, 2iscyg 17449 . 2 CycGrp .g
43simplbi 461 1 CycGrp
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1437   wcel 1870  wrex 2783   cmpt 4484   crn 4855  cfv 5601  (class class class)co 6305  cz 10937  cbs 15084  cgrp 16620  .gcmg 16623  CycGrpccyg 17447 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 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 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 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-cnv 4862  df-dm 4864  df-rn 4865  df-iota 5565  df-fv 5609  df-ov 6308  df-cyg 17448 This theorem is referenced by:  cygznlem1  19068  cygznlem2a  19069  cygznlem3  19071
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