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Theorem iscyggen 17097
 Description: The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
iscyg.1
iscyg.2 .g
iscyg3.e
Assertion
Ref Expression
iscyggen
Distinct variable groups:   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem iscyggen
StepHypRef Expression
1 simpl 455 . . . . . 6
21oveq2d 6248 . . . . 5
32mpteq2dva 4478 . . . 4
43rneqd 5170 . . 3
54eqeq1d 2402 . 2
6 iscyg3.e . 2
75, 6elrab2 3206 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 367   wceq 1403   wcel 1840  crab 2755   cmpt 4450   crn 4941  cfv 5523  (class class class)co 6232  cz 10823  cbs 14731  .gcmg 16270 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-cnv 4948  df-dm 4950  df-rn 4951  df-iota 5487  df-fv 5531  df-ov 6235 This theorem is referenced by:  iscyggen2  17098  cyggenod  17101  cyggenod2  17102  cygznlem1  18793  cygznlem3  18796
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