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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  |-  B  =  ( Base `  G
)
iscyg.2  |-  .x.  =  (.g
`  G )
iscyg3.e  |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }
Assertion
Ref Expression
iscyggen  |-  ( X  e.  E  <->  ( X  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) )  =  B ) )
Distinct variable groups:    x, n, B    n, X, x    n, G, x    .x. , n, x
Allowed substitution hints:    E( x, n)

Proof of Theorem iscyggen
StepHypRef Expression
1 simpl 455 . . . . . 6  |-  ( ( x  =  X  /\  n  e.  ZZ )  ->  x  =  X )
21oveq2d 6248 . . . . 5  |-  ( ( x  =  X  /\  n  e.  ZZ )  ->  ( n  .x.  x
)  =  ( n 
.x.  X ) )
32mpteq2dva 4478 . . . 4  |-  ( x  =  X  ->  (
n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) )
43rneqd 5170 . . 3  |-  ( x  =  X  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) ) )
54eqeq1d 2402 . 2  |-  ( x  =  X  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  =  B ) )
6 iscyg3.e . 2  |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }
75, 6elrab2 3206 1  |-  ( X  e.  E  <->  ( X  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840   {crab 2755    |-> cmpt 4450   ran crn 4941   ` cfv 5523  (class class class)co 6232   ZZcz 10823   Basecbs 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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