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Theorem iscyggen 16669
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 457 . . . . . 6  |-  ( ( x  =  X  /\  n  e.  ZZ )  ->  x  =  X )
21oveq2d 6293 . . . . 5  |-  ( ( x  =  X  /\  n  e.  ZZ )  ->  ( n  .x.  x
)  =  ( n 
.x.  X ) )
32mpteq2dva 4528 . . . 4  |-  ( x  =  X  ->  (
n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) )
43rneqd 5223 . . 3  |-  ( x  =  X  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) ) )
54eqeq1d 2464 . 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 3258 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 369    = wceq 1374    e. wcel 1762   {crab 2813    |-> cmpt 4500   ran crn 4995   ` cfv 5581  (class class class)co 6277   ZZcz 10855   Basecbs 14481  .gcmg 15722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-cnv 5002  df-dm 5004  df-rn 5005  df-iota 5544  df-fv 5589  df-ov 6280
This theorem is referenced by:  iscyggen2  16670  cyggenod  16673  cyggenod2  16674  cygznlem1  18367  cygznlem3  18370
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