MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscyggen Unicode version

Theorem iscyggen 15167
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 443 . . . . . 6  |-  ( ( x  =  X  /\  n  e.  ZZ )  ->  x  =  X )
21oveq2d 5874 . . . . 5  |-  ( ( x  =  X  /\  n  e.  ZZ )  ->  ( n  .x.  x
)  =  ( n 
.x.  X ) )
32mpteq2dva 4106 . . . 4  |-  ( x  =  X  ->  (
n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) )
43rneqd 4906 . . 3  |-  ( x  =  X  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) ) )
54eqeq1d 2291 . 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 2925 1  |-  ( X  e.  E  <->  ( X  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) )  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   ZZcz 10024   Basecbs 13148  .gcmg 14366
This theorem is referenced by:  iscyggen2  15168  cyggenod  15171  cyggenod2  15172  cygznlem1  16520  cygznlem3  16523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263  df-ov 5861
  Copyright terms: Public domain W3C validator