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

Theorem iscyggen 15492
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 445 . . . . . 6  |-  ( ( x  =  X  /\  n  e.  ZZ )  ->  x  =  X )
21oveq2d 6099 . . . . 5  |-  ( ( x  =  X  /\  n  e.  ZZ )  ->  ( n  .x.  x
)  =  ( n 
.x.  X ) )
32mpteq2dva 4297 . . . 4  |-  ( x  =  X  ->  (
n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) )
43rneqd 5099 . . 3  |-  ( x  =  X  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) ) )
54eqeq1d 2446 . 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 3096 1  |-  ( X  e.  E  <->  ( X  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) )  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711    e. cmpt 4268   ran crn 4881   ` cfv 5456  (class class class)co 6083   ZZcz 10284   Basecbs 13471  .gcmg 14691
This theorem is referenced by:  iscyggen2  15493  cyggenod  15496  cyggenod2  15497  cygznlem1  16849  cygznlem3  16852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891  df-iota 5420  df-fv 5464  df-ov 6086
  Copyright terms: Public domain W3C validator