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Theorem odzval 14329
Description: Value of the order function. This is a function of functions; the inner argument selects the base (i.e. mod  N for some  N, often prime) and the outer argument selects the integer or equivalence class (if you want to think about it that way) from the integers mod  N. In order to ensure the supremum is well-defined, we only define the expression when  A and  N are coprime. (Contributed by Mario Carneiro, 23-Feb-2014.)
Assertion
Ref Expression
odzval  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( odZ `  N ) `  A
)  =  sup ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1 ) } ,  RR ,  `'  <  ) )
Distinct variable groups:    n, N    A, n

Proof of Theorem odzval
Dummy variables  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6304 . . . . . . . . 9  |-  ( m  =  N  ->  (
x  gcd  m )  =  ( x  gcd  N ) )
21eqeq1d 2459 . . . . . . . 8  |-  ( m  =  N  ->  (
( x  gcd  m
)  =  1  <->  (
x  gcd  N )  =  1 ) )
32rabbidv 3101 . . . . . . 7  |-  ( m  =  N  ->  { x  e.  ZZ  |  ( x  gcd  m )  =  1 }  =  {
x  e.  ZZ  | 
( x  gcd  N
)  =  1 } )
4 oveq1 6303 . . . . . . . . 9  |-  ( n  =  x  ->  (
n  gcd  N )  =  ( x  gcd  N ) )
54eqeq1d 2459 . . . . . . . 8  |-  ( n  =  x  ->  (
( n  gcd  N
)  =  1  <->  (
x  gcd  N )  =  1 ) )
65cbvrabv 3108 . . . . . . 7  |-  { n  e.  ZZ  |  ( n  gcd  N )  =  1 }  =  {
x  e.  ZZ  | 
( x  gcd  N
)  =  1 }
73, 6syl6eqr 2516 . . . . . 6  |-  ( m  =  N  ->  { x  e.  ZZ  |  ( x  gcd  m )  =  1 }  =  {
n  e.  ZZ  | 
( n  gcd  N
)  =  1 } )
8 breq1 4459 . . . . . . . 8  |-  ( m  =  N  ->  (
m  ||  ( (
x ^ n )  -  1 )  <->  N  ||  (
( x ^ n
)  -  1 ) ) )
98rabbidv 3101 . . . . . . 7  |-  ( m  =  N  ->  { n  e.  NN  |  m  ||  ( ( x ^
n )  -  1 ) }  =  {
n  e.  NN  |  N  ||  ( ( x ^ n )  - 
1 ) } )
109supeq1d 7923 . . . . . 6  |-  ( m  =  N  ->  sup ( { n  e.  NN  |  m  ||  ( ( x ^ n )  -  1 ) } ,  RR ,  `'  <  )  =  sup ( { n  e.  NN  |  N  ||  ( ( x ^ n )  -  1 ) } ,  RR ,  `'  <  ) )
117, 10mpteq12dv 4535 . . . . 5  |-  ( m  =  N  ->  (
x  e.  { x  e.  ZZ  |  ( x  gcd  m )  =  1 }  |->  sup ( { n  e.  NN  |  m  ||  ( ( x ^ n )  -  1 ) } ,  RR ,  `'  <  ) )  =  ( x  e.  { n  e.  ZZ  |  ( n  gcd  N )  =  1 }  |->  sup ( { n  e.  NN  |  N  ||  ( ( x ^ n )  -  1 ) } ,  RR ,  `'  <  ) ) )
12 df-odz 14306 . . . . 5  |-  odZ 
=  ( m  e.  NN  |->  ( x  e. 
{ x  e.  ZZ  |  ( x  gcd  m )  =  1 }  |->  sup ( { n  e.  NN  |  m  ||  ( ( x ^
n )  -  1 ) } ,  RR ,  `'  <  ) ) )
13 zex 10894 . . . . . 6  |-  ZZ  e.  _V
1413mptrabex 6145 . . . . 5  |-  ( x  e.  { n  e.  ZZ  |  ( n  gcd  N )  =  1 }  |->  sup ( { n  e.  NN  |  N  ||  ( ( x ^ n )  -  1 ) } ,  RR ,  `'  <  ) )  e.  _V
1511, 12, 14fvmpt 5956 . . . 4  |-  ( N  e.  NN  ->  ( odZ `  N )  =  ( x  e. 
{ n  e.  ZZ  |  ( n  gcd  N )  =  1 } 
|->  sup ( { n  e.  NN  |  N  ||  ( ( x ^
n )  -  1 ) } ,  RR ,  `'  <  ) ) )
1615fveq1d 5874 . . 3  |-  ( N  e.  NN  ->  (
( odZ `  N ) `  A
)  =  ( ( x  e.  { n  e.  ZZ  |  ( n  gcd  N )  =  1 }  |->  sup ( { n  e.  NN  |  N  ||  ( ( x ^ n )  -  1 ) } ,  RR ,  `'  <  ) ) `  A
) )
17 oveq1 6303 . . . . . 6  |-  ( n  =  A  ->  (
n  gcd  N )  =  ( A  gcd  N ) )
1817eqeq1d 2459 . . . . 5  |-  ( n  =  A  ->  (
( n  gcd  N
)  =  1  <->  ( A  gcd  N )  =  1 ) )
1918elrab 3257 . . . 4  |-  ( A  e.  { n  e.  ZZ  |  ( n  gcd  N )  =  1 }  <->  ( A  e.  ZZ  /\  ( A  gcd  N )  =  1 ) )
20 oveq1 6303 . . . . . . . . 9  |-  ( x  =  A  ->  (
x ^ n )  =  ( A ^
n ) )
2120oveq1d 6311 . . . . . . . 8  |-  ( x  =  A  ->  (
( x ^ n
)  -  1 )  =  ( ( A ^ n )  - 
1 ) )
2221breq2d 4468 . . . . . . 7  |-  ( x  =  A  ->  ( N  ||  ( ( x ^ n )  - 
1 )  <->  N  ||  (
( A ^ n
)  -  1 ) ) )
2322rabbidv 3101 . . . . . 6  |-  ( x  =  A  ->  { n  e.  NN  |  N  ||  ( ( x ^
n )  -  1 ) }  =  {
n  e.  NN  |  N  ||  ( ( A ^ n )  - 
1 ) } )
2423supeq1d 7923 . . . . 5  |-  ( x  =  A  ->  sup ( { n  e.  NN  |  N  ||  ( ( x ^ n )  -  1 ) } ,  RR ,  `'  <  )  =  sup ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1 ) } ,  RR ,  `'  <  ) )
25 eqid 2457 . . . . 5  |-  ( x  e.  { n  e.  ZZ  |  ( n  gcd  N )  =  1 }  |->  sup ( { n  e.  NN  |  N  ||  ( ( x ^ n )  -  1 ) } ,  RR ,  `'  <  ) )  =  ( x  e.  { n  e.  ZZ  |  ( n  gcd  N )  =  1 }  |->  sup ( { n  e.  NN  |  N  ||  ( ( x ^ n )  -  1 ) } ,  RR ,  `'  <  ) )
26 gtso 9683 . . . . . 6  |-  `'  <  Or  RR
2726supex 7940 . . . . 5  |-  sup ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1 ) } ,  RR ,  `'  <  )  e.  _V
2824, 25, 27fvmpt 5956 . . . 4  |-  ( A  e.  { n  e.  ZZ  |  ( n  gcd  N )  =  1 }  ->  (
( x  e.  {
n  e.  ZZ  | 
( n  gcd  N
)  =  1 } 
|->  sup ( { n  e.  NN  |  N  ||  ( ( x ^
n )  -  1 ) } ,  RR ,  `'  <  ) ) `
 A )  =  sup ( { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) } ,  RR ,  `'  <  ) )
2919, 28sylbir 213 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  -> 
( ( x  e. 
{ n  e.  ZZ  |  ( n  gcd  N )  =  1 } 
|->  sup ( { n  e.  NN  |  N  ||  ( ( x ^
n )  -  1 ) } ,  RR ,  `'  <  ) ) `
 A )  =  sup ( { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) } ,  RR ,  `'  <  ) )
3016, 29sylan9eq 2518 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ZZ  /\  ( A  gcd  N
)  =  1 ) )  ->  ( ( odZ `  N ) `
 A )  =  sup ( { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) } ,  RR ,  `'  <  ) )
31303impb 1192 1  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( odZ `  N ) `  A
)  =  sup ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1 ) } ,  RR ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   ` cfv 5594  (class class class)co 6296   supcsup 7918   RRcr 9508   1c1 9510    < clt 9645    - cmin 9824   NNcn 10556   ZZcz 10885   ^cexp 12168    || cdvds 13997    gcd cgcd 14155   odZcodz 14304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-neg 9827  df-z 10886  df-odz 14306
This theorem is referenced by:  odzcllem  14330  odzdvds  14333
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