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Theorem odzcllem 12857
Description: - Lemma for odzcl 12858, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.)
Assertion
Ref Expression
odzcllem  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( ( od Z `  N ) `  A
)  e.  NN  /\  N  ||  ( ( A ^ ( ( od
Z `  N ) `  A ) )  - 
1 ) ) )

Proof of Theorem odzcllem
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 odzval 12856 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( od Z `  N ) `  A
)  =  sup ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1 ) } ,  RR ,  `'  <  ) )
2 ssrab2 3258 . . . . 5  |-  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  C_  NN
3 nnuz 10263 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
42, 3sseqtri 3210 . . . 4  |-  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  C_  ( ZZ>=
`  1 )
5 phicl 12837 . . . . . . 7  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
653ad2ant1 976 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( phi `  N )  e.  NN )
7 eulerth 12851 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A ^ ( phi `  N ) )  mod  N )  =  ( 1  mod  N
) )
8 simp1 955 . . . . . . . 8  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  N  e.  NN )
9 simp2 956 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  A  e.  ZZ )
106nnnn0d 10018 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( phi `  N )  e. 
NN0 )
11 zexpcl 11118 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( phi `  N )  e.  NN0 )  -> 
( A ^ ( phi `  N ) )  e.  ZZ )
129, 10, 11syl2anc 642 . . . . . . . 8  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( A ^ ( phi `  N ) )  e.  ZZ )
13 1z 10053 . . . . . . . . 9  |-  1  e.  ZZ
14 moddvds 12538 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A ^ ( phi `  N ) )  e.  ZZ  /\  1  e.  ZZ )  ->  (
( ( A ^
( phi `  N
) )  mod  N
)  =  ( 1  mod  N )  <->  N  ||  (
( A ^ ( phi `  N ) )  -  1 ) ) )
1513, 14mp3an3 1266 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A ^ ( phi `  N ) )  e.  ZZ )  ->  (
( ( A ^
( phi `  N
) )  mod  N
)  =  ( 1  mod  N )  <->  N  ||  (
( A ^ ( phi `  N ) )  -  1 ) ) )
168, 12, 15syl2anc 642 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( ( A ^
( phi `  N
) )  mod  N
)  =  ( 1  mod  N )  <->  N  ||  (
( A ^ ( phi `  N ) )  -  1 ) ) )
177, 16mpbid 201 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  N  ||  ( ( A ^
( phi `  N
) )  -  1 ) )
18 oveq2 5866 . . . . . . . . 9  |-  ( n  =  ( phi `  N )  ->  ( A ^ n )  =  ( A ^ ( phi `  N ) ) )
1918oveq1d 5873 . . . . . . . 8  |-  ( n  =  ( phi `  N )  ->  (
( A ^ n
)  -  1 )  =  ( ( A ^ ( phi `  N ) )  - 
1 ) )
2019breq2d 4035 . . . . . . 7  |-  ( n  =  ( phi `  N )  ->  ( N  ||  ( ( A ^ n )  - 
1 )  <->  N  ||  (
( A ^ ( phi `  N ) )  -  1 ) ) )
2120rspcev 2884 . . . . . 6  |-  ( ( ( phi `  N
)  e.  NN  /\  N  ||  ( ( A ^ ( phi `  N ) )  - 
1 ) )  ->  E. n  e.  NN  N  ||  ( ( A ^ n )  - 
1 ) )
226, 17, 21syl2anc 642 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  E. n  e.  NN  N  ||  (
( A ^ n
)  -  1 ) )
23 rabn0 3474 . . . . 5  |-  ( { n  e.  NN  |  N  ||  ( ( A ^ n )  - 
1 ) }  =/=  (/)  <->  E. n  e.  NN  N  ||  ( ( A ^
n )  -  1 ) )
2422, 23sylibr 203 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  =/=  (/) )
25 infmssuzcl 10301 . . . 4  |-  ( ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1 ) } 
C_  ( ZZ>= `  1
)  /\  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  =/=  (/) )  ->  sup ( { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) } ,  RR ,  `'  <  )  e. 
{ n  e.  NN  |  N  ||  ( ( A ^ n )  -  1 ) } )
264, 24, 25sylancr 644 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  sup ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1 ) } ,  RR ,  `'  <  )  e.  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) } )
271, 26eqeltrd 2357 . 2  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( od Z `  N ) `  A
)  e.  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) } )
28 oveq2 5866 . . . . 5  |-  ( n  =  ( ( od
Z `  N ) `  A )  ->  ( A ^ n )  =  ( A ^ (
( od Z `  N ) `  A
) ) )
2928oveq1d 5873 . . . 4  |-  ( n  =  ( ( od
Z `  N ) `  A )  ->  (
( A ^ n
)  -  1 )  =  ( ( A ^ ( ( od
Z `  N ) `  A ) )  - 
1 ) )
3029breq2d 4035 . . 3  |-  ( n  =  ( ( od
Z `  N ) `  A )  ->  ( N  ||  ( ( A ^ n )  - 
1 )  <->  N  ||  (
( A ^ (
( od Z `  N ) `  A
) )  -  1 ) ) )
3130elrab 2923 . 2  |-  ( ( ( od Z `  N ) `  A
)  e.  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  <->  ( (
( od Z `  N ) `  A
)  e.  NN  /\  N  ||  ( ( A ^ ( ( od
Z `  N ) `  A ) )  - 
1 ) ) )
3227, 31sylib 188 1  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( ( od Z `  N ) `  A
)  e.  NN  /\  N  ||  ( ( A ^ ( ( od
Z `  N ) `  A ) )  - 
1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   class class class wbr 4023   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   1c1 8738    < clt 8867    - cmin 9037   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230    mod cmo 10973   ^cexp 11104    || cdivides 12531    gcd cgcd 12685   od
Zcodz 12831   phicphi 12832
This theorem is referenced by:  odzcl  12858  odzid  12859
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-odz 12833  df-phi 12834
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