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Theorem odzcllem 13178
Description: - Lemma for odzcl 13179, 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 13177 . . 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 3428 . . . . 5  |-  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  C_  NN
3 nnuz 10521 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
42, 3sseqtri 3380 . . . 4  |-  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  C_  ( ZZ>=
`  1 )
5 phicl 13158 . . . . . . 7  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
653ad2ant1 978 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( phi `  N )  e.  NN )
7 eulerth 13172 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A ^ ( phi `  N ) )  mod  N )  =  ( 1  mod  N
) )
8 simp1 957 . . . . . . . 8  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  N  e.  NN )
9 simp2 958 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  A  e.  ZZ )
106nnnn0d 10274 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( phi `  N )  e. 
NN0 )
11 zexpcl 11396 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( phi `  N )  e.  NN0 )  -> 
( A ^ ( phi `  N ) )  e.  ZZ )
129, 10, 11syl2anc 643 . . . . . . . 8  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( A ^ ( phi `  N ) )  e.  ZZ )
13 1z 10311 . . . . . . . . 9  |-  1  e.  ZZ
14 moddvds 12859 . . . . . . . . 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 1268 . . . . . . . 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 643 . . . . . . 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 202 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  N  ||  ( ( A ^
( phi `  N
) )  -  1 ) )
18 oveq2 6089 . . . . . . . . 9  |-  ( n  =  ( phi `  N )  ->  ( A ^ n )  =  ( A ^ ( phi `  N ) ) )
1918oveq1d 6096 . . . . . . . 8  |-  ( n  =  ( phi `  N )  ->  (
( A ^ n
)  -  1 )  =  ( ( A ^ ( phi `  N ) )  - 
1 ) )
2019breq2d 4224 . . . . . . 7  |-  ( n  =  ( phi `  N )  ->  ( N  ||  ( ( A ^ n )  - 
1 )  <->  N  ||  (
( A ^ ( phi `  N ) )  -  1 ) ) )
2120rspcev 3052 . . . . . 6  |-  ( ( ( phi `  N
)  e.  NN  /\  N  ||  ( ( A ^ ( phi `  N ) )  - 
1 ) )  ->  E. n  e.  NN  N  ||  ( ( A ^ n )  - 
1 ) )
226, 17, 21syl2anc 643 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  E. n  e.  NN  N  ||  (
( A ^ n
)  -  1 ) )
23 rabn0 3647 . . . . 5  |-  ( { n  e.  NN  |  N  ||  ( ( A ^ n )  - 
1 ) }  =/=  (/)  <->  E. n  e.  NN  N  ||  ( ( A ^
n )  -  1 ) )
2422, 23sylibr 204 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  =/=  (/) )
25 infmssuzcl 10559 . . . 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 645 . . 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 2510 . 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 6089 . . . . 5  |-  ( n  =  ( ( od
Z `  N ) `  A )  ->  ( A ^ n )  =  ( A ^ (
( od Z `  N ) `  A
) ) )
2928oveq1d 6096 . . . 4  |-  ( n  =  ( ( od
Z `  N ) `  A )  ->  (
( A ^ n
)  -  1 )  =  ( ( A ^ ( ( od
Z `  N ) `  A ) )  - 
1 ) )
3029breq2d 4224 . . 3  |-  ( n  =  ( ( od
Z `  N ) `  A )  ->  ( N  ||  ( ( A ^ n )  - 
1 )  <->  N  ||  (
( A ^ (
( od Z `  N ) `  A
) )  -  1 ) ) )
3130elrab 3092 . 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 189 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709    C_ wss 3320   (/)c0 3628   class class class wbr 4212   `'ccnv 4877   ` cfv 5454  (class class class)co 6081   supcsup 7445   RRcr 8989   1c1 8991    < clt 9120    - cmin 9291   NNcn 10000   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488    mod cmo 11250   ^cexp 11382    || cdivides 12852    gcd cgcd 13006   od
Zcodz 13152   phicphi 13153
This theorem is referenced by:  odzcl  13179  odzid  13180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-odz 13154  df-phi 13155
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