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Theorem odzcllem 12873
Description: - Lemma for odzcl 12874, 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 12872 . . 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 3271 . . . . 5  |-  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  C_  NN
3 nnuz 10279 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
42, 3sseqtri 3223 . . . 4  |-  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  C_  ( ZZ>=
`  1 )
5 phicl 12853 . . . . . . 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 12867 . . . . . . 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 10034 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( phi `  N )  e. 
NN0 )
11 zexpcl 11134 . . . . . . . . 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 10069 . . . . . . . . 9  |-  1  e.  ZZ
14 moddvds 12554 . . . . . . . . 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 5882 . . . . . . . . 9  |-  ( n  =  ( phi `  N )  ->  ( A ^ n )  =  ( A ^ ( phi `  N ) ) )
1918oveq1d 5889 . . . . . . . 8  |-  ( n  =  ( phi `  N )  ->  (
( A ^ n
)  -  1 )  =  ( ( A ^ ( phi `  N ) )  - 
1 ) )
2019breq2d 4051 . . . . . . 7  |-  ( n  =  ( phi `  N )  ->  ( N  ||  ( ( A ^ n )  - 
1 )  <->  N  ||  (
( A ^ ( phi `  N ) )  -  1 ) ) )
2120rspcev 2897 . . . . . 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 3487 . . . . 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 10317 . . . 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 2370 . 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 5882 . . . . 5  |-  ( n  =  ( ( od
Z `  N ) `  A )  ->  ( A ^ n )  =  ( A ^ (
( od Z `  N ) `  A
) ) )
2928oveq1d 5889 . . . 4  |-  ( n  =  ( ( od
Z `  N ) `  A )  ->  (
( A ^ n
)  -  1 )  =  ( ( A ^ ( ( od
Z `  N ) `  A ) )  - 
1 ) )
3029breq2d 4051 . . 3  |-  ( n  =  ( ( od
Z `  N ) `  A )  ->  ( N  ||  ( ( A ^ n )  - 
1 )  <->  N  ||  (
( A ^ (
( od Z `  N ) `  A
) )  -  1 ) ) )
3130elrab 2936 . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   class class class wbr 4039   `'ccnv 4704   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   1c1 8754    < clt 8883    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    mod cmo 10989   ^cexp 11120    || cdivides 12547    gcd cgcd 12701   od
Zcodz 12847   phicphi 12848
This theorem is referenced by:  odzcl  12874  odzid  12875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-odz 12849  df-phi 12850
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