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Theorem odzcllem 14403
Description: - Lemma for odzcl 14404, 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 )  ->  (
( ( odZ `  N ) `  A
)  e.  NN  /\  N  ||  ( ( A ^ ( ( odZ `  N ) `  A ) )  - 
1 ) ) )

Proof of Theorem odzcllem
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 odzval 14402 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( odZ `  N ) `  A
)  =  sup ( { n  e.  NN  |  N  ||  ( ( A ^ n )  -  1 ) } ,  RR ,  `'  <  ) )
2 ssrab2 3571 . . . . 5  |-  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  C_  NN
3 nnuz 11117 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
42, 3sseqtri 3521 . . . 4  |-  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  C_  ( ZZ>=
`  1 )
5 phicl 14383 . . . . . . 7  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
653ad2ant1 1015 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( phi `  N )  e.  NN )
7 eulerth 14397 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A ^ ( phi `  N ) )  mod  N )  =  ( 1  mod  N
) )
8 simp1 994 . . . . . . . 8  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  N  e.  NN )
9 simp2 995 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  A  e.  ZZ )
106nnnn0d 10848 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( phi `  N )  e. 
NN0 )
11 zexpcl 12163 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( phi `  N )  e.  NN0 )  -> 
( A ^ ( phi `  N ) )  e.  ZZ )
129, 10, 11syl2anc 659 . . . . . . . 8  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( A ^ ( phi `  N ) )  e.  ZZ )
13 1z 10890 . . . . . . . . 9  |-  1  e.  ZZ
14 moddvds 14077 . . . . . . . . 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 1311 . . . . . . . 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 659 . . . . . . 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 210 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  N  ||  ( ( A ^
( phi `  N
) )  -  1 ) )
18 oveq2 6278 . . . . . . . . 9  |-  ( n  =  ( phi `  N )  ->  ( A ^ n )  =  ( A ^ ( phi `  N ) ) )
1918oveq1d 6285 . . . . . . . 8  |-  ( n  =  ( phi `  N )  ->  (
( A ^ n
)  -  1 )  =  ( ( A ^ ( phi `  N ) )  - 
1 ) )
2019breq2d 4451 . . . . . . 7  |-  ( n  =  ( phi `  N )  ->  ( N  ||  ( ( A ^ n )  - 
1 )  <->  N  ||  (
( A ^ ( phi `  N ) )  -  1 ) ) )
2120rspcev 3207 . . . . . 6  |-  ( ( ( phi `  N
)  e.  NN  /\  N  ||  ( ( A ^ ( phi `  N ) )  - 
1 ) )  ->  E. n  e.  NN  N  ||  ( ( A ^ n )  - 
1 ) )
226, 17, 21syl2anc 659 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  E. n  e.  NN  N  ||  (
( A ^ n
)  -  1 ) )
23 rabn0 3804 . . . . 5  |-  ( { n  e.  NN  |  N  ||  ( ( A ^ n )  - 
1 ) }  =/=  (/)  <->  E. n  e.  NN  N  ||  ( ( A ^
n )  -  1 ) )
2422, 23sylibr 212 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  =/=  (/) )
25 infmssuzcl 11166 . . . 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 661 . . 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 2542 . 2  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( odZ `  N ) `  A
)  e.  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) } )
28 oveq2 6278 . . . . 5  |-  ( n  =  ( ( odZ `  N ) `  A )  ->  ( A ^ n )  =  ( A ^ (
( odZ `  N ) `  A
) ) )
2928oveq1d 6285 . . . 4  |-  ( n  =  ( ( odZ `  N ) `  A )  ->  (
( A ^ n
)  -  1 )  =  ( ( A ^ ( ( odZ `  N ) `  A ) )  - 
1 ) )
3029breq2d 4451 . . 3  |-  ( n  =  ( ( odZ `  N ) `  A )  ->  ( N  ||  ( ( A ^ n )  - 
1 )  <->  N  ||  (
( A ^ (
( odZ `  N ) `  A
) )  -  1 ) ) )
3130elrab 3254 . 2  |-  ( ( ( odZ `  N ) `  A
)  e.  { n  e.  NN  |  N  ||  ( ( A ^
n )  -  1 ) }  <->  ( (
( odZ `  N ) `  A
)  e.  NN  /\  N  ||  ( ( A ^ ( ( odZ `  N ) `  A ) )  - 
1 ) ) )
3227, 31sylib 196 1  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( ( odZ `  N ) `  A
)  e.  NN  /\  N  ||  ( ( A ^ ( ( odZ `  N ) `  A ) )  - 
1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   {crab 2808    C_ wss 3461   (/)c0 3783   class class class wbr 4439   `'ccnv 4987   ` cfv 5570  (class class class)co 6270   supcsup 7892   RRcr 9480   1c1 9482    < clt 9617    - cmin 9796   NNcn 10531   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082    mod cmo 11978   ^cexp 12148    || cdvds 14070    gcd cgcd 14228   odZcodz 14377   phicphi 14378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-dvds 14071  df-gcd 14229  df-odz 14379  df-phi 14380
This theorem is referenced by:  odzcl  14404  odzid  14405
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