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Theorem eulerth 13172
Description: Euler's theorem, a generalization of Fermat's little theorem. If  A and  N are coprime, then  A ^ phi ( N )  ==  1, mod  N. (Contributed by Mario Carneiro, 28-Feb-2014.)
Assertion
Ref Expression
eulerth  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A ^ ( phi `  N ) )  mod  N )  =  ( 1  mod  N
) )

Proof of Theorem eulerth
Dummy variables  f 
k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 phicl 13158 . . . . . . . 8  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
21nnnn0d 10274 . . . . . . 7  |-  ( N  e.  NN  ->  ( phi `  N )  e. 
NN0 )
3 hashfz1 11630 . . . . . . 7  |-  ( ( phi `  N )  e.  NN0  ->  ( # `  ( 1 ... ( phi `  N ) ) )  =  ( phi `  N ) )
42, 3syl 16 . . . . . 6  |-  ( N  e.  NN  ->  ( # `
 ( 1 ... ( phi `  N
) ) )  =  ( phi `  N
) )
5 dfphi2 13163 . . . . . 6  |-  ( N  e.  NN  ->  ( phi `  N )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } ) )
64, 5eqtrd 2468 . . . . 5  |-  ( N  e.  NN  ->  ( # `
 ( 1 ... ( phi `  N
) ) )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } ) )
763ad2ant1 978 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( # `
 ( 1 ... ( phi `  N
) ) )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } ) )
8 fzfi 11311 . . . . 5  |-  ( 1 ... ( phi `  N ) )  e. 
Fin
9 fzofi 11313 . . . . . 6  |-  ( 0..^ N )  e.  Fin
10 ssrab2 3428 . . . . . 6  |-  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  C_  (
0..^ N )
11 ssfi 7329 . . . . . 6  |-  ( ( ( 0..^ N )  e.  Fin  /\  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  C_  ( 0..^ N ) )  ->  { k  e.  ( 0..^ N )  |  ( k  gcd 
N )  =  1 }  e.  Fin )
129, 10, 11mp2an 654 . . . . 5  |-  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  e.  Fin
13 hashen 11631 . . . . 5  |-  ( ( ( 1 ... ( phi `  N ) )  e.  Fin  /\  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  e.  Fin )  ->  ( (
# `  ( 1 ... ( phi `  N
) ) )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } )  <-> 
( 1 ... ( phi `  N ) ) 
~~  { k  e.  ( 0..^ N )  |  ( k  gcd 
N )  =  1 } ) )
148, 12, 13mp2an 654 . . . 4  |-  ( (
# `  ( 1 ... ( phi `  N
) ) )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } )  <-> 
( 1 ... ( phi `  N ) ) 
~~  { k  e.  ( 0..^ N )  |  ( k  gcd 
N )  =  1 } )
157, 14sylib 189 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
1 ... ( phi `  N ) )  ~~  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } )
16 bren 7117 . . 3  |-  ( ( 1 ... ( phi `  N ) )  ~~  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  <->  E. f 
f : ( 1 ... ( phi `  N ) ) -1-1-onto-> { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } )
1715, 16sylib 189 . 2  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  E. f 
f : ( 1 ... ( phi `  N ) ) -1-1-onto-> { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } )
18 simpl 444 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  /\  f : ( 1 ... ( phi `  N
) ) -1-1-onto-> { k  e.  ( 0..^ N )  |  ( k  gcd  N
)  =  1 } )  ->  ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 ) )
19 oveq1 6088 . . . . 5  |-  ( k  =  y  ->  (
k  gcd  N )  =  ( y  gcd 
N ) )
2019eqeq1d 2444 . . . 4  |-  ( k  =  y  ->  (
( k  gcd  N
)  =  1  <->  (
y  gcd  N )  =  1 ) )
2120cbvrabv 2955 . . 3  |-  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  =  {
y  e.  ( 0..^ N )  |  ( y  gcd  N )  =  1 }
22 eqid 2436 . . 3  |-  ( 1 ... ( phi `  N ) )  =  ( 1 ... ( phi `  N ) )
23 simpr 448 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  /\  f : ( 1 ... ( phi `  N
) ) -1-1-onto-> { k  e.  ( 0..^ N )  |  ( k  gcd  N
)  =  1 } )  ->  f :
( 1 ... ( phi `  N ) ) -1-1-onto-> { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } )
24 fveq2 5728 . . . . . 6  |-  ( k  =  x  ->  (
f `  k )  =  ( f `  x ) )
2524oveq2d 6097 . . . . 5  |-  ( k  =  x  ->  ( A  x.  ( f `  k ) )  =  ( A  x.  (
f `  x )
) )
2625oveq1d 6096 . . . 4  |-  ( k  =  x  ->  (
( A  x.  (
f `  k )
)  mod  N )  =  ( ( A  x.  ( f `  x ) )  mod 
N ) )
2726cbvmptv 4300 . . 3  |-  ( k  e.  ( 1 ... ( phi `  N
) )  |->  ( ( A  x.  ( f `
 k ) )  mod  N ) )  =  ( x  e.  ( 1 ... ( phi `  N ) ) 
|->  ( ( A  x.  ( f `  x
) )  mod  N
) )
2818, 21, 22, 23, 27eulerthlem2 13171 . 2  |-  ( ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  /\  f : ( 1 ... ( phi `  N
) ) -1-1-onto-> { k  e.  ( 0..^ N )  |  ( k  gcd  N
)  =  1 } )  ->  ( ( A ^ ( phi `  N ) )  mod 
N )  =  ( 1  mod  N ) )
2917, 28exlimddv 1648 1  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A ^ ( phi `  N ) )  mod  N )  =  ( 1  mod  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {crab 2709    C_ wss 3320   class class class wbr 4212    e. cmpt 4266   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081    ~~ cen 7106   Fincfn 7109   0cc0 8990   1c1 8991    x. cmul 8995   NNcn 10000   NN0cn0 10221   ZZcz 10282   ...cfz 11043  ..^cfzo 11135    mod cmo 11250   ^cexp 11382   #chash 11618    gcd cgcd 13006   phicphi 13153
This theorem is referenced by:  fermltl  13173  prmdiv  13174  odzcllem  13178  odzphi  13182  lgslem1  21080  lgsqrlem2  21126
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-phi 13155
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