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Theorem eulerth 12867
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 12853 . . . . . . . 8  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
21nnnn0d 10034 . . . . . . 7  |-  ( N  e.  NN  ->  ( phi `  N )  e. 
NN0 )
3 hashfz1 11361 . . . . . . 7  |-  ( ( phi `  N )  e.  NN0  ->  ( # `  ( 1 ... ( phi `  N ) ) )  =  ( phi `  N ) )
42, 3syl 15 . . . . . 6  |-  ( N  e.  NN  ->  ( # `
 ( 1 ... ( phi `  N
) ) )  =  ( phi `  N
) )
5 dfphi2 12858 . . . . . 6  |-  ( N  e.  NN  ->  ( phi `  N )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } ) )
64, 5eqtrd 2328 . . . . 5  |-  ( N  e.  NN  ->  ( # `
 ( 1 ... ( phi `  N
) ) )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } ) )
763ad2ant1 976 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( # `
 ( 1 ... ( phi `  N
) ) )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } ) )
8 fzfi 11050 . . . . 5  |-  ( 1 ... ( phi `  N ) )  e. 
Fin
9 fzofi 11052 . . . . . 6  |-  ( 0..^ N )  e.  Fin
10 ssrab2 3271 . . . . . 6  |-  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  C_  (
0..^ N )
11 ssfi 7099 . . . . . 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 653 . . . . 5  |-  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  e.  Fin
13 hashen 11362 . . . . 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 653 . . . 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 188 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
1 ... ( phi `  N ) )  ~~  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } )
16 bren 6887 . . 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 188 . 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 443 . . . . 5  |-  ( ( ( 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 5881 . . . . . . 7  |-  ( k  =  y  ->  (
k  gcd  N )  =  ( y  gcd 
N ) )
2019eqeq1d 2304 . . . . . 6  |-  ( k  =  y  ->  (
( k  gcd  N
)  =  1  <->  (
y  gcd  N )  =  1 ) )
2120cbvrabv 2800 . . . . 5  |-  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  =  {
y  e.  ( 0..^ N )  |  ( y  gcd  N )  =  1 }
22 eqid 2296 . . . . 5  |-  ( 1 ... ( phi `  N ) )  =  ( 1 ... ( phi `  N ) )
23 simpr 447 . . . . 5  |-  ( ( ( 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 5541 . . . . . . . 8  |-  ( k  =  x  ->  (
f `  k )  =  ( f `  x ) )
2524oveq2d 5890 . . . . . . 7  |-  ( k  =  x  ->  ( A  x.  ( f `  k ) )  =  ( A  x.  (
f `  x )
) )
2625oveq1d 5889 . . . . . 6  |-  ( k  =  x  ->  (
( A  x.  (
f `  k )
)  mod  N )  =  ( ( A  x.  ( f `  x ) )  mod 
N ) )
2726cbvmptv 4127 . . . . 5  |-  ( 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 12866 . . . 4  |-  ( ( ( 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 ) )
2928ex 423 . . 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 }  ->  (
( A ^ ( phi `  N ) )  mod  N )  =  ( 1  mod  N
) ) )
3029exlimdv 1626 . 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 }  ->  ( ( A ^ ( phi `  N ) )  mod  N )  =  ( 1  mod  N
) ) )
3117, 30mpd 14 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 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    ~~ cen 6876   Fincfn 6879   0cc0 8753   1c1 8754    x. cmul 8758   NNcn 9762   NN0cn0 9981   ZZcz 10040   ...cfz 10798  ..^cfzo 10886    mod cmo 10989   ^cexp 11120   #chash 11353    gcd cgcd 12701   phicphi 12848
This theorem is referenced by:  fermltl  12868  prmdiv  12869  odzcllem  12873  odzphi  12877  lgslem1  20551  lgsqrlem2  20597
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-phi 12850
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