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Theorem eulerth 13863
Description: Euler's theorem, a generalization of Fermat's little theorem. If  A and  N are coprime, then  A ^ phi ( N )  ==  1, mod  N. This is Metamath 100 proof #10. (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 13849 . . . . . . . 8  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
21nnnn0d 10641 . . . . . . 7  |-  ( N  e.  NN  ->  ( phi `  N )  e. 
NN0 )
3 hashfz1 12122 . . . . . . 7  |-  ( ( phi `  N )  e.  NN0  ->  ( # `  ( 1 ... ( phi `  N ) ) )  =  ( phi `  N ) )
42, 3syl 16 . . . . . 6  |-  ( N  e.  NN  ->  ( # `
 ( 1 ... ( phi `  N
) ) )  =  ( phi `  N
) )
5 dfphi2 13854 . . . . . 6  |-  ( N  e.  NN  ->  ( phi `  N )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } ) )
64, 5eqtrd 2475 . . . . 5  |-  ( N  e.  NN  ->  ( # `
 ( 1 ... ( phi `  N
) ) )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } ) )
763ad2ant1 1009 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( # `
 ( 1 ... ( phi `  N
) ) )  =  ( # `  {
k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } ) )
8 fzfi 11799 . . . . 5  |-  ( 1 ... ( phi `  N ) )  e. 
Fin
9 fzofi 11801 . . . . . 6  |-  ( 0..^ N )  e.  Fin
10 ssrab2 3442 . . . . . 6  |-  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  C_  (
0..^ N )
11 ssfi 7538 . . . . . 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 672 . . . . 5  |-  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  e.  Fin
13 hashen 12123 . . . . 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 672 . . . 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 196 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
1 ... ( phi `  N ) )  ~~  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 } )
16 bren 7324 . . 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 196 . 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 457 . . 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 6103 . . . . 5  |-  ( k  =  y  ->  (
k  gcd  N )  =  ( y  gcd 
N ) )
2019eqeq1d 2451 . . . 4  |-  ( k  =  y  ->  (
( k  gcd  N
)  =  1  <->  (
y  gcd  N )  =  1 ) )
2120cbvrabv 2976 . . 3  |-  { k  e.  ( 0..^ N )  |  ( k  gcd  N )  =  1 }  =  {
y  e.  ( 0..^ N )  |  ( y  gcd  N )  =  1 }
22 eqid 2443 . . 3  |-  ( 1 ... ( phi `  N ) )  =  ( 1 ... ( phi `  N ) )
23 simpr 461 . . 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 5696 . . . . . 6  |-  ( k  =  x  ->  (
f `  k )  =  ( f `  x ) )
2524oveq2d 6112 . . . . 5  |-  ( k  =  x  ->  ( A  x.  ( f `  k ) )  =  ( A  x.  (
f `  x )
) )
2625oveq1d 6111 . . . 4  |-  ( k  =  x  ->  (
( A  x.  (
f `  k )
)  mod  N )  =  ( ( A  x.  ( f `  x ) )  mod 
N ) )
2726cbvmptv 4388 . . 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 13862 . 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 1692 1  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A ^ ( phi `  N ) )  mod  N )  =  ( 1  mod  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {crab 2724    C_ wss 3333   class class class wbr 4297    e. cmpt 4355   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096    ~~ cen 7312   Fincfn 7315   0cc0 9287   1c1 9288    x. cmul 9292   NNcn 10327   NN0cn0 10584   ZZcz 10651   ...cfz 11442  ..^cfzo 11553    mod cmo 11713   ^cexp 11870   #chash 12108    gcd cgcd 13695   phicphi 13844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-dvds 13541  df-gcd 13696  df-phi 13846
This theorem is referenced by:  fermltl  13864  prmdiv  13865  odzcllem  13869  odzphi  13873  lgslem1  22640  lgsqrlem2  22686
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