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Theorem fermltl 12868
Description: Fermat's little theorem. When  P is prime,  A ^ P  ==  A, mod  P for any  A. (Contributed by Mario Carneiro, 28-Feb-2014.)
Assertion
Ref Expression
fermltl  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A ^ P
)  mod  P )  =  ( A  mod  P ) )

Proof of Theorem fermltl
StepHypRef Expression
1 prmnn 12777 . . . 4  |-  ( P  e.  Prime  ->  P  e.  NN )
2 dvdsval3 12551 . . . 4  |-  ( ( P  e.  NN  /\  A  e.  ZZ )  ->  ( P  ||  A  <->  ( A  mod  P )  =  0 ) )
31, 2sylan 457 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  A  <->  ( A  mod  P )  =  0 ) )
4 simp2 956 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  A  e.  ZZ )
5 0z 10051 . . . . . . 7  |-  0  e.  ZZ
65a1i 10 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  0  e.  ZZ )
713ad2ant1 976 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  P  e.  NN )
87nnnn0d 10034 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  P  e.  NN0 )
97nnrpd 10405 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  P  e.  RR+ )
10 simp3 957 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  ( A  mod  P )  =  0 )
11 0mod 11011 . . . . . . . 8  |-  ( P  e.  RR+  ->  ( 0  mod  P )  =  0 )
129, 11syl 15 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  (
0  mod  P )  =  0 )
1310, 12eqtr4d 2331 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  ( A  mod  P )  =  ( 0  mod  P
) )
14 modexp 11252 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  0  e.  ZZ )  /\  ( P  e. 
NN0  /\  P  e.  RR+ )  /\  ( A  mod  P )  =  ( 0  mod  P
) )  ->  (
( A ^ P
)  mod  P )  =  ( ( 0 ^ P )  mod 
P ) )
154, 6, 8, 9, 13, 14syl221anc 1193 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  (
( A ^ P
)  mod  P )  =  ( ( 0 ^ P )  mod 
P ) )
1670expd 11277 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  (
0 ^ P )  =  0 )
1716oveq1d 5889 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  (
( 0 ^ P
)  mod  P )  =  ( 0  mod 
P ) )
1813, 17eqtr4d 2331 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  ( A  mod  P )  =  ( ( 0 ^ P )  mod  P
) )
1915, 18eqtr4d 2331 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  (
( A ^ P
)  mod  P )  =  ( A  mod  P ) )
20193expia 1153 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A  mod  P
)  =  0  -> 
( ( A ^ P )  mod  P
)  =  ( A  mod  P ) ) )
213, 20sylbid 206 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  A  ->  (
( A ^ P
)  mod  P )  =  ( A  mod  P ) ) )
22 coprm 12795 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
23 prmz 12778 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
24 gcdcom 12715 . . . . . 6  |-  ( ( P  e.  ZZ  /\  A  e.  ZZ )  ->  ( P  gcd  A
)  =  ( A  gcd  P ) )
2523, 24sylan 457 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  gcd  A )  =  ( A  gcd  P
) )
2625eqeq1d 2304 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( P  gcd  A
)  =  1  <->  ( A  gcd  P )  =  1 ) )
2722, 26bitrd 244 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( A  gcd  P )  =  1 ) )
28 simp2 956 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  A  e.  ZZ )
2913ad2ant1 976 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  P  e.  NN )
3029phicld 12856 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( phi `  P )  e.  NN )
3130nnnn0d 10034 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( phi `  P )  e. 
NN0 )
32 zexpcl 11134 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( phi `  P )  e.  NN0 )  -> 
( A ^ ( phi `  P ) )  e.  ZZ )
3328, 31, 32syl2anc 642 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( A ^ ( phi `  P ) )  e.  ZZ )
3433zred 10133 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( A ^ ( phi `  P ) )  e.  RR )
35 1re 8853 . . . . . . 7  |-  1  e.  RR
3635a1i 10 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  1  e.  RR )
3729nnrpd 10405 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  P  e.  RR+ )
38 eulerth 12867 . . . . . . 7  |-  ( ( P  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
391, 38syl3an1 1215 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
40 modmul1 11018 . . . . . 6  |-  ( ( ( ( A ^
( phi `  P
) )  e.  RR  /\  1  e.  RR )  /\  ( A  e.  ZZ  /\  P  e.  RR+ )  /\  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )  ->  (
( ( A ^
( phi `  P
) )  x.  A
)  mod  P )  =  ( ( 1  x.  A )  mod 
P ) )
4134, 36, 28, 37, 39, 40syl221anc 1193 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( ( A ^
( phi `  P
) )  x.  A
)  mod  P )  =  ( ( 1  x.  A )  mod 
P ) )
42 phiprm 12861 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1 ) )
43423ad2ant1 976 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( phi `  P )  =  ( P  -  1 ) )
4443oveq2d 5890 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( A ^ ( phi `  P ) )  =  ( A ^ ( P  -  1 ) ) )
4544oveq1d 5889 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  x.  A )  =  ( ( A ^
( P  -  1 ) )  x.  A
) )
4628zcnd 10134 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  A  e.  CC )
47 expm1t 11146 . . . . . . . 8  |-  ( ( A  e.  CC  /\  P  e.  NN )  ->  ( A ^ P
)  =  ( ( A ^ ( P  -  1 ) )  x.  A ) )
4846, 29, 47syl2anc 642 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( A ^ P )  =  ( ( A ^
( P  -  1 ) )  x.  A
) )
4945, 48eqtr4d 2331 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  x.  A )  =  ( A ^ P
) )
5049oveq1d 5889 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( ( A ^
( phi `  P
) )  x.  A
)  mod  P )  =  ( ( A ^ P )  mod 
P ) )
5146mulid2d 8869 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
1  x.  A )  =  A )
5251oveq1d 5889 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( 1  x.  A
)  mod  P )  =  ( A  mod  P ) )
5341, 50, 523eqtr3d 2336 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ P
)  mod  P )  =  ( A  mod  P ) )
54533expia 1153 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A  gcd  P
)  =  1  -> 
( ( A ^ P )  mod  P
)  =  ( A  mod  P ) ) )
5527, 54sylbid 206 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  -> 
( ( A ^ P )  mod  P
)  =  ( A  mod  P ) ) )
5621, 55pm2.61d 150 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A ^ P
)  mod  P )  =  ( A  mod  P ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   RR+crp 10370    mod cmo 10989   ^cexp 11120    || cdivides 12547    gcd cgcd 12701   Primecprime 12774   phicphi 12848
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-2o 6496  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-cda 7810  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-prm 12775  df-phi 12850
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