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Theorem fermltl 13163
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 13072 . . . 4  |-  ( P  e.  Prime  ->  P  e.  NN )
2 dvdsval3 12846 . . . 4  |-  ( ( P  e.  NN  /\  A  e.  ZZ )  ->  ( P  ||  A  <->  ( A  mod  P )  =  0 ) )
31, 2sylan 458 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  A  <->  ( A  mod  P )  =  0 ) )
4 simp2 958 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  A  e.  ZZ )
5 0z 10283 . . . . . . 7  |-  0  e.  ZZ
65a1i 11 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  0  e.  ZZ )
713ad2ant1 978 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  P  e.  NN )
87nnnn0d 10264 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  P  e.  NN0 )
97nnrpd 10637 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  P  e.  RR+ )
10 simp3 959 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  ( A  mod  P )  =  0 )
11 0mod 11262 . . . . . . . 8  |-  ( P  e.  RR+  ->  ( 0  mod  P )  =  0 )
129, 11syl 16 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  (
0  mod  P )  =  0 )
1310, 12eqtr4d 2470 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  ( A  mod  P )  =  ( 0  mod  P
) )
14 modexp 11504 . . . . . 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 1195 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  (
( A ^ P
)  mod  P )  =  ( ( 0 ^ P )  mod 
P ) )
1670expd 11529 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  (
0 ^ P )  =  0 )
1716oveq1d 6088 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  (
( 0 ^ P
)  mod  P )  =  ( 0  mod 
P ) )
1813, 17eqtr4d 2470 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  ( A  mod  P )  =  ( ( 0 ^ P )  mod  P
) )
1915, 18eqtr4d 2470 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  mod  P )  =  0 )  ->  (
( A ^ P
)  mod  P )  =  ( A  mod  P ) )
20193expia 1155 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A  mod  P
)  =  0  -> 
( ( A ^ P )  mod  P
)  =  ( A  mod  P ) ) )
213, 20sylbid 207 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  A  ->  (
( A ^ P
)  mod  P )  =  ( A  mod  P ) ) )
22 coprm 13090 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
23 prmz 13073 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
24 gcdcom 13010 . . . . . 6  |-  ( ( P  e.  ZZ  /\  A  e.  ZZ )  ->  ( P  gcd  A
)  =  ( A  gcd  P ) )
2523, 24sylan 458 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  gcd  A )  =  ( A  gcd  P
) )
2625eqeq1d 2443 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( P  gcd  A
)  =  1  <->  ( A  gcd  P )  =  1 ) )
2722, 26bitrd 245 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( A  gcd  P )  =  1 ) )
28 simp2 958 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  A  e.  ZZ )
2913ad2ant1 978 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  P  e.  NN )
3029phicld 13151 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( phi `  P )  e.  NN )
3130nnnn0d 10264 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( phi `  P )  e. 
NN0 )
32 zexpcl 11386 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( phi `  P )  e.  NN0 )  -> 
( A ^ ( phi `  P ) )  e.  ZZ )
3328, 31, 32syl2anc 643 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( A ^ ( phi `  P ) )  e.  ZZ )
3433zred 10365 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( A ^ ( phi `  P ) )  e.  RR )
35 1re 9080 . . . . . . 7  |-  1  e.  RR
3635a1i 11 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  1  e.  RR )
3729nnrpd 10637 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  P  e.  RR+ )
38 eulerth 13162 . . . . . . 7  |-  ( ( P  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
391, 38syl3an1 1217 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
40 modmul1 11269 . . . . . 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 1195 . . . . 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 13156 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1 ) )
43423ad2ant1 978 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( phi `  P )  =  ( P  -  1 ) )
4443oveq2d 6089 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( A ^ ( phi `  P ) )  =  ( A ^ ( P  -  1 ) ) )
4544oveq1d 6088 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  x.  A )  =  ( ( A ^
( P  -  1 ) )  x.  A
) )
4628zcnd 10366 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  A  e.  CC )
47 expm1t 11398 . . . . . . . 8  |-  ( ( A  e.  CC  /\  P  e.  NN )  ->  ( A ^ P
)  =  ( ( A ^ ( P  -  1 ) )  x.  A ) )
4846, 29, 47syl2anc 643 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  ( A ^ P )  =  ( ( A ^
( P  -  1 ) )  x.  A
) )
4945, 48eqtr4d 2470 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  x.  A )  =  ( A ^ P
) )
5049oveq1d 6088 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( ( A ^
( phi `  P
) )  x.  A
)  mod  P )  =  ( ( A ^ P )  mod 
P ) )
5146mulid2d 9096 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
1  x.  A )  =  A )
5251oveq1d 6088 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( 1  x.  A
)  mod  P )  =  ( A  mod  P ) )
5341, 50, 523eqtr3d 2475 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ P
)  mod  P )  =  ( A  mod  P ) )
54533expia 1155 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A  gcd  P
)  =  1  -> 
( ( A ^ P )  mod  P
)  =  ( A  mod  P ) ) )
5527, 54sylbid 207 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  -> 
( ( A ^ P )  mod  P
)  =  ( A  mod  P ) ) )
5621, 55pm2.61d 152 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8978   RRcr 8979   0cc0 8980   1c1 8981    x. cmul 8985    - cmin 9281   NNcn 9990   NN0cn0 10211   ZZcz 10272   RR+crp 10602    mod cmo 11240   ^cexp 11372    || cdivides 12842    gcd cgcd 12996   Primecprime 13069   phicphi 13143
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7816  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-fz 11034  df-fzo 11126  df-fl 11192  df-mod 11241  df-seq 11314  df-exp 11373  df-hash 11609  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-dvds 12843  df-gcd 12997  df-prm 13070  df-phi 13145
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