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Theorem wilth 22414
Description: Wilson's theorem. A number is prime iff it is greater or equal to  2 and  ( N  - 
1 ) ! is congruent to  -u 1,  mod  N, or alternatively if  N divides  ( N  - 
1 ) !  + 
1. In this part of the proof we show the relatively simple reverse implication; see wilthlem3 22413 for the forward implication. This is Metamath 100 proof #51. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
Assertion
Ref Expression
wilth  |-  ( N  e.  Prime  <->  ( N  e.  ( ZZ>= `  2 )  /\  N  ||  ( ( ! `  ( N  -  1 ) )  +  1 ) ) )

Proof of Theorem wilth
Dummy variables  x  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prmuz2 13786 . . 3  |-  ( N  e.  Prime  ->  N  e.  ( ZZ>= `  2 )
)
2 eqid 2443 . . . 4  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
3 eleq2 2504 . . . . . 6  |-  ( z  =  x  ->  (
( N  -  1 )  e.  z  <->  ( N  -  1 )  e.  x ) )
4 oveq1 6103 . . . . . . . . . 10  |-  ( n  =  y  ->  (
n ^ ( N  -  2 ) )  =  ( y ^
( N  -  2 ) ) )
54oveq1d 6111 . . . . . . . . 9  |-  ( n  =  y  ->  (
( n ^ ( N  -  2 ) )  mod  N )  =  ( ( y ^ ( N  - 
2 ) )  mod 
N ) )
65eleq1d 2509 . . . . . . . 8  |-  ( n  =  y  ->  (
( ( n ^
( N  -  2 ) )  mod  N
)  e.  z  <->  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  z ) )
76cbvralv 2952 . . . . . . 7  |-  ( A. n  e.  z  (
( n ^ ( N  -  2 ) )  mod  N )  e.  z  <->  A. y  e.  z  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  z )
8 eleq2 2504 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( y ^
( N  -  2 ) )  mod  N
)  e.  z  <->  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  x ) )
98raleqbi1dv 2930 . . . . . . 7  |-  ( z  =  x  ->  ( A. y  e.  z 
( ( y ^
( N  -  2 ) )  mod  N
)  e.  z  <->  A. y  e.  x  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  x ) )
107, 9syl5bb 257 . . . . . 6  |-  ( z  =  x  ->  ( A. n  e.  z 
( ( n ^
( N  -  2 ) )  mod  N
)  e.  z  <->  A. y  e.  x  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  x ) )
113, 10anbi12d 710 . . . . 5  |-  ( z  =  x  ->  (
( ( N  - 
1 )  e.  z  /\  A. n  e.  z  ( ( n ^ ( N  - 
2 ) )  mod 
N )  e.  z )  <->  ( ( N  -  1 )  e.  x  /\  A. y  e.  x  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  x ) ) )
1211cbvrabv 2976 . . . 4  |-  { z  e.  ~P ( 1 ... ( N  - 
1 ) )  |  ( ( N  - 
1 )  e.  z  /\  A. n  e.  z  ( ( n ^ ( N  - 
2 ) )  mod 
N )  e.  z ) }  =  {
x  e.  ~P (
1 ... ( N  - 
1 ) )  |  ( ( N  - 
1 )  e.  x  /\  A. y  e.  x  ( ( y ^
( N  -  2 ) )  mod  N
)  e.  x ) }
132, 12wilthlem3 22413 . . 3  |-  ( N  e.  Prime  ->  N  ||  ( ( ! `  ( N  -  1
) )  +  1 ) )
141, 13jca 532 . 2  |-  ( N  e.  Prime  ->  ( N  e.  ( ZZ>= `  2
)  /\  N  ||  (
( ! `  ( N  -  1 ) )  +  1 ) ) )
15 simpl 457 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  ||  ( ( ! `  ( N  -  1
) )  +  1 ) )  ->  N  e.  ( ZZ>= `  2 )
)
16 elfzuz 11454 . . . . . . . . 9  |-  ( n  e.  ( 2 ... ( N  -  1 ) )  ->  n  e.  ( ZZ>= `  2 )
)
1716adantl 466 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  n  e.  (
ZZ>= `  2 ) )
18 eluz2b2 10932 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  2
)  <->  ( n  e.  NN  /\  1  < 
n ) )
1918simplbi 460 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  2
)  ->  n  e.  NN )
2017, 19syl 16 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  n  e.  NN )
21 elfzuz3 11455 . . . . . . . 8  |-  ( n  e.  ( 2 ... ( N  -  1 ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  n
) )
2221adantl 466 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( N  - 
1 )  e.  (
ZZ>= `  n ) )
23 dvdsfac 13593 . . . . . . 7  |-  ( ( n  e.  NN  /\  ( N  -  1
)  e.  ( ZZ>= `  n ) )  ->  n  ||  ( ! `  ( N  -  1
) ) )
2420, 22, 23syl2anc 661 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  n  ||  ( ! `  ( N  -  1 ) ) )
25 eluz2b2 10932 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  1  < 
N ) )
2625simplbi 460 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
2726ad2antrr 725 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  N  e.  NN )
28 nnm1nn0 10626 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
29 faccl 12066 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  NN0  ->  ( ! `
 ( N  - 
1 ) )  e.  NN )
3027, 28, 293syl 20 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( ! `  ( N  -  1
) )  e.  NN )
3130nnzd 10751 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( ! `  ( N  -  1
) )  e.  ZZ )
3218simprbi 464 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  2
)  ->  1  <  n )
3317, 32syl 16 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  1  <  n
)
34 ndvdsp1 13618 . . . . . . 7  |-  ( ( ( ! `  ( N  -  1 ) )  e.  ZZ  /\  n  e.  NN  /\  1  <  n )  ->  (
n  ||  ( ! `  ( N  -  1 ) )  ->  -.  n  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) ) )
3531, 20, 33, 34syl3anc 1218 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( n  ||  ( ! `  ( N  -  1 ) )  ->  -.  n  ||  (
( ! `  ( N  -  1 ) )  +  1 ) ) )
3624, 35mpd 15 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  -.  n  ||  (
( ! `  ( N  -  1 ) )  +  1 ) )
37 simplr 754 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  N  ||  (
( ! `  ( N  -  1 ) )  +  1 ) )
3820nnzd 10751 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  n  e.  ZZ )
3927nnzd 10751 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  N  e.  ZZ )
4031peano2zd 10755 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( ( ! `
 ( N  - 
1 ) )  +  1 )  e.  ZZ )
41 dvdstr 13571 . . . . . . 7  |-  ( ( n  e.  ZZ  /\  N  e.  ZZ  /\  (
( ! `  ( N  -  1 ) )  +  1 )  e.  ZZ )  -> 
( ( n  ||  N  /\  N  ||  (
( ! `  ( N  -  1 ) )  +  1 ) )  ->  n  ||  (
( ! `  ( N  -  1 ) )  +  1 ) ) )
4238, 39, 40, 41syl3anc 1218 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( ( n 
||  N  /\  N  ||  ( ( ! `  ( N  -  1
) )  +  1 ) )  ->  n  ||  ( ( ! `  ( N  -  1
) )  +  1 ) ) )
4337, 42mpan2d 674 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( n  ||  N  ->  n  ||  (
( ! `  ( N  -  1 ) )  +  1 ) ) )
4436, 43mtod 177 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  -.  n  ||  N
)
4544ralrimiva 2804 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  ||  ( ( ! `  ( N  -  1
) )  +  1 ) )  ->  A. n  e.  ( 2 ... ( N  -  1 ) )  -.  n  ||  N )
46 isprm3 13777 . . 3  |-  ( N  e.  Prime  <->  ( N  e.  ( ZZ>= `  2 )  /\  A. n  e.  ( 2 ... ( N  -  1 ) )  -.  n  ||  N
) )
4715, 45, 46sylanbrc 664 . 2  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  ||  ( ( ! `  ( N  -  1
) )  +  1 ) )  ->  N  e.  Prime )
4814, 47impbii 188 1  |-  ( N  e.  Prime  <->  ( N  e.  ( ZZ>= `  2 )  /\  N  ||  ( ( ! `  ( N  -  1 ) )  +  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756   A.wral 2720   {crab 2724   ~Pcpw 3865   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   1c1 9288    + caddc 9290    < clt 9423    - cmin 9600   NNcn 10327   2c2 10376   NN0cn0 10584   ZZcz 10651   ZZ>=cuz 10866   ...cfz 11442    mod cmo 11713   ^cexp 11870   !cfa 12056    || cdivides 13540   Primecprime 13768  mulGrpcmgp 16596  ℂfldccnfld 17823
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-inf2 7852  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  ax-addf 9366  ax-mulf 9367
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-iin 4179  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-se 4685  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-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  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-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  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-fac 12057  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-prm 13769  df-phi 13846  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-0g 14385  df-gsum 14386  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-mulg 15553  df-subg 15683  df-cntz 15840  df-cmn 16284  df-mgp 16597  df-ur 16609  df-rng 16652  df-cring 16653  df-subrg 16868  df-cnfld 17824
This theorem is referenced by: (None)
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