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Theorem pockthi 14273
Description: Pocklington's theorem, which gives a sufficient criterion for a number  N to be prime. This is the preferred method for verifying large primes, being much more efficient to compute than trial division. This form has been optimized for application to specific large primes; see pockthg 14272 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.)
Hypotheses
Ref Expression
pockthi.p  |-  P  e. 
Prime
pockthi.g  |-  G  e.  NN
pockthi.m  |-  M  =  ( G  x.  P
)
pockthi.n  |-  N  =  ( M  +  1 )
pockthi.d  |-  D  e.  NN
pockthi.e  |-  E  e.  NN
pockthi.a  |-  A  e.  NN
pockthi.fac  |-  M  =  ( D  x.  ( P ^ E ) )
pockthi.gt  |-  D  < 
( P ^ E
)
pockthi.mod  |-  ( ( A ^ M )  mod  N )  =  ( 1  mod  N
)
pockthi.gcd  |-  ( ( ( A ^ G
)  -  1 )  gcd  N )  =  1
Assertion
Ref Expression
pockthi  |-  N  e. 
Prime

Proof of Theorem pockthi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pockthi.d . 2  |-  D  e.  NN
2 pockthi.p . . . . . 6  |-  P  e. 
Prime
3 prmnn 14068 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
42, 3ax-mp 5 . . . . 5  |-  P  e.  NN
5 pockthi.e . . . . . 6  |-  E  e.  NN
65nnnn0i 10792 . . . . 5  |-  E  e. 
NN0
7 nnexpcl 12135 . . . . 5  |-  ( ( P  e.  NN  /\  E  e.  NN0 )  -> 
( P ^ E
)  e.  NN )
84, 6, 7mp2an 672 . . . 4  |-  ( P ^ E )  e.  NN
98a1i 11 . . 3  |-  ( D  e.  NN  ->  ( P ^ E )  e.  NN )
10 id 22 . . 3  |-  ( D  e.  NN  ->  D  e.  NN )
11 pockthi.gt . . . 4  |-  D  < 
( P ^ E
)
1211a1i 11 . . 3  |-  ( D  e.  NN  ->  D  <  ( P ^ E
) )
13 pockthi.n . . . . 5  |-  N  =  ( M  +  1 )
14 pockthi.fac . . . . . . 7  |-  M  =  ( D  x.  ( P ^ E ) )
151nncni 10535 . . . . . . . 8  |-  D  e.  CC
168nncni 10535 . . . . . . . 8  |-  ( P ^ E )  e.  CC
1715, 16mulcomi 9591 . . . . . . 7  |-  ( D  x.  ( P ^ E ) )  =  ( ( P ^ E )  x.  D
)
1814, 17eqtri 2489 . . . . . 6  |-  M  =  ( ( P ^ E )  x.  D
)
1918oveq1i 6285 . . . . 5  |-  ( M  +  1 )  =  ( ( ( P ^ E )  x.  D )  +  1 )
2013, 19eqtri 2489 . . . 4  |-  N  =  ( ( ( P ^ E )  x.  D )  +  1 )
2120a1i 11 . . 3  |-  ( D  e.  NN  ->  N  =  ( ( ( P ^ E )  x.  D )  +  1 ) )
22 prmdvdsexpb 14104 . . . . . . 7  |-  ( ( x  e.  Prime  /\  P  e.  Prime  /\  E  e.  NN )  ->  ( x 
||  ( P ^ E )  <->  x  =  P ) )
232, 5, 22mp3an23 1311 . . . . . 6  |-  ( x  e.  Prime  ->  ( x 
||  ( P ^ E )  <->  x  =  P ) )
2413eqcomi 2473 . . . . . . . . . . 11  |-  ( M  +  1 )  =  N
25 pockthi.m . . . . . . . . . . . . . . . 16  |-  M  =  ( G  x.  P
)
26 pockthi.g . . . . . . . . . . . . . . . . 17  |-  G  e.  NN
2726, 4nnmulcli 10549 . . . . . . . . . . . . . . . 16  |-  ( G  x.  P )  e.  NN
2825, 27eqeltri 2544 . . . . . . . . . . . . . . 15  |-  M  e.  NN
29 peano2nn 10537 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  ( M  +  1 )  e.  NN )
3028, 29ax-mp 5 . . . . . . . . . . . . . 14  |-  ( M  +  1 )  e.  NN
3113, 30eqeltri 2544 . . . . . . . . . . . . 13  |-  N  e.  NN
3231nncni 10535 . . . . . . . . . . . 12  |-  N  e.  CC
33 ax-1cn 9539 . . . . . . . . . . . 12  |-  1  e.  CC
3428nncni 10535 . . . . . . . . . . . 12  |-  M  e.  CC
3532, 33, 34subadd2i 9896 . . . . . . . . . . 11  |-  ( ( N  -  1 )  =  M  <->  ( M  +  1 )  =  N )
3624, 35mpbir 209 . . . . . . . . . 10  |-  ( N  -  1 )  =  M
3736oveq2i 6286 . . . . . . . . 9  |-  ( A ^ ( N  - 
1 ) )  =  ( A ^ M
)
3837oveq1i 6285 . . . . . . . 8  |-  ( ( A ^ ( N  -  1 ) )  mod  N )  =  ( ( A ^ M )  mod  N
)
39 pockthi.mod . . . . . . . . 9  |-  ( ( A ^ M )  mod  N )  =  ( 1  mod  N
)
4031nnrei 10534 . . . . . . . . . 10  |-  N  e.  RR
4128nngt0i 10558 . . . . . . . . . . . 12  |-  0  <  M
4228nnrei 10534 . . . . . . . . . . . . 13  |-  M  e.  RR
43 1re 9584 . . . . . . . . . . . . 13  |-  1  e.  RR
44 ltaddpos2 10032 . . . . . . . . . . . . 13  |-  ( ( M  e.  RR  /\  1  e.  RR )  ->  ( 0  <  M  <->  1  <  ( M  + 
1 ) ) )
4542, 43, 44mp2an 672 . . . . . . . . . . . 12  |-  ( 0  <  M  <->  1  <  ( M  +  1 ) )
4641, 45mpbi 208 . . . . . . . . . . 11  |-  1  <  ( M  +  1 )
4746, 13breqtrri 4465 . . . . . . . . . 10  |-  1  <  N
48 1mod 11984 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  1  <  N )  -> 
( 1  mod  N
)  =  1 )
4940, 47, 48mp2an 672 . . . . . . . . 9  |-  ( 1  mod  N )  =  1
5039, 49eqtri 2489 . . . . . . . 8  |-  ( ( A ^ M )  mod  N )  =  1
5138, 50eqtri 2489 . . . . . . 7  |-  ( ( A ^ ( N  -  1 ) )  mod  N )  =  1
52 oveq2 6283 . . . . . . . . . . . 12  |-  ( x  =  P  ->  (
( N  -  1 )  /  x )  =  ( ( N  -  1 )  /  P ) )
5326nncni 10535 . . . . . . . . . . . . . . 15  |-  G  e.  CC
544nncni 10535 . . . . . . . . . . . . . . 15  |-  P  e.  CC
5553, 54mulcomi 9591 . . . . . . . . . . . . . 14  |-  ( G  x.  P )  =  ( P  x.  G
)
5636, 25, 553eqtrri 2494 . . . . . . . . . . . . 13  |-  ( P  x.  G )  =  ( N  -  1 )
5732, 33subcli 9884 . . . . . . . . . . . . . 14  |-  ( N  -  1 )  e.  CC
584nnne0i 10559 . . . . . . . . . . . . . 14  |-  P  =/=  0
5957, 54, 53, 58divmuli 10287 . . . . . . . . . . . . 13  |-  ( ( ( N  -  1 )  /  P )  =  G  <->  ( P  x.  G )  =  ( N  -  1 ) )
6056, 59mpbir 209 . . . . . . . . . . . 12  |-  ( ( N  -  1 )  /  P )  =  G
6152, 60syl6eq 2517 . . . . . . . . . . 11  |-  ( x  =  P  ->  (
( N  -  1 )  /  x )  =  G )
6261oveq2d 6291 . . . . . . . . . 10  |-  ( x  =  P  ->  ( A ^ ( ( N  -  1 )  /  x ) )  =  ( A ^ G
) )
6362oveq1d 6290 . . . . . . . . 9  |-  ( x  =  P  ->  (
( A ^ (
( N  -  1 )  /  x ) )  -  1 )  =  ( ( A ^ G )  - 
1 ) )
6463oveq1d 6290 . . . . . . . 8  |-  ( x  =  P  ->  (
( ( A ^
( ( N  - 
1 )  /  x
) )  -  1 )  gcd  N )  =  ( ( ( A ^ G )  -  1 )  gcd 
N ) )
65 pockthi.gcd . . . . . . . 8  |-  ( ( ( A ^ G
)  -  1 )  gcd  N )  =  1
6664, 65syl6eq 2517 . . . . . . 7  |-  ( x  =  P  ->  (
( ( A ^
( ( N  - 
1 )  /  x
) )  -  1 )  gcd  N )  =  1 )
67 pockthi.a . . . . . . . . 9  |-  A  e.  NN
6867nnzi 10877 . . . . . . . 8  |-  A  e.  ZZ
69 oveq1 6282 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
y ^ ( N  -  1 ) )  =  ( A ^
( N  -  1 ) ) )
7069oveq1d 6290 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
( y ^ ( N  -  1 ) )  mod  N )  =  ( ( A ^ ( N  - 
1 ) )  mod 
N ) )
7170eqeq1d 2462 . . . . . . . . . 10  |-  ( y  =  A  ->  (
( ( y ^
( N  -  1 ) )  mod  N
)  =  1  <->  (
( A ^ ( N  -  1 ) )  mod  N )  =  1 ) )
72 oveq1 6282 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  (
y ^ ( ( N  -  1 )  /  x ) )  =  ( A ^
( ( N  - 
1 )  /  x
) ) )
7372oveq1d 6290 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
( y ^ (
( N  -  1 )  /  x ) )  -  1 )  =  ( ( A ^ ( ( N  -  1 )  /  x ) )  - 
1 ) )
7473oveq1d 6290 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
( ( y ^
( ( N  - 
1 )  /  x
) )  -  1 )  gcd  N )  =  ( ( ( A ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N ) )
7574eqeq1d 2462 . . . . . . . . . 10  |-  ( y  =  A  ->  (
( ( ( y ^ ( ( N  -  1 )  /  x ) )  - 
1 )  gcd  N
)  =  1  <->  (
( ( A ^
( ( N  - 
1 )  /  x
) )  -  1 )  gcd  N )  =  1 ) )
7671, 75anbi12d 710 . . . . . . . . 9  |-  ( y  =  A  ->  (
( ( ( y ^ ( N  - 
1 ) )  mod 
N )  =  1  /\  ( ( ( y ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N )  =  1 )  <->  ( ( ( A ^ ( N  -  1 ) )  mod  N )  =  1  /\  ( ( ( A ^ (
( N  -  1 )  /  x ) )  -  1 )  gcd  N )  =  1 ) ) )
7776rspcev 3207 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( ( ( A ^ ( N  - 
1 ) )  mod 
N )  =  1  /\  ( ( ( A ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N )  =  1 ) )  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  -  1 ) )  mod  N )  =  1  /\  ( ( ( y ^ (
( N  -  1 )  /  x ) )  -  1 )  gcd  N )  =  1 ) )
7868, 77mpan 670 . . . . . . 7  |-  ( ( ( ( A ^
( N  -  1 ) )  mod  N
)  =  1  /\  ( ( ( A ^ ( ( N  -  1 )  /  x ) )  - 
1 )  gcd  N
)  =  1 )  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  - 
1 ) )  mod 
N )  =  1  /\  ( ( ( y ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N )  =  1 ) )
7951, 66, 78sylancr 663 . . . . . 6  |-  ( x  =  P  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  -  1 ) )  mod  N )  =  1  /\  ( ( ( y ^ (
( N  -  1 )  /  x ) )  -  1 )  gcd  N )  =  1 ) )
8023, 79syl6bi 228 . . . . 5  |-  ( x  e.  Prime  ->  ( x 
||  ( P ^ E )  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  -  1 ) )  mod  N )  =  1  /\  ( ( ( y ^ (
( N  -  1 )  /  x ) )  -  1 )  gcd  N )  =  1 ) ) )
8180rgen 2817 . . . 4  |-  A. x  e.  Prime  ( x  ||  ( P ^ E )  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  - 
1 ) )  mod 
N )  =  1  /\  ( ( ( y ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N )  =  1 ) )
8281a1i 11 . . 3  |-  ( D  e.  NN  ->  A. x  e.  Prime  ( x  ||  ( P ^ E )  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  - 
1 ) )  mod 
N )  =  1  /\  ( ( ( y ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N )  =  1 ) ) )
839, 10, 12, 21, 82pockthg 14272 . 2  |-  ( D  e.  NN  ->  N  e.  Prime )
841, 83ax-mp 5 1  |-  N  e. 
Prime
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   class class class wbr 4440  (class class class)co 6275   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    - cmin 9794    / cdiv 10195   NNcn 10525   NN0cn0 10784   ZZcz 10853    mod cmo 11952   ^cexp 12122    || cdivides 13836    gcd cgcd 13992   Primecprime 14065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-dvds 13837  df-gcd 13993  df-prm 14066  df-odz 14143  df-phi 14144  df-pc 14209
This theorem is referenced by:  1259prm  14465  2503prm  14469  4001prm  14474
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