Theorem pockthi 12552

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 12551 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.

Step	Hyp	Ref	Expression
1		pockthi.d	. 2 |- D e. NN
2		pockthi.p	. . . . . 6 |- P e. Prime
3		prmnn 12303	. . . . . 6 |- ( P e. Prime -> P e. NN )
4	2, 3	ax-mp 5	. . . . 5 |- P e. NN
5		pockthi.e	. . . . . 6 |- E e. NN
6	5	nnnn0i 9274	. . . . 5 |- E e. NN0
7		nnexpcl 10661	. . . . 5 |- ( ( P e. NN /\ E e. NN0 ) -> ( P ^ E ) e. NN )
8	4, 6, 7	mp2an 426	. . . 4 |- ( P ^ E ) e. NN
9	8	a1i 9	. . 3 |- ( D e. NN -> ( P ^ E ) e. NN )
10		id 19	. . 3 |- ( D e. NN -> D e. NN )
11		pockthi.gt	. . . 4 |- D < ( P ^ E )
12	11	a1i 9	. . 3 |- ( D e. NN -> D < ( P ^ E ) )
13		pockthi.n	. . . . 5 |- N = ( M + 1 )
14		pockthi.fac	. . . . . . 7 |- M = ( D x. ( P ^ E ) )
15	1	nncni 9017	. . . . . . . 8 |- D e. CC
16	8	nncni 9017	. . . . . . . 8 |- ( P ^ E ) e. CC
17	15, 16	mulcomi 8049	. . . . . . 7 |- ( D x. ( P ^ E ) ) = ( ( P ^ E ) x. D )
18	14, 17	eqtri 2217	. . . . . 6 |- M = ( ( P ^ E ) x. D )
19	18	oveq1i 5935	. . . . 5 |- ( M + 1 ) = ( ( ( P ^ E ) x. D ) + 1 )
20	13, 19	eqtri 2217	. . . 4 |- N = ( ( ( P ^ E ) x. D ) + 1 )
21	20	a1i 9	. . 3 |- ( D e. NN -> N = ( ( ( P ^ E ) x. D ) + 1 ) )
22		prmdvdsexpb 12342	. . . . . . 7 |- ( ( x e. Prime /\ P e. Prime /\ E e. NN ) -> ( x || ( P ^ E ) <-> x = P ) )
23	2, 5, 22	mp3an23 1340	. . . . . 6 |- ( x e. Prime -> ( x || ( P ^ E ) <-> x = P ) )
24		pockthi.m	. . . . . . . . . . . . 13 |- M = ( G x. P )
25		pockthi.g	. . . . . . . . . . . . . 14 |- G e. NN
26	25, 4	nnmulcli 9029	. . . . . . . . . . . . 13 |- ( G x. P ) e. NN
27	24, 26	eqeltri 2269	. . . . . . . . . . . 12 |- M e. NN
28	27	nncni 9017	. . . . . . . . . . 11 |- M e. CC
29		ax-1cn 7989	. . . . . . . . . . 11 |- 1 e. CC
30	28, 29, 13	mvrraddi 8260	. . . . . . . . . 10 |- ( N - 1 ) = M
31	30	oveq2i 5936	. . . . . . . . 9 |- ( A ^ ( N - 1 ) ) = ( A ^ M )
32	31	oveq1i 5935	. . . . . . . 8 |- ( ( A ^ ( N - 1 ) ) mod N ) = ( ( A ^ M ) mod N )
33		pockthi.mod	. . . . . . . . 9 |- ( ( A ^ M ) mod N ) = ( 1 mod N )
34		peano2nn 9019	. . . . . . . . . . . . 13 |- ( M e. NN -> ( M + 1 ) e. NN )
35	27, 34	ax-mp 5	. . . . . . . . . . . 12 |- ( M + 1 ) e. NN
36	13, 35	eqeltri 2269	. . . . . . . . . . 11 |- N e. NN
37		nnq 9724	. . . . . . . . . . 11 |- ( N e. NN -> N e. QQ )
38	36, 37	ax-mp 5	. . . . . . . . . 10 |- N e. QQ
39	27	nngt0i 9037	. . . . . . . . . . . 12 |- 0 < M
40	27	nnrei 9016	. . . . . . . . . . . . 13 |- M e. RR
41		1re 8042	. . . . . . . . . . . . 13 |- 1 e. RR
42		ltaddpos2 8497	. . . . . . . . . . . . 13 |- ( ( M e. RR /\ 1 e. RR ) -> ( 0 < M <-> 1 < ( M + 1 ) ) )
43	40, 41, 42	mp2an 426	. . . . . . . . . . . 12 |- ( 0 < M <-> 1 < ( M + 1 ) )
44	39, 43	mpbi 145	. . . . . . . . . . 11 |- 1 < ( M + 1 )
45	44, 13	breqtrri 4061	. . . . . . . . . 10 |- 1 < N
46		q1mod 10465	. . . . . . . . . 10 |- ( ( N e. QQ /\ 1 < N ) -> ( 1 mod N ) = 1 )
47	38, 45, 46	mp2an 426	. . . . . . . . 9 |- ( 1 mod N ) = 1
48	33, 47	eqtri 2217	. . . . . . . 8 |- ( ( A ^ M ) mod N ) = 1
49	32, 48	eqtri 2217	. . . . . . 7 |- ( ( A ^ ( N - 1 ) ) mod N ) = 1
50		oveq2 5933	. . . . . . . . . . . 12 |- ( x = P -> ( ( N - 1 ) / x ) = ( ( N - 1 ) / P ) )
51	25	nncni 9017	. . . . . . . . . . . . . . 15 |- G e. CC
52	4	nncni 9017	. . . . . . . . . . . . . . 15 |- P e. CC
53	51, 52	mulcomi 8049	. . . . . . . . . . . . . 14 |- ( G x. P ) = ( P x. G )
54	30, 24, 53	3eqtrri 2222	. . . . . . . . . . . . 13 |- ( P x. G ) = ( N - 1 )
55	36	nncni 9017	. . . . . . . . . . . . . . 15 |- N e. CC
56	55, 29	subcli 8319	. . . . . . . . . . . . . 14 |- ( N - 1 ) e. CC
57	4	nnap0i 9038	. . . . . . . . . . . . . 14 |- P # 0
58	56, 52, 51, 57	divmulapi 8810	. . . . . . . . . . . . 13 |- ( ( ( N - 1 ) / P ) = G <-> ( P x. G ) = ( N - 1 ) )
59	54, 58	mpbir 146	. . . . . . . . . . . 12 |- ( ( N - 1 ) / P ) = G
60	50, 59	eqtrdi 2245	. . . . . . . . . . 11 |- ( x = P -> ( ( N - 1 ) / x ) = G )
61	60	oveq2d 5941	. . . . . . . . . 10 |- ( x = P -> ( A ^ ( ( N - 1 ) / x ) ) = ( A ^ G ) )
62	61	oveq1d 5940	. . . . . . . . 9 |- ( x = P -> ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ G ) - 1 ) )
63	62	oveq1d 5940	. . . . . . . 8 |- ( x = P -> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = ( ( ( A ^ G ) - 1 ) gcd N ) )
64		pockthi.gcd	. . . . . . . 8 |- ( ( ( A ^ G ) - 1 ) gcd N ) = 1
65	63, 64	eqtrdi 2245	. . . . . . 7 |- ( x = P -> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 )
66		pockthi.a	. . . . . . . . 9 |- A e. NN
67	66	nnzi 9364	. . . . . . . 8 |- A e. ZZ
68		oveq1 5932	. . . . . . . . . . . 12 |- ( y = A -> ( y ^ ( N - 1 ) ) = ( A ^ ( N - 1 ) ) )
69	68	oveq1d 5940	. . . . . . . . . . 11 |- ( y = A -> ( ( y ^ ( N - 1 ) ) mod N ) = ( ( A ^ ( N - 1 ) ) mod N ) )
70	69	eqeq1d 2205	. . . . . . . . . 10 |- ( y = A -> ( ( ( y ^ ( N - 1 ) ) mod N ) = 1 <-> ( ( A ^ ( N - 1 ) ) mod N ) = 1 ) )
71		oveq1 5932	. . . . . . . . . . . . 13 |- ( y = A -> ( y ^ ( ( N - 1 ) / x ) ) = ( A ^ ( ( N - 1 ) / x ) ) )
72	71	oveq1d 5940	. . . . . . . . . . . 12 |- ( y = A -> ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) )
73	72	oveq1d 5940	. . . . . . . . . . 11 |- ( y = A -> ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) )
74	73	eqeq1d 2205	. . . . . . . . . 10 |- ( y = A -> ( ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 <-> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 ) )
75	70, 74	anbi12d 473	. . . . . . . . 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 ) ) )
76	75	rspcev 2868	. . . . . . . 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 ) )
77	67, 76	mpan 424	. . . . . . 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 ) )
78	49, 65, 77	sylancr 414	. . . . . 6 |- ( x = P -> E. y e. ZZ ( ( ( y ^ ( N - 1 ) ) mod N ) = 1 /\ ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 ) )
79	23, 78	biimtrdi 163	. . . . 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 ) ) )
80	79	rgen 2550	. . . 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 ) )
81	80	a1i 9	. . 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 ) ) )
82	9, 10, 12, 21, 81	pockthg 12551	. 2 |- ( D e. NN -> N e. Prime )
83	1, 82	ax-mp 5	1 |- N e. Prime

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   class class class wbr 4034  (class class class)co 5925   RRcr 7895   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   < clt 8078   - cmin 8214   / cdiv 8716   NNcn 9007   NN0cn0 9266   ZZcz 9343   QQcq 9710   mod cmo 10431   ^cexp 10647   || cdvds 11969   gcd cgcd 12145   Primecprime 12300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-2o 6484  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-xnn0 9330  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-ihash 10885  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-proddc 11733  df-dvds 11970  df-gcd 12146  df-prm 12301  df-odz 12403  df-phi 12404  df-pc 12479

This theorem is referenced by: (None)