Theorem pockthi 13081

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 13080 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 12832	. . . . . 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 9521	. . . . 5 |- E e. NN0
7		nnexpcl 10938	. . . . 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 9264	. . . . . . . 8 |- D e. CC
16	8	nncni 9264	. . . . . . . 8 |- ( P ^ E ) e. CC
17	15, 16	mulcomi 8296	. . . . . . 7 |- ( D x. ( P ^ E ) ) = ( ( P ^ E ) x. D )
18	14, 17	eqtri 2255	. . . . . 6 |- M = ( ( P ^ E ) x. D )
19	18	oveq1i 6068	. . . . 5 |- ( M + 1 ) = ( ( ( P ^ E ) x. D ) + 1 )
20	13, 19	eqtri 2255	. . . 4 |- N = ( ( ( P ^ E ) x. D ) + 1 )
21	20	a1i 9	. . 3 |- ( D e. NN -> N = ( ( ( P ^ E ) x. D ) + 1 ) )
22		prmdvdsexpb 12871	. . . . . . 7 |- ( ( x e. Prime /\ P e. Prime /\ E e. NN ) -> ( x || ( P ^ E ) <-> x = P ) )
23	2, 5, 22	mp3an23 1366	. . . . . 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 9276	. . . . . . . . . . . . 13 |- ( G x. P ) e. NN
27	24, 26	eqeltri 2307	. . . . . . . . . . . 12 |- M e. NN
28	27	nncni 9264	. . . . . . . . . . 11 |- M e. CC
29		ax-1cn 8236	. . . . . . . . . . 11 |- 1 e. CC
30	28, 29, 13	mvrraddi 8506	. . . . . . . . . 10 |- ( N - 1 ) = M
31	30	oveq2i 6069	. . . . . . . . 9 |- ( A ^ ( N - 1 ) ) = ( A ^ M )
32	31	oveq1i 6068	. . . . . . . 8 |- ( ( A ^ ( N - 1 ) ) mod N ) = ( ( A ^ M ) mod N )
33		pockthi.mod	. . . . . . . . 9 |- ( ( A ^ M ) mod N ) = ( 1 mod N )
34		peano2nn 9266	. . . . . . . . . . . . 13 |- ( M e. NN -> ( M + 1 ) e. NN )
35	27, 34	ax-mp 5	. . . . . . . . . . . 12 |- ( M + 1 ) e. NN
36	13, 35	eqeltri 2307	. . . . . . . . . . 11 |- N e. NN
37		nnq 9983	. . . . . . . . . . 11 |- ( N e. NN -> N e. QQ )
38	36, 37	ax-mp 5	. . . . . . . . . 10 |- N e. QQ
39	27	nngt0i 9284	. . . . . . . . . . . 12 |- 0 < M
40	27	nnrei 9263	. . . . . . . . . . . . 13 |- M e. RR
41		1re 8289	. . . . . . . . . . . . 13 |- 1 e. RR
42		ltaddpos2 8744	. . . . . . . . . . . . 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 4141	. . . . . . . . . 10 |- 1 < N
46		q1mod 10742	. . . . . . . . . 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 2255	. . . . . . . 8 |- ( ( A ^ M ) mod N ) = 1
49	32, 48	eqtri 2255	. . . . . . 7 |- ( ( A ^ ( N - 1 ) ) mod N ) = 1
50		oveq2 6066	. . . . . . . . . . . 12 |- ( x = P -> ( ( N - 1 ) / x ) = ( ( N - 1 ) / P ) )
51	25	nncni 9264	. . . . . . . . . . . . . . 15 |- G e. CC
52	4	nncni 9264	. . . . . . . . . . . . . . 15 |- P e. CC
53	51, 52	mulcomi 8296	. . . . . . . . . . . . . 14 |- ( G x. P ) = ( P x. G )
54	30, 24, 53	3eqtrri 2260	. . . . . . . . . . . . 13 |- ( P x. G ) = ( N - 1 )
55	36	nncni 9264	. . . . . . . . . . . . . . 15 |- N e. CC
56	55, 29	subcli 8565	. . . . . . . . . . . . . 14 |- ( N - 1 ) e. CC
57	4	nnap0i 9285	. . . . . . . . . . . . . 14 |- P # 0
58	56, 52, 51, 57	divmulapi 9057	. . . . . . . . . . . . 13 |- ( ( ( N - 1 ) / P ) = G <-> ( P x. G ) = ( N - 1 ) )
59	54, 58	mpbir 146	. . . . . . . . . . . 12 |- ( ( N - 1 ) / P ) = G
60	50, 59	eqtrdi 2283	. . . . . . . . . . 11 |- ( x = P -> ( ( N - 1 ) / x ) = G )
61	60	oveq2d 6074	. . . . . . . . . 10 |- ( x = P -> ( A ^ ( ( N - 1 ) / x ) ) = ( A ^ G ) )
62	61	oveq1d 6073	. . . . . . . . 9 |- ( x = P -> ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ G ) - 1 ) )
63	62	oveq1d 6073	. . . . . . . 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 2283	. . . . . . 7 |- ( x = P -> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 )
66		pockthi.a	. . . . . . . . 9 |- A e. NN
67	66	nnzi 9615	. . . . . . . 8 |- A e. ZZ
68		oveq1 6065	. . . . . . . . . . . 12 |- ( y = A -> ( y ^ ( N - 1 ) ) = ( A ^ ( N - 1 ) ) )
69	68	oveq1d 6073	. . . . . . . . . . 11 |- ( y = A -> ( ( y ^ ( N - 1 ) ) mod N ) = ( ( A ^ ( N - 1 ) ) mod N ) )
70	69	eqeq1d 2243	. . . . . . . . . 10 |- ( y = A -> ( ( ( y ^ ( N - 1 ) ) mod N ) = 1 <-> ( ( A ^ ( N - 1 ) ) mod N ) = 1 ) )
71		oveq1 6065	. . . . . . . . . . . . 13 |- ( y = A -> ( y ^ ( ( N - 1 ) / x ) ) = ( A ^ ( ( N - 1 ) / x ) ) )
72	71	oveq1d 6073	. . . . . . . . . . . 12 |- ( y = A -> ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) )
73	72	oveq1d 6073	. . . . . . . . . . 11 |- ( y = A -> ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) )
74	73	eqeq1d 2243	. . . . . . . . . 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 2923	. . . . . . . 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 2597	. . . 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 13080	. 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 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   - cmin 8460   / cdiv 8963   NNcn 9254   NN0cn0 9513   ZZcz 9594   QQcq 9969   mod cmo 10708   ^cexp 10924   || cdvds 12498   gcd cgcd 12674   Primecprime 12829

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-xnn0 9581  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-proddc 12262  df-dvds 12499  df-gcd 12675  df-prm 12830  df-odz 12932  df-phi 12933  df-pc 13008

This theorem is referenced by: (None)