Theorem pockthi 12876

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 12875 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 12627	. . . . . 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 9373	. . . . 5 |- E e. NN0
7		nnexpcl 10769	. . . . 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 9116	. . . . . . . 8 |- D e. CC
16	8	nncni 9116	. . . . . . . 8 |- ( P ^ E ) e. CC
17	15, 16	mulcomi 8148	. . . . . . 7 |- ( D x. ( P ^ E ) ) = ( ( P ^ E ) x. D )
18	14, 17	eqtri 2250	. . . . . 6 |- M = ( ( P ^ E ) x. D )
19	18	oveq1i 6010	. . . . 5 |- ( M + 1 ) = ( ( ( P ^ E ) x. D ) + 1 )
20	13, 19	eqtri 2250	. . . 4 |- N = ( ( ( P ^ E ) x. D ) + 1 )
21	20	a1i 9	. . 3 |- ( D e. NN -> N = ( ( ( P ^ E ) x. D ) + 1 ) )
22		prmdvdsexpb 12666	. . . . . . 7 |- ( ( x e. Prime /\ P e. Prime /\ E e. NN ) -> ( x || ( P ^ E ) <-> x = P ) )
23	2, 5, 22	mp3an23 1363	. . . . . 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 9128	. . . . . . . . . . . . 13 |- ( G x. P ) e. NN
27	24, 26	eqeltri 2302	. . . . . . . . . . . 12 |- M e. NN
28	27	nncni 9116	. . . . . . . . . . 11 |- M e. CC
29		ax-1cn 8088	. . . . . . . . . . 11 |- 1 e. CC
30	28, 29, 13	mvrraddi 8359	. . . . . . . . . 10 |- ( N - 1 ) = M
31	30	oveq2i 6011	. . . . . . . . 9 |- ( A ^ ( N - 1 ) ) = ( A ^ M )
32	31	oveq1i 6010	. . . . . . . 8 |- ( ( A ^ ( N - 1 ) ) mod N ) = ( ( A ^ M ) mod N )
33		pockthi.mod	. . . . . . . . 9 |- ( ( A ^ M ) mod N ) = ( 1 mod N )
34		peano2nn 9118	. . . . . . . . . . . . 13 |- ( M e. NN -> ( M + 1 ) e. NN )
35	27, 34	ax-mp 5	. . . . . . . . . . . 12 |- ( M + 1 ) e. NN
36	13, 35	eqeltri 2302	. . . . . . . . . . 11 |- N e. NN
37		nnq 9824	. . . . . . . . . . 11 |- ( N e. NN -> N e. QQ )
38	36, 37	ax-mp 5	. . . . . . . . . 10 |- N e. QQ
39	27	nngt0i 9136	. . . . . . . . . . . 12 |- 0 < M
40	27	nnrei 9115	. . . . . . . . . . . . 13 |- M e. RR
41		1re 8141	. . . . . . . . . . . . 13 |- 1 e. RR
42		ltaddpos2 8596	. . . . . . . . . . . . 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 4109	. . . . . . . . . 10 |- 1 < N
46		q1mod 10573	. . . . . . . . . 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 2250	. . . . . . . 8 |- ( ( A ^ M ) mod N ) = 1
49	32, 48	eqtri 2250	. . . . . . 7 |- ( ( A ^ ( N - 1 ) ) mod N ) = 1
50		oveq2 6008	. . . . . . . . . . . 12 |- ( x = P -> ( ( N - 1 ) / x ) = ( ( N - 1 ) / P ) )
51	25	nncni 9116	. . . . . . . . . . . . . . 15 |- G e. CC
52	4	nncni 9116	. . . . . . . . . . . . . . 15 |- P e. CC
53	51, 52	mulcomi 8148	. . . . . . . . . . . . . 14 |- ( G x. P ) = ( P x. G )
54	30, 24, 53	3eqtrri 2255	. . . . . . . . . . . . 13 |- ( P x. G ) = ( N - 1 )
55	36	nncni 9116	. . . . . . . . . . . . . . 15 |- N e. CC
56	55, 29	subcli 8418	. . . . . . . . . . . . . 14 |- ( N - 1 ) e. CC
57	4	nnap0i 9137	. . . . . . . . . . . . . 14 |- P # 0
58	56, 52, 51, 57	divmulapi 8909	. . . . . . . . . . . . 13 |- ( ( ( N - 1 ) / P ) = G <-> ( P x. G ) = ( N - 1 ) )
59	54, 58	mpbir 146	. . . . . . . . . . . 12 |- ( ( N - 1 ) / P ) = G
60	50, 59	eqtrdi 2278	. . . . . . . . . . 11 |- ( x = P -> ( ( N - 1 ) / x ) = G )
61	60	oveq2d 6016	. . . . . . . . . 10 |- ( x = P -> ( A ^ ( ( N - 1 ) / x ) ) = ( A ^ G ) )
62	61	oveq1d 6015	. . . . . . . . 9 |- ( x = P -> ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ G ) - 1 ) )
63	62	oveq1d 6015	. . . . . . . 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 2278	. . . . . . 7 |- ( x = P -> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 )
66		pockthi.a	. . . . . . . . 9 |- A e. NN
67	66	nnzi 9463	. . . . . . . 8 |- A e. ZZ
68		oveq1 6007	. . . . . . . . . . . 12 |- ( y = A -> ( y ^ ( N - 1 ) ) = ( A ^ ( N - 1 ) ) )
69	68	oveq1d 6015	. . . . . . . . . . 11 |- ( y = A -> ( ( y ^ ( N - 1 ) ) mod N ) = ( ( A ^ ( N - 1 ) ) mod N ) )
70	69	eqeq1d 2238	. . . . . . . . . 10 |- ( y = A -> ( ( ( y ^ ( N - 1 ) ) mod N ) = 1 <-> ( ( A ^ ( N - 1 ) ) mod N ) = 1 ) )
71		oveq1 6007	. . . . . . . . . . . . 13 |- ( y = A -> ( y ^ ( ( N - 1 ) / x ) ) = ( A ^ ( ( N - 1 ) / x ) ) )
72	71	oveq1d 6015	. . . . . . . . . . . 12 |- ( y = A -> ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) )
73	72	oveq1d 6015	. . . . . . . . . . 11 |- ( y = A -> ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) )
74	73	eqeq1d 2238	. . . . . . . . . 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 2907	. . . . . . . 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 2583	. . . 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 12875	. 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 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   class class class wbr 4082  (class class class)co 6000   RRcr 7994   0cc0 7995   1c1 7996   + caddc 7998   x. cmul 8000   < clt 8177   - cmin 8313   / cdiv 8815   NNcn 9106   NN0cn0 9365   ZZcz 9442   QQcq 9810   mod cmo 10539   ^cexp 10755   || cdvds 12293   gcd cgcd 12469   Primecprime 12624

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-2o 6561  df-oadd 6564  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-xnn0 9429  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-ihash 10993  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-proddc 12057  df-dvds 12294  df-gcd 12470  df-prm 12625  df-odz 12727  df-phi 12728  df-pc 12803

This theorem is referenced by: (None)