Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > pockthi | Unicode version |
Description: Pocklington's theorem, which gives a sufficient criterion for a number 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 12296 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.) |
Ref | Expression |
---|---|
pockthi.p | |
pockthi.g | |
pockthi.m | |
pockthi.n | |
pockthi.d | |
pockthi.e | |
pockthi.a | |
pockthi.fac | |
pockthi.gt | |
pockthi.mod | |
pockthi.gcd |
Ref | Expression |
---|---|
pockthi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pockthi.d | . 2 | |
2 | pockthi.p | . . . . . 6 | |
3 | prmnn 12051 | . . . . . 6 | |
4 | 2, 3 | ax-mp 5 | . . . . 5 |
5 | pockthi.e | . . . . . 6 | |
6 | 5 | nnnn0i 9130 | . . . . 5 |
7 | nnexpcl 10476 | . . . . 5 | |
8 | 4, 6, 7 | mp2an 424 | . . . 4 |
9 | 8 | a1i 9 | . . 3 |
10 | id 19 | . . 3 | |
11 | pockthi.gt | . . . 4 | |
12 | 11 | a1i 9 | . . 3 |
13 | pockthi.n | . . . . 5 | |
14 | pockthi.fac | . . . . . . 7 | |
15 | 1 | nncni 8875 | . . . . . . . 8 |
16 | 8 | nncni 8875 | . . . . . . . 8 |
17 | 15, 16 | mulcomi 7913 | . . . . . . 7 |
18 | 14, 17 | eqtri 2191 | . . . . . 6 |
19 | 18 | oveq1i 5860 | . . . . 5 |
20 | 13, 19 | eqtri 2191 | . . . 4 |
21 | 20 | a1i 9 | . . 3 |
22 | prmdvdsexpb 12090 | . . . . . . 7 | |
23 | 2, 5, 22 | mp3an23 1324 | . . . . . 6 |
24 | pockthi.m | . . . . . . . . . . . . 13 | |
25 | pockthi.g | . . . . . . . . . . . . . 14 | |
26 | 25, 4 | nnmulcli 8887 | . . . . . . . . . . . . 13 |
27 | 24, 26 | eqeltri 2243 | . . . . . . . . . . . 12 |
28 | 27 | nncni 8875 | . . . . . . . . . . 11 |
29 | ax-1cn 7854 | . . . . . . . . . . 11 | |
30 | 28, 29, 13 | mvrraddi 8123 | . . . . . . . . . 10 |
31 | 30 | oveq2i 5861 | . . . . . . . . 9 |
32 | 31 | oveq1i 5860 | . . . . . . . 8 |
33 | pockthi.mod | . . . . . . . . 9 | |
34 | peano2nn 8877 | . . . . . . . . . . . . 13 | |
35 | 27, 34 | ax-mp 5 | . . . . . . . . . . . 12 |
36 | 13, 35 | eqeltri 2243 | . . . . . . . . . . 11 |
37 | nnq 9579 | . . . . . . . . . . 11 | |
38 | 36, 37 | ax-mp 5 | . . . . . . . . . 10 |
39 | 27 | nngt0i 8895 | . . . . . . . . . . . 12 |
40 | 27 | nnrei 8874 | . . . . . . . . . . . . 13 |
41 | 1re 7906 | . . . . . . . . . . . . 13 | |
42 | ltaddpos2 8359 | . . . . . . . . . . . . 13 | |
43 | 40, 41, 42 | mp2an 424 | . . . . . . . . . . . 12 |
44 | 39, 43 | mpbi 144 | . . . . . . . . . . 11 |
45 | 44, 13 | breqtrri 4014 | . . . . . . . . . 10 |
46 | q1mod 10299 | . . . . . . . . . 10 | |
47 | 38, 45, 46 | mp2an 424 | . . . . . . . . 9 |
48 | 33, 47 | eqtri 2191 | . . . . . . . 8 |
49 | 32, 48 | eqtri 2191 | . . . . . . 7 |
50 | oveq2 5858 | . . . . . . . . . . . 12 | |
51 | 25 | nncni 8875 | . . . . . . . . . . . . . . 15 |
52 | 4 | nncni 8875 | . . . . . . . . . . . . . . 15 |
53 | 51, 52 | mulcomi 7913 | . . . . . . . . . . . . . 14 |
54 | 30, 24, 53 | 3eqtrri 2196 | . . . . . . . . . . . . 13 |
55 | 36 | nncni 8875 | . . . . . . . . . . . . . . 15 |
56 | 55, 29 | subcli 8182 | . . . . . . . . . . . . . 14 |
57 | 4 | nnap0i 8896 | . . . . . . . . . . . . . 14 # |
58 | 56, 52, 51, 57 | divmulapi 8670 | . . . . . . . . . . . . 13 |
59 | 54, 58 | mpbir 145 | . . . . . . . . . . . 12 |
60 | 50, 59 | eqtrdi 2219 | . . . . . . . . . . 11 |
61 | 60 | oveq2d 5866 | . . . . . . . . . 10 |
62 | 61 | oveq1d 5865 | . . . . . . . . 9 |
63 | 62 | oveq1d 5865 | . . . . . . . 8 |
64 | pockthi.gcd | . . . . . . . 8 | |
65 | 63, 64 | eqtrdi 2219 | . . . . . . 7 |
66 | pockthi.a | . . . . . . . . 9 | |
67 | 66 | nnzi 9220 | . . . . . . . 8 |
68 | oveq1 5857 | . . . . . . . . . . . 12 | |
69 | 68 | oveq1d 5865 | . . . . . . . . . . 11 |
70 | 69 | eqeq1d 2179 | . . . . . . . . . 10 |
71 | oveq1 5857 | . . . . . . . . . . . . 13 | |
72 | 71 | oveq1d 5865 | . . . . . . . . . . . 12 |
73 | 72 | oveq1d 5865 | . . . . . . . . . . 11 |
74 | 73 | eqeq1d 2179 | . . . . . . . . . 10 |
75 | 70, 74 | anbi12d 470 | . . . . . . . . 9 |
76 | 75 | rspcev 2834 | . . . . . . . 8 |
77 | 67, 76 | mpan 422 | . . . . . . 7 |
78 | 49, 65, 77 | sylancr 412 | . . . . . 6 |
79 | 23, 78 | syl6bi 162 | . . . . 5 |
80 | 79 | rgen 2523 | . . . 4 |
81 | 80 | a1i 9 | . . 3 |
82 | 9, 10, 12, 21, 81 | pockthg 12296 | . 2 |
83 | 1, 82 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 wrex 2449 class class class wbr 3987 (class class class)co 5850 cr 7760 cc0 7761 c1 7762 caddc 7764 cmul 7766 clt 7941 cmin 8077 cdiv 8576 cn 8865 cn0 9122 cz 9199 cq 9565 cmo 10265 cexp 10462 cdvds 11736 cgcd 11884 cprime 12048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 ax-arch 7880 ax-caucvg 7881 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-frec 6367 df-1o 6392 df-2o 6393 df-oadd 6396 df-er 6509 df-en 6715 df-dom 6716 df-fin 6717 df-sup 6957 df-inf 6958 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-n0 9123 df-xnn0 9186 df-z 9200 df-uz 9475 df-q 9566 df-rp 9598 df-fz 9953 df-fzo 10086 df-fl 10213 df-mod 10266 df-seqfrec 10389 df-exp 10463 df-ihash 10697 df-cj 10793 df-re 10794 df-im 10795 df-rsqrt 10949 df-abs 10950 df-clim 11229 df-proddc 11501 df-dvds 11737 df-gcd 11885 df-prm 12049 df-odz 12151 df-phi 12152 df-pc 12226 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |