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Mirrors > Home > ILE Home > Th. List > isprm5 | Unicode version |
Description: One need only check prime divisors of up to in order to ensure primality. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
isprm5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isprm3 12029 | . 2 | |
2 | breq1 3979 | . . . . . . . 8 | |
3 | 2 | notbid 657 | . . . . . . 7 |
4 | simpllr 524 | . . . . . . 7 | |
5 | 2z 9210 | . . . . . . . . . 10 | |
6 | 5 | a1i 9 | . . . . . . . . 9 |
7 | eluzelz 9466 | . . . . . . . . . . 11 | |
8 | 7 | ad3antrrr 484 | . . . . . . . . . 10 |
9 | peano2zm 9220 | . . . . . . . . . 10 | |
10 | 8, 9 | syl 14 | . . . . . . . . 9 |
11 | prmz 12022 | . . . . . . . . . 10 | |
12 | 11 | ad2antlr 481 | . . . . . . . . 9 |
13 | 6, 10, 12 | 3jca 1166 | . . . . . . . 8 |
14 | prmuz2 12042 | . . . . . . . . . . 11 | |
15 | eluzle 9469 | . . . . . . . . . . 11 | |
16 | 14, 15 | syl 14 | . . . . . . . . . 10 |
17 | 16 | ad2antlr 481 | . . . . . . . . 9 |
18 | eluzelre 9467 | . . . . . . . . . . . . 13 | |
19 | 14, 18 | syl 14 | . . . . . . . . . . . 12 |
20 | 19 | ad2antlr 481 | . . . . . . . . . . 11 |
21 | 20 | resqcld 10603 | . . . . . . . . . . 11 |
22 | eluzelre 9467 | . . . . . . . . . . . 12 | |
23 | 22 | ad3antrrr 484 | . . . . . . . . . . 11 |
24 | prmnn 12021 | . . . . . . . . . . . . . . 15 | |
25 | 24 | nncnd 8862 | . . . . . . . . . . . . . 14 |
26 | 25 | exp1d 10572 | . . . . . . . . . . . . 13 |
27 | 1lt2 9017 | . . . . . . . . . . . . . 14 | |
28 | 1nn0 9121 | . . . . . . . . . . . . . . . 16 | |
29 | 28 | a1i 9 | . . . . . . . . . . . . . . 15 |
30 | 2nn0 9122 | . . . . . . . . . . . . . . . 16 | |
31 | 30 | a1i 9 | . . . . . . . . . . . . . . 15 |
32 | prmgt1 12043 | . . . . . . . . . . . . . . 15 | |
33 | nn0ltexp2 10612 | . . . . . . . . . . . . . . 15 | |
34 | 19, 29, 31, 32, 33 | syl31anc 1230 | . . . . . . . . . . . . . 14 |
35 | 27, 34 | mpbii 147 | . . . . . . . . . . . . 13 |
36 | 26, 35 | eqbrtrrd 4000 | . . . . . . . . . . . 12 |
37 | 36 | ad2antlr 481 | . . . . . . . . . . 11 |
38 | simpr 109 | . . . . . . . . . . 11 | |
39 | 20, 21, 23, 37, 38 | ltletrd 8312 | . . . . . . . . . 10 |
40 | zltlem1 9239 | . . . . . . . . . . 11 | |
41 | 12, 8, 40 | syl2anc 409 | . . . . . . . . . 10 |
42 | 39, 41 | mpbid 146 | . . . . . . . . 9 |
43 | 17, 42 | jca 304 | . . . . . . . 8 |
44 | elfz2 9942 | . . . . . . . 8 | |
45 | 13, 43, 44 | sylanbrc 414 | . . . . . . 7 |
46 | 3, 4, 45 | rspcdva 2830 | . . . . . 6 |
47 | 46 | ex 114 | . . . . 5 |
48 | 47 | ralrimiva 2537 | . . . 4 |
49 | simpll 519 | . . . . . 6 | |
50 | simplr 520 | . . . . . 6 | |
51 | simpr 109 | . . . . . 6 | |
52 | 49, 50, 51 | isprm5lem 12052 | . . . . 5 |
53 | 52 | ralrimiva 2537 | . . . 4 |
54 | 48, 53 | impbida 586 | . . 3 |
55 | 54 | pm5.32i 450 | . 2 |
56 | 1, 55 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 967 wcel 2135 wral 2442 class class class wbr 3976 cfv 5182 (class class class)co 5836 cr 7743 c1 7745 clt 7924 cle 7925 cmin 8060 c2 8899 cn0 9105 cz 9182 cuz 9457 cfz 9935 cexp 10444 cdvds 11713 cprime 12018 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-1o 6375 df-2o 6376 df-er 6492 df-en 6698 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 df-prm 12019 |
This theorem is referenced by: pockthg 12266 |
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