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Mirrors > Home > MPE Home > Th. List > 1259prm | Structured version Visualization version GIF version |
Description: 1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
1259prm.1 | ⊢ 𝑁 = ;;;1259 |
Ref | Expression |
---|---|
1259prm | ⊢ 𝑁 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 37prm 16442 | . 2 ⊢ ;37 ∈ ℙ | |
2 | 3nn0 11903 | . . 3 ⊢ 3 ∈ ℕ0 | |
3 | 4nn 11708 | . . 3 ⊢ 4 ∈ ℕ | |
4 | 2, 3 | decnncl 12106 | . 2 ⊢ ;34 ∈ ℕ |
5 | 1nn0 11901 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
6 | 2nn0 11902 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
7 | 5, 6 | deccl 12101 | . . . . . . 7 ⊢ ;12 ∈ ℕ0 |
8 | 5nn0 11905 | . . . . . . 7 ⊢ 5 ∈ ℕ0 | |
9 | 7, 8 | deccl 12101 | . . . . . 6 ⊢ ;;125 ∈ ℕ0 |
10 | 8nn0 11908 | . . . . . 6 ⊢ 8 ∈ ℕ0 | |
11 | 9, 10 | deccl 12101 | . . . . 5 ⊢ ;;;1258 ∈ ℕ0 |
12 | 11 | nn0cni 11897 | . . . 4 ⊢ ;;;1258 ∈ ℂ |
13 | ax-1cn 10583 | . . . 4 ⊢ 1 ∈ ℂ | |
14 | 1259prm.1 | . . . . 5 ⊢ 𝑁 = ;;;1259 | |
15 | eqid 2818 | . . . . . 6 ⊢ ;;;1258 = ;;;1258 | |
16 | 8p1e9 11775 | . . . . . 6 ⊢ (8 + 1) = 9 | |
17 | 9, 10, 5, 15, 16 | decaddi 12146 | . . . . 5 ⊢ (;;;1258 + 1) = ;;;1259 |
18 | 14, 17 | eqtr4i 2844 | . . . 4 ⊢ 𝑁 = (;;;1258 + 1) |
19 | 12, 13, 18 | mvrraddi 10891 | . . 3 ⊢ (𝑁 − 1) = ;;;1258 |
20 | 4nn0 11904 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
21 | 2, 20 | deccl 12101 | . . . 4 ⊢ ;34 ∈ ℕ0 |
22 | 7nn0 11907 | . . . 4 ⊢ 7 ∈ ℕ0 | |
23 | eqid 2818 | . . . 4 ⊢ ;37 = ;37 | |
24 | 6, 2 | deccl 12101 | . . . 4 ⊢ ;23 ∈ ℕ0 |
25 | eqid 2818 | . . . . 5 ⊢ ;34 = ;34 | |
26 | eqid 2818 | . . . . 5 ⊢ ;23 = ;23 | |
27 | 3t3e9 11792 | . . . . . . 7 ⊢ (3 · 3) = 9 | |
28 | 2p1e3 11767 | . . . . . . 7 ⊢ (2 + 1) = 3 | |
29 | 27, 28 | oveq12i 7157 | . . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3) |
30 | 9p3e12 12174 | . . . . . 6 ⊢ (9 + 3) = ;12 | |
31 | 29, 30 | eqtri 2841 | . . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12 |
32 | 4t3e12 12184 | . . . . . 6 ⊢ (4 · 3) = ;12 | |
33 | 3cn 11706 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
34 | 2cn 11700 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
35 | 3p2e5 11776 | . . . . . . 7 ⊢ (3 + 2) = 5 | |
36 | 33, 34, 35 | addcomli 10820 | . . . . . 6 ⊢ (2 + 3) = 5 |
37 | 5, 6, 2, 32, 36 | decaddi 12146 | . . . . 5 ⊢ ((4 · 3) + 3) = ;15 |
38 | 2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37 | decmac 12138 | . . . 4 ⊢ ((;34 · 3) + ;23) = ;;125 |
39 | 7cn 11719 | . . . . . . 7 ⊢ 7 ∈ ℂ | |
40 | 7t3e21 12196 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
41 | 39, 33, 40 | mulcomli 10638 | . . . . . 6 ⊢ (3 · 7) = ;21 |
42 | 1p2e3 11768 | . . . . . 6 ⊢ (1 + 2) = 3 | |
43 | 6, 5, 6, 41, 42 | decaddi 12146 | . . . . 5 ⊢ ((3 · 7) + 2) = ;23 |
44 | 4cn 11710 | . . . . . 6 ⊢ 4 ∈ ℂ | |
45 | 7t4e28 12197 | . . . . . 6 ⊢ (7 · 4) = ;28 | |
46 | 39, 44, 45 | mulcomli 10638 | . . . . 5 ⊢ (4 · 7) = ;28 |
47 | 22, 2, 20, 25, 10, 6, 43, 46 | decmul1c 12151 | . . . 4 ⊢ (;34 · 7) = ;;238 |
48 | 21, 2, 22, 23, 10, 24, 38, 47 | decmul2c 12152 | . . 3 ⊢ (;34 · ;37) = ;;;1258 |
49 | 19, 48 | eqtr4i 2844 | . 2 ⊢ (𝑁 − 1) = (;34 · ;37) |
50 | 9nn0 11909 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
51 | 9, 50 | deccl 12101 | . . . . . 6 ⊢ ;;;1259 ∈ ℕ0 |
52 | 14, 51 | eqeltri 2906 | . . . . 5 ⊢ 𝑁 ∈ ℕ0 |
53 | 52 | nn0cni 11897 | . . . 4 ⊢ 𝑁 ∈ ℂ |
54 | npcan 10883 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
55 | 53, 13, 54 | mp2an 688 | . . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁 |
56 | 55 | eqcomi 2827 | . 2 ⊢ 𝑁 = ((𝑁 − 1) + 1) |
57 | 1nn 11637 | . 2 ⊢ 1 ∈ ℕ | |
58 | 2nn 11698 | . 2 ⊢ 2 ∈ ℕ | |
59 | 2, 22 | deccl 12101 | . . . . 5 ⊢ ;37 ∈ ℕ0 |
60 | 59 | numexp1 16401 | . . . 4 ⊢ (;37↑1) = ;37 |
61 | 60 | oveq2i 7156 | . . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37) |
62 | 49, 61 | eqtr4i 2844 | . 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1)) |
63 | 7nn 11717 | . . . 4 ⊢ 7 ∈ ℕ | |
64 | 4lt7 11813 | . . . 4 ⊢ 4 < 7 | |
65 | 2, 20, 63, 64 | declt 12114 | . . 3 ⊢ ;34 < ;37 |
66 | 65, 60 | breqtrri 5084 | . 2 ⊢ ;34 < (;37↑1) |
67 | 14 | 1259lem4 16455 | . 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁) |
68 | 14 | 1259lem5 16456 | . 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1 |
69 | 1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68 | pockthi 16231 | 1 ⊢ 𝑁 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 1c1 10526 + caddc 10528 · cmul 10530 < clt 10663 − cmin 10858 2c2 11680 3c3 11681 4c4 11682 5c5 11683 7c7 11685 8c8 11686 9c9 11687 ℕ0cn0 11885 ;cdc 12086 ↑cexp 13417 ℙcprime 16003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 df-prm 16004 df-odz 16090 df-phi 16091 df-pc 16162 |
This theorem is referenced by: (None) |
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