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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 17053 | . 2 ⊢ ;37 ∈ ℙ | |
2 | 3nn0 12487 | . . 3 ⊢ 3 ∈ ℕ0 | |
3 | 4nn 12292 | . . 3 ⊢ 4 ∈ ℕ | |
4 | 2, 3 | decnncl 12694 | . 2 ⊢ ;34 ∈ ℕ |
5 | 1nn0 12485 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
6 | 2nn0 12486 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
7 | 5, 6 | deccl 12689 | . . . . . . 7 ⊢ ;12 ∈ ℕ0 |
8 | 5nn0 12489 | . . . . . . 7 ⊢ 5 ∈ ℕ0 | |
9 | 7, 8 | deccl 12689 | . . . . . 6 ⊢ ;;125 ∈ ℕ0 |
10 | 8nn0 12492 | . . . . . 6 ⊢ 8 ∈ ℕ0 | |
11 | 9, 10 | deccl 12689 | . . . . 5 ⊢ ;;;1258 ∈ ℕ0 |
12 | 11 | nn0cni 12481 | . . . 4 ⊢ ;;;1258 ∈ ℂ |
13 | ax-1cn 11164 | . . . 4 ⊢ 1 ∈ ℂ | |
14 | 1259prm.1 | . . . . 5 ⊢ 𝑁 = ;;;1259 | |
15 | eqid 2724 | . . . . . 6 ⊢ ;;;1258 = ;;;1258 | |
16 | 8p1e9 12359 | . . . . . 6 ⊢ (8 + 1) = 9 | |
17 | 9, 10, 5, 15, 16 | decaddi 12734 | . . . . 5 ⊢ (;;;1258 + 1) = ;;;1259 |
18 | 14, 17 | eqtr4i 2755 | . . . 4 ⊢ 𝑁 = (;;;1258 + 1) |
19 | 12, 13, 18 | mvrraddi 11474 | . . 3 ⊢ (𝑁 − 1) = ;;;1258 |
20 | 4nn0 12488 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
21 | 2, 20 | deccl 12689 | . . . 4 ⊢ ;34 ∈ ℕ0 |
22 | 7nn0 12491 | . . . 4 ⊢ 7 ∈ ℕ0 | |
23 | eqid 2724 | . . . 4 ⊢ ;37 = ;37 | |
24 | 6, 2 | deccl 12689 | . . . 4 ⊢ ;23 ∈ ℕ0 |
25 | eqid 2724 | . . . . 5 ⊢ ;34 = ;34 | |
26 | eqid 2724 | . . . . 5 ⊢ ;23 = ;23 | |
27 | 3t3e9 12376 | . . . . . . 7 ⊢ (3 · 3) = 9 | |
28 | 2p1e3 12351 | . . . . . . 7 ⊢ (2 + 1) = 3 | |
29 | 27, 28 | oveq12i 7413 | . . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3) |
30 | 9p3e12 12762 | . . . . . 6 ⊢ (9 + 3) = ;12 | |
31 | 29, 30 | eqtri 2752 | . . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12 |
32 | 4t3e12 12772 | . . . . . 6 ⊢ (4 · 3) = ;12 | |
33 | 3cn 12290 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
34 | 2cn 12284 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
35 | 3p2e5 12360 | . . . . . . 7 ⊢ (3 + 2) = 5 | |
36 | 33, 34, 35 | addcomli 11403 | . . . . . 6 ⊢ (2 + 3) = 5 |
37 | 5, 6, 2, 32, 36 | decaddi 12734 | . . . . 5 ⊢ ((4 · 3) + 3) = ;15 |
38 | 2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37 | decmac 12726 | . . . 4 ⊢ ((;34 · 3) + ;23) = ;;125 |
39 | 7cn 12303 | . . . . . . 7 ⊢ 7 ∈ ℂ | |
40 | 7t3e21 12784 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
41 | 39, 33, 40 | mulcomli 11220 | . . . . . 6 ⊢ (3 · 7) = ;21 |
42 | 1p2e3 12352 | . . . . . 6 ⊢ (1 + 2) = 3 | |
43 | 6, 5, 6, 41, 42 | decaddi 12734 | . . . . 5 ⊢ ((3 · 7) + 2) = ;23 |
44 | 4cn 12294 | . . . . . 6 ⊢ 4 ∈ ℂ | |
45 | 7t4e28 12785 | . . . . . 6 ⊢ (7 · 4) = ;28 | |
46 | 39, 44, 45 | mulcomli 11220 | . . . . 5 ⊢ (4 · 7) = ;28 |
47 | 22, 2, 20, 25, 10, 6, 43, 46 | decmul1c 12739 | . . . 4 ⊢ (;34 · 7) = ;;238 |
48 | 21, 2, 22, 23, 10, 24, 38, 47 | decmul2c 12740 | . . 3 ⊢ (;34 · ;37) = ;;;1258 |
49 | 19, 48 | eqtr4i 2755 | . 2 ⊢ (𝑁 − 1) = (;34 · ;37) |
50 | 9nn0 12493 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
51 | 9, 50 | deccl 12689 | . . . . . 6 ⊢ ;;;1259 ∈ ℕ0 |
52 | 14, 51 | eqeltri 2821 | . . . . 5 ⊢ 𝑁 ∈ ℕ0 |
53 | 52 | nn0cni 12481 | . . . 4 ⊢ 𝑁 ∈ ℂ |
54 | npcan 11466 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
55 | 53, 13, 54 | mp2an 689 | . . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁 |
56 | 55 | eqcomi 2733 | . 2 ⊢ 𝑁 = ((𝑁 − 1) + 1) |
57 | 1nn 12220 | . 2 ⊢ 1 ∈ ℕ | |
58 | 2nn 12282 | . 2 ⊢ 2 ∈ ℕ | |
59 | 2, 22 | deccl 12689 | . . . . 5 ⊢ ;37 ∈ ℕ0 |
60 | 59 | numexp1 17009 | . . . 4 ⊢ (;37↑1) = ;37 |
61 | 60 | oveq2i 7412 | . . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37) |
62 | 49, 61 | eqtr4i 2755 | . 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1)) |
63 | 7nn 12301 | . . . 4 ⊢ 7 ∈ ℕ | |
64 | 4lt7 12397 | . . . 4 ⊢ 4 < 7 | |
65 | 2, 20, 63, 64 | declt 12702 | . . 3 ⊢ ;34 < ;37 |
66 | 65, 60 | breqtrri 5165 | . 2 ⊢ ;34 < (;37↑1) |
67 | 14 | 1259lem4 17066 | . 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁) |
68 | 14 | 1259lem5 17067 | . 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1 |
69 | 1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68 | pockthi 16839 | 1 ⊢ 𝑁 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7401 ℂcc 11104 1c1 11107 + caddc 11109 · cmul 11111 < clt 11245 − cmin 11441 2c2 12264 3c3 12265 4c4 12266 5c5 12267 7c7 12269 8c8 12270 9c9 12271 ℕ0cn0 12469 ;cdc 12674 ↑cexp 14024 ℙcprime 16605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-dju 9892 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-xnn0 12542 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-dvds 16195 df-gcd 16433 df-prm 16606 df-odz 16697 df-phi 16698 df-pc 16769 |
This theorem is referenced by: (None) |
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