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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 16226 | . 2 ⊢ ;37 ∈ ℙ | |
2 | 3nn0 11662 | . . 3 ⊢ 3 ∈ ℕ0 | |
3 | 4nn 11459 | . . 3 ⊢ 4 ∈ ℕ | |
4 | 2, 3 | decnncl 11866 | . 2 ⊢ ;34 ∈ ℕ |
5 | 1nn0 11660 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
6 | 2nn0 11661 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
7 | 5, 6 | deccl 11860 | . . . . . . 7 ⊢ ;12 ∈ ℕ0 |
8 | 5nn0 11664 | . . . . . . 7 ⊢ 5 ∈ ℕ0 | |
9 | 7, 8 | deccl 11860 | . . . . . 6 ⊢ ;;125 ∈ ℕ0 |
10 | 8nn0 11667 | . . . . . 6 ⊢ 8 ∈ ℕ0 | |
11 | 9, 10 | deccl 11860 | . . . . 5 ⊢ ;;;1258 ∈ ℕ0 |
12 | 11 | nn0cni 11655 | . . . 4 ⊢ ;;;1258 ∈ ℂ |
13 | ax-1cn 10330 | . . . 4 ⊢ 1 ∈ ℂ | |
14 | 1259prm.1 | . . . . 5 ⊢ 𝑁 = ;;;1259 | |
15 | eqid 2778 | . . . . . 6 ⊢ ;;;1258 = ;;;1258 | |
16 | 8p1e9 11532 | . . . . . 6 ⊢ (8 + 1) = 9 | |
17 | 9, 10, 5, 15, 16 | decaddi 11906 | . . . . 5 ⊢ (;;;1258 + 1) = ;;;1259 |
18 | 14, 17 | eqtr4i 2805 | . . . 4 ⊢ 𝑁 = (;;;1258 + 1) |
19 | 12, 13, 18 | mvrraddi 10640 | . . 3 ⊢ (𝑁 − 1) = ;;;1258 |
20 | 4nn0 11663 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
21 | 2, 20 | deccl 11860 | . . . 4 ⊢ ;34 ∈ ℕ0 |
22 | 7nn0 11666 | . . . 4 ⊢ 7 ∈ ℕ0 | |
23 | eqid 2778 | . . . 4 ⊢ ;37 = ;37 | |
24 | 6, 2 | deccl 11860 | . . . 4 ⊢ ;23 ∈ ℕ0 |
25 | eqid 2778 | . . . . 5 ⊢ ;34 = ;34 | |
26 | eqid 2778 | . . . . 5 ⊢ ;23 = ;23 | |
27 | 3t3e9 11549 | . . . . . . 7 ⊢ (3 · 3) = 9 | |
28 | 2p1e3 11524 | . . . . . . 7 ⊢ (2 + 1) = 3 | |
29 | 27, 28 | oveq12i 6934 | . . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3) |
30 | 9p3e12 11935 | . . . . . 6 ⊢ (9 + 3) = ;12 | |
31 | 29, 30 | eqtri 2802 | . . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12 |
32 | 4t3e12 11945 | . . . . . 6 ⊢ (4 · 3) = ;12 | |
33 | 3cn 11456 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
34 | 2cn 11450 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
35 | 3p2e5 11533 | . . . . . . 7 ⊢ (3 + 2) = 5 | |
36 | 33, 34, 35 | addcomli 10568 | . . . . . 6 ⊢ (2 + 3) = 5 |
37 | 5, 6, 2, 32, 36 | decaddi 11906 | . . . . 5 ⊢ ((4 · 3) + 3) = ;15 |
38 | 2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37 | decmac 11898 | . . . 4 ⊢ ((;34 · 3) + ;23) = ;;125 |
39 | 7cn 11473 | . . . . . . 7 ⊢ 7 ∈ ℂ | |
40 | 7t3e21 11957 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
41 | 39, 33, 40 | mulcomli 10386 | . . . . . 6 ⊢ (3 · 7) = ;21 |
42 | 1p2e3 11525 | . . . . . 6 ⊢ (1 + 2) = 3 | |
43 | 6, 5, 6, 41, 42 | decaddi 11906 | . . . . 5 ⊢ ((3 · 7) + 2) = ;23 |
44 | 4cn 11461 | . . . . . 6 ⊢ 4 ∈ ℂ | |
45 | 7t4e28 11958 | . . . . . 6 ⊢ (7 · 4) = ;28 | |
46 | 39, 44, 45 | mulcomli 10386 | . . . . 5 ⊢ (4 · 7) = ;28 |
47 | 22, 2, 20, 25, 10, 6, 43, 46 | decmul1c 11912 | . . . 4 ⊢ (;34 · 7) = ;;238 |
48 | 21, 2, 22, 23, 10, 24, 38, 47 | decmul2c 11913 | . . 3 ⊢ (;34 · ;37) = ;;;1258 |
49 | 19, 48 | eqtr4i 2805 | . 2 ⊢ (𝑁 − 1) = (;34 · ;37) |
50 | 9nn0 11668 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
51 | 9, 50 | deccl 11860 | . . . . . 6 ⊢ ;;;1259 ∈ ℕ0 |
52 | 14, 51 | eqeltri 2855 | . . . . 5 ⊢ 𝑁 ∈ ℕ0 |
53 | 52 | nn0cni 11655 | . . . 4 ⊢ 𝑁 ∈ ℂ |
54 | npcan 10632 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
55 | 53, 13, 54 | mp2an 682 | . . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁 |
56 | 55 | eqcomi 2787 | . 2 ⊢ 𝑁 = ((𝑁 − 1) + 1) |
57 | 1nn 11387 | . 2 ⊢ 1 ∈ ℕ | |
58 | 2nn 11448 | . 2 ⊢ 2 ∈ ℕ | |
59 | 2, 22 | deccl 11860 | . . . . 5 ⊢ ;37 ∈ ℕ0 |
60 | 59 | numexp1 16185 | . . . 4 ⊢ (;37↑1) = ;37 |
61 | 60 | oveq2i 6933 | . . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37) |
62 | 49, 61 | eqtr4i 2805 | . 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1)) |
63 | 7nn 11471 | . . . 4 ⊢ 7 ∈ ℕ | |
64 | 4lt7 11570 | . . . 4 ⊢ 4 < 7 | |
65 | 2, 20, 63, 64 | declt 11874 | . . 3 ⊢ ;34 < ;37 |
66 | 65, 60 | breqtrri 4913 | . 2 ⊢ ;34 < (;37↑1) |
67 | 14 | 1259lem4 16239 | . 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁) |
68 | 14 | 1259lem5 16240 | . 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1 |
69 | 1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68 | pockthi 16015 | 1 ⊢ 𝑁 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 1c1 10273 + caddc 10275 · cmul 10277 < clt 10411 − cmin 10606 2c2 11430 3c3 11431 4c4 11432 5c5 11433 7c7 11435 8c8 11436 9c9 11437 ℕ0cn0 11642 ;cdc 11845 ↑cexp 13178 ℙcprime 15790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-xnn0 11715 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-dvds 15388 df-gcd 15623 df-prm 15791 df-odz 15874 df-phi 15875 df-pc 15946 |
This theorem is referenced by: (None) |
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