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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 17158 | . 2 ⊢ ;37 ∈ ℙ | |
| 2 | 3nn0 12544 | . . 3 ⊢ 3 ∈ ℕ0 | |
| 3 | 4nn 12349 | . . 3 ⊢ 4 ∈ ℕ | |
| 4 | 2, 3 | decnncl 12753 | . 2 ⊢ ;34 ∈ ℕ |
| 5 | 1nn0 12542 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
| 6 | 2nn0 12543 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
| 7 | 5, 6 | deccl 12748 | . . . . . . 7 ⊢ ;12 ∈ ℕ0 |
| 8 | 5nn0 12546 | . . . . . . 7 ⊢ 5 ∈ ℕ0 | |
| 9 | 7, 8 | deccl 12748 | . . . . . 6 ⊢ ;;125 ∈ ℕ0 |
| 10 | 8nn0 12549 | . . . . . 6 ⊢ 8 ∈ ℕ0 | |
| 11 | 9, 10 | deccl 12748 | . . . . 5 ⊢ ;;;1258 ∈ ℕ0 |
| 12 | 11 | nn0cni 12538 | . . . 4 ⊢ ;;;1258 ∈ ℂ |
| 13 | ax-1cn 11213 | . . . 4 ⊢ 1 ∈ ℂ | |
| 14 | 1259prm.1 | . . . . 5 ⊢ 𝑁 = ;;;1259 | |
| 15 | eqid 2737 | . . . . . 6 ⊢ ;;;1258 = ;;;1258 | |
| 16 | 8p1e9 12416 | . . . . . 6 ⊢ (8 + 1) = 9 | |
| 17 | 9, 10, 5, 15, 16 | decaddi 12793 | . . . . 5 ⊢ (;;;1258 + 1) = ;;;1259 |
| 18 | 14, 17 | eqtr4i 2768 | . . . 4 ⊢ 𝑁 = (;;;1258 + 1) |
| 19 | 12, 13, 18 | mvrraddi 11525 | . . 3 ⊢ (𝑁 − 1) = ;;;1258 |
| 20 | 4nn0 12545 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
| 21 | 2, 20 | deccl 12748 | . . . 4 ⊢ ;34 ∈ ℕ0 |
| 22 | 7nn0 12548 | . . . 4 ⊢ 7 ∈ ℕ0 | |
| 23 | eqid 2737 | . . . 4 ⊢ ;37 = ;37 | |
| 24 | 6, 2 | deccl 12748 | . . . 4 ⊢ ;23 ∈ ℕ0 |
| 25 | eqid 2737 | . . . . 5 ⊢ ;34 = ;34 | |
| 26 | eqid 2737 | . . . . 5 ⊢ ;23 = ;23 | |
| 27 | 3t3e9 12433 | . . . . . . 7 ⊢ (3 · 3) = 9 | |
| 28 | 2p1e3 12408 | . . . . . . 7 ⊢ (2 + 1) = 3 | |
| 29 | 27, 28 | oveq12i 7443 | . . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3) |
| 30 | 9p3e12 12821 | . . . . . 6 ⊢ (9 + 3) = ;12 | |
| 31 | 29, 30 | eqtri 2765 | . . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12 |
| 32 | 4t3e12 12831 | . . . . . 6 ⊢ (4 · 3) = ;12 | |
| 33 | 3cn 12347 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
| 34 | 2cn 12341 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
| 35 | 3p2e5 12417 | . . . . . . 7 ⊢ (3 + 2) = 5 | |
| 36 | 33, 34, 35 | addcomli 11453 | . . . . . 6 ⊢ (2 + 3) = 5 |
| 37 | 5, 6, 2, 32, 36 | decaddi 12793 | . . . . 5 ⊢ ((4 · 3) + 3) = ;15 |
| 38 | 2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37 | decmac 12785 | . . . 4 ⊢ ((;34 · 3) + ;23) = ;;125 |
| 39 | 7cn 12360 | . . . . . . 7 ⊢ 7 ∈ ℂ | |
| 40 | 7t3e21 12843 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
| 41 | 39, 33, 40 | mulcomli 11270 | . . . . . 6 ⊢ (3 · 7) = ;21 |
| 42 | 1p2e3 12409 | . . . . . 6 ⊢ (1 + 2) = 3 | |
| 43 | 6, 5, 6, 41, 42 | decaddi 12793 | . . . . 5 ⊢ ((3 · 7) + 2) = ;23 |
| 44 | 4cn 12351 | . . . . . 6 ⊢ 4 ∈ ℂ | |
| 45 | 7t4e28 12844 | . . . . . 6 ⊢ (7 · 4) = ;28 | |
| 46 | 39, 44, 45 | mulcomli 11270 | . . . . 5 ⊢ (4 · 7) = ;28 |
| 47 | 22, 2, 20, 25, 10, 6, 43, 46 | decmul1c 12798 | . . . 4 ⊢ (;34 · 7) = ;;238 |
| 48 | 21, 2, 22, 23, 10, 24, 38, 47 | decmul2c 12799 | . . 3 ⊢ (;34 · ;37) = ;;;1258 |
| 49 | 19, 48 | eqtr4i 2768 | . 2 ⊢ (𝑁 − 1) = (;34 · ;37) |
| 50 | 9nn0 12550 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
| 51 | 9, 50 | deccl 12748 | . . . . . 6 ⊢ ;;;1259 ∈ ℕ0 |
| 52 | 14, 51 | eqeltri 2837 | . . . . 5 ⊢ 𝑁 ∈ ℕ0 |
| 53 | 52 | nn0cni 12538 | . . . 4 ⊢ 𝑁 ∈ ℂ |
| 54 | npcan 11517 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
| 55 | 53, 13, 54 | mp2an 692 | . . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁 |
| 56 | 55 | eqcomi 2746 | . 2 ⊢ 𝑁 = ((𝑁 − 1) + 1) |
| 57 | 1nn 12277 | . 2 ⊢ 1 ∈ ℕ | |
| 58 | 2nn 12339 | . 2 ⊢ 2 ∈ ℕ | |
| 59 | 2, 22 | deccl 12748 | . . . . 5 ⊢ ;37 ∈ ℕ0 |
| 60 | 59 | numexp1 17114 | . . . 4 ⊢ (;37↑1) = ;37 |
| 61 | 60 | oveq2i 7442 | . . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37) |
| 62 | 49, 61 | eqtr4i 2768 | . 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1)) |
| 63 | 7nn 12358 | . . . 4 ⊢ 7 ∈ ℕ | |
| 64 | 4lt7 12454 | . . . 4 ⊢ 4 < 7 | |
| 65 | 2, 20, 63, 64 | declt 12761 | . . 3 ⊢ ;34 < ;37 |
| 66 | 65, 60 | breqtrri 5170 | . 2 ⊢ ;34 < (;37↑1) |
| 67 | 14 | 1259lem4 17171 | . 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁) |
| 68 | 14 | 1259lem5 17172 | . 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1 |
| 69 | 1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68 | pockthi 16945 | 1 ⊢ 𝑁 ∈ ℙ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 − cmin 11492 2c2 12321 3c3 12322 4c4 12323 5c5 12324 7c7 12326 8c8 12327 9c9 12328 ℕ0cn0 12526 ;cdc 12733 ↑cexp 14102 ℙcprime 16708 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 df-odz 16802 df-phi 16803 df-pc 16875 |
| This theorem is referenced by: (None) |
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