1259prm - Metamath Proof Explorer

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Theorem 1259prm 17082

Description: 1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

**Hypothesis**
Ref	Expression
1259prm.1	⊢ 𝑁 = ;;;1259

**Assertion**
Ref	Expression
1259prm	⊢ 𝑁 ∈ ℙ

**Proof of Theorem 1259prm**
Step	Hyp	Ref	Expression
1		37prm 17067	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12436	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12245	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12645	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12434	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12435	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12640	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12438	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12640	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12441	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12640	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12430	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11102	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2729	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12307	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12685	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2755	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11414	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12437	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12640	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12440	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2729	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12640	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2729	. . . . 5 ⊢ ;34 = ;34
26		eqid 2729	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12324	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12299	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7381	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12713	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2752	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12723	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12243	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12237	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12308	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11342	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12685	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12677	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12256	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12735	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11159	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12300	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12685	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12247	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12736	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11159	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12690	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12691	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2755	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12442	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12640	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2824	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12430	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11406	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2738	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12173	. 2 ⊢ 1 ∈ ℕ
58		2nn 12235	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12640	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17023	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7380	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2755	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12254	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12345	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12653	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5129	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17080	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17081	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16854	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 (class class class)co 7369 ℂcc 11042 1c1 11045 + caddc 11047 · cmul 11049 < clt 11184 − cmin 11381 2c2 12217 3c3 12218 4c4 12219 5c5 12220 7c7 12222 8c8 12223 9c9 12224 ℕ₀cn0 12418 cdc 12625 ↑cexp 14002 ℙcprime 16617

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9369 df-inf 9370 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-dvds 16199 df-gcd 16441 df-prm 16618 df-odz 16711 df-phi 16712 df-pc 16784

This theorem is referenced by: (None)

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