1259prm - Metamath Proof Explorer

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Theorem 1259prm 17173

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 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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