1259prm - Metamath Proof Explorer

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Theorem 1259prm 17169

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 17154	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12541	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12346	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12750	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12539	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12540	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12745	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12543	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12745	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12546	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12745	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12535	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11210	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2734	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12413	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12790	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2765	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11522	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12542	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12745	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12545	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2734	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12745	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2734	. . . . 5 ⊢ ;34 = ;34
26		eqid 2734	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12430	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12405	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7442	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12818	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2762	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12828	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12344	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12338	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12414	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11450	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12790	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12782	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12357	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12840	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11267	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12406	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12790	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12348	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12841	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11267	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12795	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12796	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2765	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12547	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12745	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2834	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12535	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11514	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2743	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12274	. 2 ⊢ 1 ∈ ℕ
58		2nn 12336	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12745	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17110	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7441	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2765	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12355	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12451	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12758	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5174	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17167	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17168	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16940	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1536 ∈ wcel 2105 (class class class)co 7430 ℂcc 11150 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 − cmin 11489 2c2 12318 3c3 12319 4c4 12320 5c5 12321 7c7 12323 8c8 12324 9c9 12325 ℕ₀cn0 12523 cdc 12730 ↑cexp 14098 ℙcprime 16704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-gcd 16528 df-prm 16705 df-odz 16798 df-phi 16799 df-pc 16870

This theorem is referenced by: (None)

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