1259prm - Metamath Proof Explorer

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Theorem 1259prm 17106

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 17091	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12460	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12269	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12669	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12458	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12459	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12664	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12462	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12664	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12465	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12664	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12454	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11126	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2729	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12331	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12709	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2755	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11438	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12461	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12664	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12464	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2729	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12664	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2729	. . . . 5 ⊢ ;34 = ;34
26		eqid 2729	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12348	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12323	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7399	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12737	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2752	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12747	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12267	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12261	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12332	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11366	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12709	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12701	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12280	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12759	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11183	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12324	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12709	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12271	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12760	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11183	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12714	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12715	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2755	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12466	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12664	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2824	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12454	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11430	. . . 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 12197	. 2 ⊢ 1 ∈ ℕ
58		2nn 12259	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12664	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17047	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7398	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2755	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12278	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12369	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12677	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5134	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17104	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17105	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16878	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 (class class class)co 7387 ℂcc 11066 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 − cmin 11405 2c2 12241 3c3 12242 4c4 12243 5c5 12244 7c7 12246 8c8 12247 9c9 12248 ℕ₀cn0 12442 cdc 12649 ↑cexp 14026 ℙcprime 16641

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-gcd 16465 df-prm 16642 df-odz 16735 df-phi 16736 df-pc 16808

This theorem is referenced by: (None)

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