1259prm - Metamath Proof Explorer

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Theorem 1259prm 16837

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 16822	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12251	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12056	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12457	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12249	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12250	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12452	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12253	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12452	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12256	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12452	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12245	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 10929	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2738	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12123	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12497	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2769	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11238	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12252	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12452	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12255	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2738	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12452	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2738	. . . . 5 ⊢ ;34 = ;34
26		eqid 2738	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12140	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12115	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7287	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12525	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2766	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12535	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12054	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12048	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12124	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11167	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12497	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12489	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12067	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12547	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 10984	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12116	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12497	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12058	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12548	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 10984	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12502	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12503	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2769	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12257	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12452	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2835	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12245	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11230	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 689	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2747	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 11984	. 2 ⊢ 1 ∈ ℕ
58		2nn 12046	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12452	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 16778	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7286	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2769	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12065	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12161	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12465	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5101	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 16835	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 16836	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16608	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2106 (class class class)co 7275 ℂcc 10869 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 − cmin 11205 2c2 12028 3c3 12029 4c4 12030 5c5 12031 7c7 12033 8c8 12034 9c9 12035 ℕ₀cn0 12233 cdc 12437 ↑cexp 13782 ℙcprime 16376

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-dvds 15964 df-gcd 16202 df-prm 16377 df-odz 16466 df-phi 16467 df-pc 16538

This theorem is referenced by: (None)

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