1259prm - Metamath Proof Explorer

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Theorem 1259prm 16765

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 16750	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12181	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 11986	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12386	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12179	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12180	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12381	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12183	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12381	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12186	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12381	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12175	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 10860	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2738	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12053	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12426	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2769	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11168	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12182	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12381	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12185	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2738	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12381	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2738	. . . . 5 ⊢ ;34 = ;34
26		eqid 2738	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12070	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12045	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7267	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12454	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2766	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12464	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 11984	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 11978	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12054	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11097	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12426	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12418	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 11997	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12476	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 10915	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12046	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12426	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 11988	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12477	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 10915	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12431	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12432	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2769	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12187	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12381	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2835	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12175	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11160	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 688	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2747	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 11914	. 2 ⊢ 1 ∈ ℕ
58		2nn 11976	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12381	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 16706	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7266	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2769	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 11995	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12091	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12394	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5097	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 16763	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 16764	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16536	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2108 (class class class)co 7255 ℂcc 10800 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 − cmin 11135 2c2 11958 3c3 11959 4c4 11960 5c5 11961 7c7 11963 8c8 11964 9c9 11965 ℕ₀cn0 12163 cdc 12366 ↑cexp 13710 ℙcprime 16304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-prm 16305 df-odz 16394 df-phi 16395 df-pc 16466

This theorem is referenced by: (None)

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