1259prm - Metamath Proof Explorer

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Theorem 1259prm 17192

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 17177	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12518	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12320	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12731	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12516	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12517	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12722	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12520	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12722	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12523	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12722	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12512	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11154	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2769	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12386	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12772	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2795	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11470	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12519	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12722	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12522	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2769	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12722	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2769	. . . . 5 ⊢ ;34 = ;34
26		eqid 2769	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12404	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12378	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7420	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12800	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2792	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12810	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12318	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12312	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12387	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11398	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12772	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12764	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12331	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12822	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11214	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12379	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12772	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12322	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12823	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11214	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12777	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12778	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2795	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12524	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12722	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2865	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12512	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11462	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 704	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2778	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12240	. 2 ⊢ 1 ∈ ℕ
58		2nn 12310	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12722	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17132	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7419	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2795	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12329	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12427	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12740	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5139	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17190	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17191	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16963	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1567 ∈ wcel 2149 (class class class)co 7408 ℂcc 11094 1c1 11097 + caddc 11099 · cmul 11101 < clt 11239 − cmin 11437 2c2 12291 3c3 12292 4c4 12293 5c5 12294 7c7 12296 8c8 12297 9c9 12298 ℕ₀cn0 12500 cdc 12707 ↑cexp 14093 ℙcprime 16725

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-dju 9883 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-xnn0 12574 df-z 12588 df-dec 12708 df-uz 12859 df-q 12969 df-rp 13013 df-fz 13532 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-dvds 16307 df-gcd 16549 df-prm 16726 df-odz 16820 df-phi 16821 df-pc 16893

This theorem is referenced by: (None)

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