1259prm - Metamath Proof Explorer

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Theorem 1259prm 17042

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 17027	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12394	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12203	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12603	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12392	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12393	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12598	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12396	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12598	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12399	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12598	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12388	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11059	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2731	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12265	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12643	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2757	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11372	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12395	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12598	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12398	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2731	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12598	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2731	. . . . 5 ⊢ ;34 = ;34
26		eqid 2731	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12282	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12257	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7353	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12671	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2754	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12681	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12201	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12195	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12266	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11300	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12643	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12635	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12214	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12693	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11116	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12258	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12643	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12205	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12694	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11116	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12648	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12649	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2757	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12400	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12598	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2827	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12388	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11364	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2740	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12131	. 2 ⊢ 1 ∈ ℕ
58		2nn 12193	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12598	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 16983	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7352	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2757	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12212	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12303	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12611	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5113	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17040	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17041	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16814	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2111 (class class class)co 7341 ℂcc 10999 1c1 11002 + caddc 11004 · cmul 11006 < clt 11141 − cmin 11339 2c2 12175 3c3 12176 4c4 12177 5c5 12178 7c7 12180 8c8 12181 9c9 12182 ℕ₀cn0 12376 cdc 12583 ↑cexp 13963 ℙcprime 16577

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-sup 9321 df-inf 9322 df-dju 9789 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-xnn0 12450 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-fz 13403 df-fzo 13550 df-fl 13691 df-mod 13769 df-seq 13904 df-exp 13964 df-hash 14233 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-dvds 16159 df-gcd 16401 df-prm 16578 df-odz 16671 df-phi 16672 df-pc 16744

This theorem is referenced by: (None)

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