1259prm - Metamath Proof Explorer

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Theorem 1259prm 17321

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 17306	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12629	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12431	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12843	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12627	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12628	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12834	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12631	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12834	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12634	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12834	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12623	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11263	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2760	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12497	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12884	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2786	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11579	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12630	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12834	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12633	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2760	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12834	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2760	. . . . 5 ⊢ ;34 = ;34
26		eqid 2760	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12515	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12489	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7434	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12912	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2783	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12922	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12429	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12423	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12498	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11507	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12884	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12876	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12442	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12934	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11323	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12490	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12884	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12433	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12935	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11323	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12889	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12890	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2786	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12635	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12834	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2856	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12623	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11571	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 705	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2769	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12351	. 2 ⊢ 1 ∈ ℕ
58		2nn 12421	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12834	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17261	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7433	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2786	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12440	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12538	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12852	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5131	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17319	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17320	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 17092	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1569 ∈ wcel 2144 (class class class)co 7422 ℂcc 11203 1c1 11206 + caddc 11208 · cmul 11210 < clt 11348 − cmin 11546 2c2 12402 3c3 12403 4c4 12404 5c5 12405 7c7 12407 8c8 12408 9c9 12409 ℕ₀cn0 12611 cdc 12819 ↑cexp 14209 ℙcprime 16853

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-pre-sup 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-om 7880 df-1st 8003 df-2nd 8004 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-2o 8482 df-oadd 8485 df-er 8722 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-sup 9439 df-inf 9440 df-dju 9987 df-card 10025 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-div 11979 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-7 12415 df-8 12416 df-9 12417 df-n0 12612 df-xnn0 12685 df-z 12699 df-dec 12820 df-uz 12971 df-q 13081 df-rp 13126 df-fz 13645 df-fzo 13794 df-fl 13937 df-mod 14015 df-seq 14150 df-exp 14210 df-hash 14480 df-cj 15273 df-re 15274 df-im 15275 df-sqrt 15409 df-abs 15410 df-dvds 16430 df-gcd 16672 df-prm 16854 df-odz 16949 df-phi 16950 df-pc 17022

This theorem is used by: (None)

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