1259prm - Metamath Proof Explorer

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Theorem 1259prm 17153

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 17138	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12517	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12321	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12726	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12515	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12516	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12721	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12519	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12721	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12522	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12721	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12511	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11185	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2735	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12388	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12766	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2761	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11497	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12518	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12721	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12521	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2735	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12721	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2735	. . . . 5 ⊢ ;34 = ;34
26		eqid 2735	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12405	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12380	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7415	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12794	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2758	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12804	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12319	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12313	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12389	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11425	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12766	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12758	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12332	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12816	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11242	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12381	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12766	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12323	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12817	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11242	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12771	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12772	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2761	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12523	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12721	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2830	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12511	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11489	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2744	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12249	. 2 ⊢ 1 ∈ ℕ
58		2nn 12311	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12721	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17094	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7414	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2761	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12330	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12426	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12734	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5146	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17151	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17152	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16925	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7403 ℂcc 11125 1c1 11128 + caddc 11130 · cmul 11132 < clt 11267 − cmin 11464 2c2 12293 3c3 12294 4c4 12295 5c5 12296 7c7 12298 8c8 12299 9c9 12300 ℕ₀cn0 12499 cdc 12706 ↑cexp 14077 ℙcprime 16688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9452 df-inf 9453 df-dju 9913 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-hash 14347 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-dvds 16271 df-gcd 16512 df-prm 16689 df-odz 16782 df-phi 16783 df-pc 16855

This theorem is referenced by: (None)

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