1259prm - Metamath Proof Explorer

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Theorem 1259prm 17061

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 17046	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12417	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12226	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12625	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12415	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12416	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12620	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12419	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12620	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12422	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12620	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12411	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11082	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2734	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12288	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12665	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2760	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11395	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12418	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12620	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12421	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2734	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12620	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2734	. . . . 5 ⊢ ;34 = ;34
26		eqid 2734	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12305	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12280	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7368	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12693	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2757	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12703	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12224	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12218	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12289	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11323	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12665	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12657	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12237	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12715	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11139	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12281	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12665	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12228	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12716	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11139	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12670	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12671	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2760	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12423	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12620	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2830	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12411	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11387	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2743	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12154	. 2 ⊢ 1 ∈ ℕ
58		2nn 12216	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12620	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17002	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7367	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2760	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12235	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12326	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12633	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5123	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17059	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17060	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16833	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2113 (class class class)co 7356 ℂcc 11022 1c1 11025 + caddc 11027 · cmul 11029 < clt 11164 − cmin 11362 2c2 12198 3c3 12199 4c4 12200 5c5 12201 7c7 12203 8c8 12204 9c9 12205 ℕ₀cn0 12399 cdc 12605 ↑cexp 13982 ℙcprime 16596

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-inf 9344 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-xnn0 12473 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-dvds 16178 df-gcd 16420 df-prm 16597 df-odz 16690 df-phi 16691 df-pc 16763

This theorem is referenced by: (None)

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