1259prm - Metamath Proof Explorer

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Theorem 1259prm 16461

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 16446	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 11903	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 11708	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12106	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 11901	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 11902	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12101	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 11905	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12101	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 11908	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12101	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 11897	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 10584	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2798	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 11775	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12146	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2824	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 10892	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 11904	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12101	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 11907	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2798	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12101	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2798	. . . . 5 ⊢ ;34 = ;34
26		eqid 2798	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 11792	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 11767	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7147	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12174	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2821	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12184	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 11706	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 11700	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 11776	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 10821	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12146	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12138	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 11719	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12196	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 10639	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 11768	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12146	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 11710	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12197	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 10639	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12151	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12152	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2824	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 11909	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12101	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2886	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 11897	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 10884	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 691	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2807	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 11636	. 2 ⊢ 1 ∈ ℕ
58		2nn 11698	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12101	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 16403	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7146	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2824	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 11717	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 11813	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12114	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5057	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 16459	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 16460	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16233	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 − cmin 10859 2c2 11680 3c3 11681 4c4 11682 5c5 11683 7c7 11685 8c8 11686 9c9 11687 ℕ₀cn0 11885 cdc 12086 ↑cexp 13425 ℙcprime 16005

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-prm 16006 df-odz 16092 df-phi 16093 df-pc 16164

This theorem is referenced by: (None)

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