1259prm - Metamath Proof Explorer

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Theorem 1259prm 17183

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 17168	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12571	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12376	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12778	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12569	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12570	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12773	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12573	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12773	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12576	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12773	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12565	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11242	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2740	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12443	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12818	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2771	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11553	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12572	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12773	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12575	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2740	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12773	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2740	. . . . 5 ⊢ ;34 = ;34
26		eqid 2740	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12460	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12435	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7460	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12846	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2768	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12856	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12374	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12368	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12444	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11482	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12818	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12810	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12387	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12868	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11299	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12436	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12818	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12378	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12869	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11299	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12823	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12824	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2771	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12577	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12773	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2840	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12565	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11545	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 691	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2749	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12304	. 2 ⊢ 1 ∈ ℕ
58		2nn 12366	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12773	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17124	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7459	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2771	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12385	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12481	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12786	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5193	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17181	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17182	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16954	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2108 (class class class)co 7448 ℂcc 11182 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 − cmin 11520 2c2 12348 3c3 12349 4c4 12350 5c5 12351 7c7 12353 8c8 12354 9c9 12355 ℕ₀cn0 12553 cdc 12758 ↑cexp 14112 ℙcprime 16718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541 df-prm 16719 df-odz 16812 df-phi 16813 df-pc 16884

This theorem is referenced by: (None)

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