1259prm - Metamath Proof Explorer

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Theorem 1259prm 17191

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 17176	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12517	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12319	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12730	. 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 11153	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2763	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12385	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12771	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2789	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11469	. . 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 2763	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12721	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2763	. . . . 5 ⊢ ;34 = ;34
26		eqid 2763	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12403	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12377	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7422	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12799	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2786	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12809	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12317	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12311	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12386	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11397	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12771	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12763	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12330	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12821	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11213	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12378	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12771	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12321	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12822	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11213	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12776	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12777	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2789	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12523	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12721	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2859	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12511	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11461	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 704	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2772	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12239	. 2 ⊢ 1 ∈ ℕ
58		2nn 12309	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12721	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17131	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7421	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2789	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12328	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12426	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12739	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5138	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17189	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17190	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16962	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1570 ∈ wcel 2143 (class class class)co 7410 ℂcc 11093 1c1 11096 + caddc 11098 · cmul 11100 < clt 11238 − cmin 11436 2c2 12290 3c3 12291 4c4 12292 5c5 12293 7c7 12295 8c8 12296 9c9 12297 ℕ₀cn0 12499 cdc 12706 ↑cexp 14093 ℙcprime 16724

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-dvds 16306 df-gcd 16548 df-prm 16725 df-odz 16819 df-phi 16820 df-pc 16892

This theorem is referenced by: (None)

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