1259prm - Metamath Proof Explorer

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Theorem 1259prm 17068

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 17053	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12487	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12292	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12694	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12485	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12486	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12689	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12489	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12689	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12492	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12689	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12481	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11164	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2724	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12359	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12734	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2755	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11474	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12488	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12689	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12491	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2724	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12689	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2724	. . . . 5 ⊢ ;34 = ;34
26		eqid 2724	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12376	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12351	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7413	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12762	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2752	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12772	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12290	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12284	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12360	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11403	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12734	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12726	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12303	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12784	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11220	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12352	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12734	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12294	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12785	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11220	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12739	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12740	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2755	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12493	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12689	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2821	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12481	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11466	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 689	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2733	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12220	. 2 ⊢ 1 ∈ ℕ
58		2nn 12282	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12689	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17009	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7412	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2755	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12301	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12397	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12702	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5165	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17066	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17067	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16839	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7401 ℂcc 11104 1c1 11107 + caddc 11109 · cmul 11111 < clt 11245 − cmin 11441 2c2 12264 3c3 12265 4c4 12266 5c5 12267 7c7 12269 8c8 12270 9c9 12271 ℕ₀cn0 12469 cdc 12674 ↑cexp 14024 ℙcprime 16605

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-dju 9892 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-xnn0 12542 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-dvds 16195 df-gcd 16433 df-prm 16606 df-odz 16697 df-phi 16698 df-pc 16769

This theorem is referenced by: (None)

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