1259prm - Metamath Proof Explorer

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Theorem 1259prm 17106

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 17091	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12455	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12264	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12664	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12453	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12454	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12659	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12457	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12659	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12460	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12659	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12449	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11096	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2736	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12326	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12704	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2762	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11410	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12456	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12659	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12459	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2736	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12659	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2736	. . . . 5 ⊢ ;34 = ;34
26		eqid 2736	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12343	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12318	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7379	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12732	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2759	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12742	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12262	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12256	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12327	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11338	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12704	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12696	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12275	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12754	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11154	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12319	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12704	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12266	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12755	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11154	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12709	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12710	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2762	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12461	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12659	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2832	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12449	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11402	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 693	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2745	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12185	. 2 ⊢ 1 ∈ ℕ
58		2nn 12254	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12659	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17047	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7378	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2762	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12273	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12364	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12672	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5112	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17104	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17105	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16878	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 (class class class)co 7367 ℂcc 11036 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 − cmin 11377 2c2 12236 3c3 12237 4c4 12238 5c5 12239 7c7 12241 8c8 12242 9c9 12243 ℕ₀cn0 12437 cdc 12644 ↑cexp 14023 ℙcprime 16640

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-dvds 16222 df-gcd 16464 df-prm 16641 df-odz 16735 df-phi 16736 df-pc 16808

This theorem is referenced by: (None)

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