1259prm - Metamath Proof Explorer

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Theorem 1259prm 17063

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 17048	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12419	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12228	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12627	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12417	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12418	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12622	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12421	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12622	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12424	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12622	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12413	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11084	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2736	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12290	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12667	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2762	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11397	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12420	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12622	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12423	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2736	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12622	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2736	. . . . 5 ⊢ ;34 = ;34
26		eqid 2736	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12307	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12282	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7370	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12695	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2759	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12705	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12226	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12220	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12291	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11325	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12667	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12659	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12239	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12717	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11141	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12283	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12667	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12230	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12718	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11141	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12672	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12673	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2762	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12425	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12622	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2832	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12413	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11389	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2745	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12156	. 2 ⊢ 1 ∈ ℕ
58		2nn 12218	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12622	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17004	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7369	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2762	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12237	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12328	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12635	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5125	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17061	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17062	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16835	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2113 (class class class)co 7358 ℂcc 11024 1c1 11027 + caddc 11029 · cmul 11031 < clt 11166 − cmin 11364 2c2 12200 3c3 12201 4c4 12202 5c5 12203 7c7 12205 8c8 12206 9c9 12207 ℕ₀cn0 12401 cdc 12607 ↑cexp 13984 ℙcprime 16598

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 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 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-dvds 16180 df-gcd 16422 df-prm 16599 df-odz 16692 df-phi 16693 df-pc 16765

This theorem is referenced by: (None)

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