1259prm - Metamath Proof Explorer

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Theorem 1259prm 17066

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 17051	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12421	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12230	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12630	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12419	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12420	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12625	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12423	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12625	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12426	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12625	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12415	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11086	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2729	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12292	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12670	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2755	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11399	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12422	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12625	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12425	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2729	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12625	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2729	. . . . 5 ⊢ ;34 = ;34
26		eqid 2729	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12309	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12284	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7365	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12698	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2752	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12708	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12228	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12222	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12293	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11327	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12670	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12662	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12241	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12720	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11143	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12285	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12670	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12232	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12721	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11143	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12675	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12676	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2755	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12427	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12625	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2824	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12415	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11391	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2738	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12158	. 2 ⊢ 1 ∈ ℕ
58		2nn 12220	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12625	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17007	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7364	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2755	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12239	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12330	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12638	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5122	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17064	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17065	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16838	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 (class class class)co 7353 ℂcc 11026 1c1 11029 + caddc 11031 · cmul 11033 < clt 11168 − cmin 11366 2c2 12202 3c3 12203 4c4 12204 5c5 12205 7c7 12207 8c8 12208 9c9 12209 ℕ₀cn0 12403 cdc 12610 ↑cexp 13987 ℙcprime 16601

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 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 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-dju 9816 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-xnn0 12477 df-z 12491 df-dec 12611 df-uz 12755 df-q 12869 df-rp 12913 df-fz 13430 df-fzo 13577 df-fl 13715 df-mod 13793 df-seq 13928 df-exp 13988 df-hash 14257 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-dvds 16183 df-gcd 16425 df-prm 16602 df-odz 16695 df-phi 16696 df-pc 16768

This theorem is referenced by: (None)

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