1259prm - Metamath Proof Explorer

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Theorem 1259prm 17358

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 17343	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12664	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12466	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12878	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12662	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12663	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12869	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12666	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12869	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12669	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12869	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12658	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11297	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2763	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12532	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12919	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2789	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11613	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12665	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12869	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12668	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2763	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12869	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2763	. . . . 5 ⊢ ;34 = ;34
26		eqid 2763	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12550	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12524	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7463	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12947	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2786	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12957	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12464	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12458	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12533	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11541	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12919	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12911	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12477	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12969	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11357	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12525	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12919	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12468	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12970	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11357	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12924	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12925	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2789	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12670	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12869	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2860	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12658	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11605	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 705	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2772	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12386	. 2 ⊢ 1 ∈ ℕ
58		2nn 12456	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12869	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17298	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7462	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2789	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12475	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12573	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12887	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5138	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17356	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17357	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 17129	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2147 (class class class)co 7451 ℂcc 11237 1c1 11240 + caddc 11242 · cmul 11244 < clt 11382 − cmin 11580 2c2 12437 3c3 12438 4c4 12439 5c5 12440 7c7 12442 8c8 12443 9c9 12444 ℕ₀cn0 12646 cdc 12854 ↑cexp 14244 ℙcprime 16890

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-inf 9472 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-dvds 16467 df-gcd 16709 df-prm 16891 df-odz 16986 df-phi 16987 df-pc 17059

This theorem is used by: (None)

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