1259prm - Metamath Proof Explorer

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Theorem 1259prm 17228

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 17213	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12546	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12348	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12760	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12544	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12545	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12751	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12548	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12751	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12551	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12751	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12540	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11182	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2760	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12414	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12801	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2786	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11498	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12547	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12751	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12550	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2760	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12751	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2760	. . . . 5 ⊢ ;34 = ;34
26		eqid 2760	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12432	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12406	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7425	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12829	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2783	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12839	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12346	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12340	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12415	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11426	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12801	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12793	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12359	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12851	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11242	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12407	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12801	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12350	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12852	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11242	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12806	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12807	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2786	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12552	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12751	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2856	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12540	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11490	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 705	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2769	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12268	. 2 ⊢ 1 ∈ ℕ
58		2nn 12338	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12751	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17168	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7424	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2786	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12357	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12455	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12769	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5132	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17226	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17227	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16999	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2145 (class class class)co 7413 ℂcc 11122 1c1 11125 + caddc 11127 · cmul 11129 < clt 11267 − cmin 11465 2c2 12319 3c3 12320 4c4 12321 5c5 12322 7c7 12324 8c8 12325 9c9 12326 ℕ₀cn0 12528 cdc 12736 ↑cexp 14125 ℙcprime 16761

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-dju 9906 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-dvds 16343 df-gcd 16585 df-prm 16762 df-odz 16856 df-phi 16857 df-pc 16929

This theorem is used by: (None)

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