1259prm - Metamath Proof Explorer

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Theorem 1259prm 16241

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 16226	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 11662	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 11459	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 11866	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 11660	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 11661	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 11860	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 11664	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 11860	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 11667	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 11860	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 11655	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 10330	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2778	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 11532	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 11906	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2805	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 10640	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 11663	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 11860	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 11666	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2778	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 11860	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2778	. . . . 5 ⊢ ;34 = ;34
26		eqid 2778	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 11549	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 11524	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 6934	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 11935	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2802	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 11945	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 11456	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 11450	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 11533	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 10568	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 11906	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 11898	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 11473	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 11957	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 10386	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 11525	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 11906	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 11461	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 11958	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 10386	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 11912	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 11913	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2805	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 11668	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 11860	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2855	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 11655	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 10632	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 682	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2787	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 11387	. 2 ⊢ 1 ∈ ℕ
58		2nn 11448	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 11860	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 16185	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 6933	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2805	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 11471	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 11570	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 11874	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 4913	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 16239	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 16240	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16015	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 1c1 10273 + caddc 10275 · cmul 10277 < clt 10411 − cmin 10606 2c2 11430 3c3 11431 4c4 11432 5c5 11433 7c7 11435 8c8 11436 9c9 11437 ℕ₀cn0 11642 cdc 11845 ↑cexp 13178 ℙcprime 15790

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-xnn0 11715 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-dvds 15388 df-gcd 15623 df-prm 15791 df-odz 15874 df-phi 15875 df-pc 15946

This theorem is referenced by: (None)

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