1259prm - Metamath Proof Explorer

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Theorem 1259prm 17113

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 17098	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12467	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12276	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12676	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12465	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12466	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12671	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12469	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12671	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12472	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12671	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12461	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11133	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2730	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12338	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12716	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2756	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11445	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12468	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12671	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12471	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2730	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12671	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2730	. . . . 5 ⊢ ;34 = ;34
26		eqid 2730	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12355	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12330	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7402	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12744	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2753	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12754	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12274	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12268	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12339	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11373	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12716	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12708	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12287	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12766	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11190	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12331	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12716	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12278	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12767	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11190	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12721	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12722	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2756	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12473	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12671	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2825	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12461	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11437	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2739	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12204	. 2 ⊢ 1 ∈ ℕ
58		2nn 12266	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12671	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17054	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7401	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2756	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12285	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12376	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12684	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5137	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17111	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17112	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16885	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 (class class class)co 7390 ℂcc 11073 1c1 11076 + caddc 11078 · cmul 11080 < clt 11215 − cmin 11412 2c2 12248 3c3 12249 4c4 12250 5c5 12251 7c7 12253 8c8 12254 9c9 12255 ℕ₀cn0 12449 cdc 12656 ↑cexp 14033 ℙcprime 16648

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-xnn0 12523 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16230 df-gcd 16472 df-prm 16649 df-odz 16742 df-phi 16743 df-pc 16815

This theorem is referenced by: (None)

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