1259prm - Metamath Proof Explorer

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Theorem 1259prm 16469

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 16454	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 11916	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 11721	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12119	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 11914	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 11915	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12114	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 11918	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12114	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 11921	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12114	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 11910	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 10595	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2821	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 11788	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12159	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2847	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 10903	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 11917	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12114	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 11920	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2821	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12114	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2821	. . . . 5 ⊢ ;34 = ;34
26		eqid 2821	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 11805	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 11780	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7168	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12187	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2844	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12197	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 11719	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 11713	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 11789	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 10832	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12159	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12151	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 11732	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12209	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 10650	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 11781	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12159	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 11723	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12210	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 10650	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12164	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12165	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2847	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 11922	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12114	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2909	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 11910	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 10895	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 690	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2830	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 11649	. 2 ⊢ 1 ∈ ℕ
58		2nn 11711	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12114	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 16413	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7167	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2847	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 11730	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 11826	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12127	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5093	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 16467	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 16468	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16243	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 − cmin 10870 2c2 11693 3c3 11694 4c4 11695 5c5 11696 7c7 11698 8c8 11699 9c9 11700 ℕ₀cn0 11898 cdc 12099 ↑cexp 13430 ℙcprime 16015

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 df-prm 16016 df-odz 16102 df-phi 16103 df-pc 16174

This theorem is referenced by: (None)

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