1259prm - Metamath Proof Explorer

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Theorem 1259prm 17218

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 17203	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12537	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12339	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12751	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12535	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12536	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12742	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12539	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12742	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12542	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12742	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12531	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11173	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2765	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12405	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12792	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2791	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11489	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12538	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12742	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12541	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2765	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12742	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2765	. . . . 5 ⊢ ;34 = ;34
26		eqid 2765	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12423	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12397	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7431	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12820	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2788	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12830	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12337	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12331	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12406	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11417	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12792	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12784	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12350	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12842	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11233	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12398	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12792	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12341	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12843	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11233	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12797	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12798	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2791	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12543	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12742	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2861	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12531	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11481	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 705	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2774	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12259	. 2 ⊢ 1 ∈ ℕ
58		2nn 12329	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12742	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17158	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7430	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2791	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12348	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12446	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12760	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5140	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17216	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17217	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16989	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2146 (class class class)co 7419 ℂcc 11113 1c1 11116 + caddc 11118 · cmul 11120 < clt 11258 − cmin 11456 2c2 12310 3c3 12311 4c4 12312 5c5 12313 7c7 12315 8c8 12316 9c9 12317 ℕ₀cn0 12519 cdc 12727 ↑cexp 14115 ℙcprime 16751

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-dju 9903 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-xnn0 12593 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-fz 13552 df-fzo 13700 df-fl 13843 df-mod 13921 df-seq 14056 df-exp 14116 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-dvds 16333 df-gcd 16575 df-prm 16752 df-odz 16846 df-phi 16847 df-pc 16919

This theorem is used by: (None)

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