1259prm - Metamath Proof Explorer

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Theorem 1259prm 17097

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 17082	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12446	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12255	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12655	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12444	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12445	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12650	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12448	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12650	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12451	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12650	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12440	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11087	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2739	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12317	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12695	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2765	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11401	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12447	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12650	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12450	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2739	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12650	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2739	. . . . 5 ⊢ ;34 = ;34
26		eqid 2739	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12334	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12309	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7368	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12723	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2762	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12733	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12253	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12247	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12318	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11329	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12695	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12687	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12266	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12745	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11145	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12310	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12695	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12257	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12746	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11145	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12700	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12701	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2765	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12452	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12650	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2835	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12440	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11393	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 698	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2748	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12176	. 2 ⊢ 1 ∈ ℕ
58		2nn 12245	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12650	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17038	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7367	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2765	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12264	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12355	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12663	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5099	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17095	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17096	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16869	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1547 ∈ wcel 2119 (class class class)co 7356 ℂcc 11027 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 − cmin 11368 2c2 12227 3c3 12228 4c4 12229 5c5 12230 7c7 12232 8c8 12233 9c9 12234 ℕ₀cn0 12428 cdc 12635 ↑cexp 14014 ℙcprime 16631

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16213 df-gcd 16455 df-prm 16632 df-odz 16726 df-phi 16727 df-pc 16799

This theorem is referenced by: (None)

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