1259prm - Metamath Proof Explorer

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Theorem 1259prm 17075

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 17060	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12431	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12240	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12639	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12429	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12430	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12634	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12433	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12634	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12436	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12634	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12425	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11096	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2737	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12302	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12679	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2763	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11409	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12432	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12634	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12435	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2737	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12634	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2737	. . . . 5 ⊢ ;34 = ;34
26		eqid 2737	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12319	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12294	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7380	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12707	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2760	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12717	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12238	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12232	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12303	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11337	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12679	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12671	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12251	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12729	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11153	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12295	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12679	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12242	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12730	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11153	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12684	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12685	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2763	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12437	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12634	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2833	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12425	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11401	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 693	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2746	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12168	. 2 ⊢ 1 ∈ ℕ
58		2nn 12230	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12634	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17016	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7379	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2763	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12249	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12340	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12647	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5127	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17073	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17074	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16847	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 (class class class)co 7368 ℂcc 11036 1c1 11039 + caddc 11041 · cmul 11043 < clt 11178 − cmin 11376 2c2 12212 3c3 12213 4c4 12214 5c5 12215 7c7 12217 8c8 12218 9c9 12219 ℕ₀cn0 12413 cdc 12619 ↑cexp 13996 ℙcprime 16610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-dvds 16192 df-gcd 16434 df-prm 16611 df-odz 16704 df-phi 16705 df-pc 16777

This theorem is referenced by: (None)

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