1259prm - Metamath Proof Explorer

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Theorem 1259prm 17307

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 17292	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12617	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12419	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12831	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12615	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12616	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12822	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12619	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12822	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12622	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12822	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12611	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11251	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2761	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12485	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12872	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2787	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11567	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12618	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12822	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12621	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2761	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12822	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2761	. . . . 5 ⊢ ;34 = ;34
26		eqid 2761	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12503	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12477	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7430	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12900	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2784	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12910	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12417	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12411	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12486	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11495	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12872	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12864	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12430	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12922	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11311	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12478	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12872	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12421	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12923	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11311	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12877	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12878	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2787	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12623	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12822	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2857	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12611	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11559	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 705	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2770	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12339	. 2 ⊢ 1 ∈ ℕ
58		2nn 12409	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12822	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17247	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7429	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2787	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12428	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12526	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12840	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5132	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17305	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17306	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 17078	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2145 (class class class)co 7418 ℂcc 11191 1c1 11194 + caddc 11196 · cmul 11198 < clt 11336 − cmin 11534 2c2 12390 3c3 12391 4c4 12392 5c5 12393 7c7 12395 8c8 12396 9c9 12397 ℕ₀cn0 12599 cdc 12807 ↑cexp 14197 ℙcprime 16839

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-oadd 8473 df-er 8710 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9427 df-inf 9428 df-dju 9975 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-xnn0 12673 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-fz 13633 df-fzo 13782 df-fl 13925 df-mod 14003 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-dvds 16416 df-gcd 16658 df-prm 16840 df-odz 16935 df-phi 16936 df-pc 17008

This theorem is used by: (None)

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