1259prm - Intuitionistic Logic Explorer

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Theorem 1259prm 13276

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 13264	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 9592	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 9479	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 9811	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 9590	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 9591	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 9802	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 9594	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 9802	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 9597	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 9802	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 9586	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 8278	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2238	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 9454	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 9852	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2262	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 8550	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 9593	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 9802	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 9596	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2238	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 9802	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2238	. . . . 5 ⊢ ;34 = ;34
26		eqid 2238	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 9472	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 9447	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 6100	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 9880	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2259	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 9890	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 9388	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 9384	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 9455	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 8478	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 9852	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 9844	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 9397	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 9902	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 8339	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 9448	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 9852	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 9391	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 9903	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 8339	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 9857	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 9858	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2262	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 9598	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 9802	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2311	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 9586	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 8542	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 430	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2242	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 9324	. 2 ⊢ 1 ∈ ℕ
58		2nn 9477	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 9802	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 13232	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 6099	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2262	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 9482	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 9502	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 9820	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 4158	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 13274	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 13275	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 13166	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff set class

This proof depends on syntax axioms: = wceq 1402 ∈ wcel 2209 (class class class)co 6088 ℂcc 8183 1c1 8186 + caddc 8188 · cmul 8190 < clt 8366 − cmin 8504 2c2 9364 3c3 9365 4c4 9366 5c5 9367 7c7 9369 8c8 9370 9c9 9371 ℕ₀cn0 9574 cdc 9788 ↑cexp 10996 ℙcprime 12910

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-2o 6692 df-oadd 6695 df-er 6811 df-en 7027 df-dom 7028 df-fin 7029 df-sup 7330 df-inf 7331 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-5 9375 df-6 9376 df-7 9377 df-8 9378 df-9 9379 df-n0 9575 df-xnn0 9642 df-z 9656 df-dec 9789 df-uz 9938 df-q 10036 df-rp 10072 df-fz 10429 df-fzo 10567 df-fl 10722 df-mod 10781 df-seqfrec 10906 df-exp 10997 df-ihash 11237 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-clim 12070 df-proddc 12343 df-dvds 12580 df-gcd 12756 df-prm 12911 df-odz 13017 df-phi 13018 df-pc 13093

This theorem is used by: (None)

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