1259prm - Metamath Proof Explorer

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Theorem 1259prm 17163

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 17148	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12493	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12295	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12706	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12491	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12492	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12697	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12495	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12697	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12498	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12697	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12487	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11125	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2761	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12361	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12747	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2787	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11441	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12494	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12697	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12497	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2761	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12697	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2761	. . . . 5 ⊢ ;34 = ;34
26		eqid 2761	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12379	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12353	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7403	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12775	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2784	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12785	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12293	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12287	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12362	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11369	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12747	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12739	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12306	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12797	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11185	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12354	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12747	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12297	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12798	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11185	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12752	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12753	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2787	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12499	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12697	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2857	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12487	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11433	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 702	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2770	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12215	. 2 ⊢ 1 ∈ ℕ
58		2nn 12285	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12697	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 17103	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7402	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2787	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12304	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12402	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12715	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5124	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17161	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17162	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16934	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1559 ∈ wcel 2141 (class class class)co 7391 ℂcc 11065 1c1 11068 + caddc 11070 · cmul 11072 < clt 11210 − cmin 11408 2c2 12266 3c3 12267 4c4 12268 5c5 12269 7c7 12271 8c8 12272 9c9 12273 ℕ₀cn0 12475 cdc 12682 ↑cexp 14068 ℙcprime 16696

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-oadd 8435 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9382 df-inf 9383 df-dju 9853 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-xnn0 12549 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-fz 13507 df-fzo 13654 df-fl 13796 df-mod 13874 df-seq 14009 df-exp 14069 df-hash 14338 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-dvds 16278 df-gcd 16520 df-prm 16697 df-odz 16791 df-phi 16792 df-pc 16864

This theorem is referenced by: (None)

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