1259prm - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 1259prm		Structured version Visualization version GIF version

Theorem 1259prm 17053

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 17038	. 2 ⊢ ;37 ∈ ℙ
2		3nn0 12405	. . 3 ⊢ 3 ∈ ℕ₀
3		4nn 12214	. . 3 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12614	. 2 ⊢ ;34 ∈ ℕ
5		1nn0 12403	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
6		2nn0 12404	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7	5, 6	deccl 12609	. . . . . . 7 ⊢ ;12 ∈ ℕ₀
8		5nn0 12407	. . . . . . 7 ⊢ 5 ∈ ℕ₀
9	7, 8	deccl 12609	. . . . . 6 ⊢ ;;125 ∈ ℕ₀
10		8nn0 12410	. . . . . 6 ⊢ 8 ∈ ℕ₀
11	9, 10	deccl 12609	. . . . 5 ⊢ ;;;1258 ∈ ℕ₀
12	11	nn0cni 12399	. . . 4 ⊢ ;;;1258 ∈ ℂ
13		ax-1cn 11070	. . . 4 ⊢ 1 ∈ ℂ
14		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
15		eqid 2731	. . . . . 6 ⊢ ;;;1258 = ;;;1258
16		8p1e9 12276	. . . . . 6 ⊢ (8 + 1) = 9
17	9, 10, 5, 15, 16	decaddi 12654	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
18	14, 17	eqtr4i 2757	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
19	12, 13, 18	mvrraddi 11383	. . 3 ⊢ (𝑁 − 1) = ;;;1258
20		4nn0 12406	. . . . 5 ⊢ 4 ∈ ℕ₀
21	2, 20	deccl 12609	. . . 4 ⊢ ;34 ∈ ℕ₀
22		7nn0 12409	. . . 4 ⊢ 7 ∈ ℕ₀
23		eqid 2731	. . . 4 ⊢ ;37 = ;37
24	6, 2	deccl 12609	. . . 4 ⊢ ;23 ∈ ℕ₀
25		eqid 2731	. . . . 5 ⊢ ;34 = ;34
26		eqid 2731	. . . . 5 ⊢ ;23 = ;23
27		3t3e9 12293	. . . . . . 7 ⊢ (3 · 3) = 9
28		2p1e3 12268	. . . . . . 7 ⊢ (2 + 1) = 3
29	27, 28	oveq12i 7364	. . . . . 6 ⊢ ((3 · 3) + (2 + 1)) = (9 + 3)
30		9p3e12 12682	. . . . . 6 ⊢ (9 + 3) = ;12
31	29, 30	eqtri 2754	. . . . 5 ⊢ ((3 · 3) + (2 + 1)) = ;12
32		4t3e12 12692	. . . . . 6 ⊢ (4 · 3) = ;12
33		3cn 12212	. . . . . . 7 ⊢ 3 ∈ ℂ
34		2cn 12206	. . . . . . 7 ⊢ 2 ∈ ℂ
35		3p2e5 12277	. . . . . . 7 ⊢ (3 + 2) = 5
36	33, 34, 35	addcomli 11311	. . . . . 6 ⊢ (2 + 3) = 5
37	5, 6, 2, 32, 36	decaddi 12654	. . . . 5 ⊢ ((4 · 3) + 3) = ;15
38	2, 20, 6, 2, 25, 26, 2, 8, 5, 31, 37	decmac 12646	. . . 4 ⊢ ((;34 · 3) + ;23) = ;;125
39		7cn 12225	. . . . . . 7 ⊢ 7 ∈ ℂ
40		7t3e21 12704	. . . . . . 7 ⊢ (7 · 3) = ;21
41	39, 33, 40	mulcomli 11127	. . . . . 6 ⊢ (3 · 7) = ;21
42		1p2e3 12269	. . . . . 6 ⊢ (1 + 2) = 3
43	6, 5, 6, 41, 42	decaddi 12654	. . . . 5 ⊢ ((3 · 7) + 2) = ;23
44		4cn 12216	. . . . . 6 ⊢ 4 ∈ ℂ
45		7t4e28 12705	. . . . . 6 ⊢ (7 · 4) = ;28
46	39, 44, 45	mulcomli 11127	. . . . 5 ⊢ (4 · 7) = ;28
47	22, 2, 20, 25, 10, 6, 43, 46	decmul1c 12659	. . . 4 ⊢ (;34 · 7) = ;;238
48	21, 2, 22, 23, 10, 24, 38, 47	decmul2c 12660	. . 3 ⊢ (;34 · ;37) = ;;;1258
49	19, 48	eqtr4i 2757	. 2 ⊢ (𝑁 − 1) = (;34 · ;37)
50		9nn0 12411	. . . . . . 7 ⊢ 9 ∈ ℕ₀
51	9, 50	deccl 12609	. . . . . 6 ⊢ ;;;1259 ∈ ℕ₀
52	14, 51	eqeltri 2827	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
53	52	nn0cni 12399	. . . 4 ⊢ 𝑁 ∈ ℂ
54		npcan 11375	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
55	53, 13, 54	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
56	55	eqcomi 2740	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
57		1nn 12142	. 2 ⊢ 1 ∈ ℕ
58		2nn 12204	. 2 ⊢ 2 ∈ ℕ
59	2, 22	deccl 12609	. . . . 5 ⊢ ;37 ∈ ℕ₀
60	59	numexp1 16994	. . . 4 ⊢ (;37↑1) = ;37
61	60	oveq2i 7363	. . 3 ⊢ (;34 · (;37↑1)) = (;34 · ;37)
62	49, 61	eqtr4i 2757	. 2 ⊢ (𝑁 − 1) = (;34 · (;37↑1))
63		7nn 12223	. . . 4 ⊢ 7 ∈ ℕ
64		4lt7 12314	. . . 4 ⊢ 4 < 7
65	2, 20, 63, 64	declt 12622	. . 3 ⊢ ;34 < ;37
66	65, 60	breqtrri 5120	. 2 ⊢ ;34 < (;37↑1)
67	14	1259lem4 17051	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
68	14	1259lem5 17052	. 2 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
69	1, 4, 49, 56, 4, 57, 58, 62, 66, 67, 68	pockthi 16825	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2111 (class class class)co 7352 ℂcc 11010 1c1 11013 + caddc 11015 · cmul 11017 < clt 11152 − cmin 11350 2c2 12186 3c3 12187 4c4 12188 5c5 12189 7c7 12191 8c8 12192 9c9 12193 ℕ₀cn0 12387 cdc 12594 ↑cexp 13974 ℙcprime 16588

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-oadd 8395 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9332 df-inf 9333 df-dju 9800 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-div 11781 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-xnn0 12461 df-z 12475 df-dec 12595 df-uz 12739 df-q 12853 df-rp 12897 df-fz 13414 df-fzo 13561 df-fl 13702 df-mod 13780 df-seq 13915 df-exp 13975 df-hash 14244 df-cj 15012 df-re 15013 df-im 15014 df-sqrt 15148 df-abs 15149 df-dvds 16170 df-gcd 16412 df-prm 16589 df-odz 16682 df-phi 16683 df-pc 16755

This theorem is referenced by: (None)

W3C validator