2503prm - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 2503prm		Structured version Visualization version GIF version

Theorem 2503prm 16841

Description: 2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

**Hypothesis**
Ref	Expression
2503prm.1	⊢ 𝑁 = ;;;2503

**Assertion**
Ref	Expression
2503prm	⊢ 𝑁 ∈ ℙ

**Proof of Theorem 2503prm**
Step	Hyp	Ref	Expression
1		139prm 16825	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12249	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12068	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12457	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12250	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12253	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12452	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12248	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12452	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12115	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2738	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12468	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2769	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7285	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12256	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12452	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12251	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12452	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12257	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2738	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12254	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12452	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2738	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2738	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12255	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2738	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12064	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 10929	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12121	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11167	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12459	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2766	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulid1i 10979	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addid2i 11163	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7287	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12098	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2766	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12070	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulid1i 10979	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7285	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12522	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2766	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12489	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12459	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12054	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mulid2i 10980	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addid2i 11163	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7287	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12125	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2766	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12252	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12553	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12058	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12509	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11167	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12499	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12489	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12490	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12073	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mulid2i 10980	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7285	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12529	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2766	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12565	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 10984	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12502	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12503	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12452	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12245	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11237	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2769	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2769	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12452	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2835	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12245	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11230	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 689	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2747	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 11984	. 2 ⊢ 1 ∈ ℕ
81		2nn 12046	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12452	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 16778	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7286	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2769	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12569	. . . 4 ⊢ 8 < ;10
87		1lt10 12576	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12475	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12466	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5101	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 16839	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 16840	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16608	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2106 (class class class)co 7275 ℂcc 10869 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 − cmin 11205 2c2 12028 3c3 12029 4c4 12030 5c5 12031 6c6 12032 7c7 12033 8c8 12034 9c9 12035 ℕ₀cn0 12233 cdc 12437 ↑cexp 13782 ℙcprime 16376

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-dvds 15964 df-gcd 16202 df-prm 16377 df-odz 16466 df-phi 16467 df-pc 16538

This theorem is referenced by: (None)

W3C validator