2503prm - Metamath Proof Explorer

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Theorem 2503prm 16930

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 16914	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12342	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12161	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12550	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12343	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12346	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12545	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12341	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12545	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12208	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2736	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12561	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2767	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7339	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12349	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12545	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12344	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12545	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12350	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2736	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12347	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12545	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2736	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2736	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12348	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2736	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12157	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11022	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12214	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11260	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12552	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2764	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulid1i 11072	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addid2i 11256	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7341	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12191	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2764	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12163	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulid1i 11072	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7339	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12615	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2764	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12582	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12552	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12147	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mulid2i 11073	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addid2i 11256	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7341	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12218	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2764	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12345	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12646	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12151	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12602	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11260	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12592	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12582	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12583	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12166	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mulid2i 11073	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7339	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12622	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2764	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12658	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11077	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12595	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12596	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12545	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12338	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11330	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2767	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2767	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12545	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2833	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12338	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11323	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 689	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2745	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12077	. 2 ⊢ 1 ∈ ℕ
81		2nn 12139	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12545	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 16867	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7340	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2767	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12662	. . . 4 ⊢ 8 < ;10
87		1lt10 12669	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12568	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12559	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5116	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 16928	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 16929	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16697	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2105 (class class class)co 7329 ℂcc 10962 0cc0 10964 1c1 10965 + caddc 10967 · cmul 10969 < clt 11102 − cmin 11298 2c2 12121 3c3 12122 4c4 12123 5c5 12124 6c6 12125 7c7 12126 8c8 12127 9c9 12128 ℕ₀cn0 12326 cdc 12530 ↑cexp 13875 ℙcprime 16465

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-1st 7891 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-2o 8360 df-oadd 8363 df-er 8561 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-sup 9291 df-inf 9292 df-dju 9750 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-xnn0 12399 df-z 12413 df-dec 12531 df-uz 12676 df-q 12782 df-rp 12824 df-fz 13333 df-fzo 13476 df-fl 13605 df-mod 13683 df-seq 13815 df-exp 13876 df-hash 14138 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-dvds 16055 df-gcd 16293 df-prm 16466 df-odz 16555 df-phi 16556 df-pc 16627

This theorem is referenced by: (None)

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