2503prm - Metamath Proof Explorer

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Theorem 2503prm 17157

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 17141	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12515	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12333	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12726	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12516	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12519	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12721	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12514	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12721	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12380	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2735	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12737	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2761	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7413	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12522	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12721	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12517	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12721	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12523	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2735	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12520	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12721	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2735	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2735	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12521	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2735	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12329	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11185	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12386	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11425	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12728	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2758	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11237	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11421	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7415	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12363	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2758	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12335	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11237	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7413	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12791	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2758	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12758	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12728	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12319	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11238	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11421	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7415	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12390	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2758	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12518	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12822	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12323	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12778	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11425	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12768	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12758	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12759	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12338	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11238	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7413	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12798	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2758	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12834	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11242	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12771	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12772	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12721	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12511	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11496	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2761	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2761	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12721	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2830	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12511	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11489	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2744	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12249	. 2 ⊢ 1 ∈ ℕ
81		2nn 12311	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12721	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17094	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7414	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2761	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12838	. . . 4 ⊢ 8 < ;10
87		1lt10 12845	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12744	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12735	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5146	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17155	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17156	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16925	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7403 ℂcc 11125 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 < clt 11267 − cmin 11464 2c2 12293 3c3 12294 4c4 12295 5c5 12296 6c6 12297 7c7 12298 8c8 12299 9c9 12300 ℕ₀cn0 12499 cdc 12706 ↑cexp 14077 ℙcprime 16688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9452 df-inf 9453 df-dju 9913 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-hash 14347 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-dvds 16271 df-gcd 16512 df-prm 16689 df-odz 16782 df-phi 16783 df-pc 16855

This theorem is referenced by: (None)

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