2503prm - Metamath Proof Explorer

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Theorem 2503prm 17177

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 17161	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12542	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12361	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12753	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12543	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12546	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12748	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12541	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12748	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12408	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2737	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12764	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2768	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7441	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12549	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12748	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12544	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12748	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12550	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2737	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12547	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12748	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2737	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2737	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12548	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2737	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12357	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11213	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12414	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11453	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12755	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2765	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11265	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11449	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7443	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12391	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2765	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12363	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11265	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7441	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12818	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2765	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12785	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12755	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12347	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11266	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11449	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7443	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12418	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2765	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12545	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12849	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12351	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12805	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11453	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12795	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12785	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12786	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12366	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11266	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7441	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12825	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2765	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12861	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11270	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12798	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12799	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12748	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12538	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11524	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2768	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2768	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12748	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2837	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12538	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11517	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2746	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12277	. 2 ⊢ 1 ∈ ℕ
81		2nn 12339	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12748	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17114	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7442	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2768	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12865	. . . 4 ⊢ 8 < ;10
87		1lt10 12872	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12771	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12762	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5170	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17175	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17176	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16945	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 − cmin 11492 2c2 12321 3c3 12322 4c4 12323 5c5 12324 6c6 12325 7c7 12326 8c8 12327 9c9 12328 ℕ₀cn0 12526 cdc 12733 ↑cexp 14102 ℙcprime 16708

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 df-odz 16802 df-phi 16803 df-pc 16875

This theorem is referenced by: (None)

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