2503prm - Metamath Proof Explorer

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Theorem 2503prm 17012

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 16996	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12429	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12248	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12638	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12430	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12433	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12633	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12428	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12633	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12295	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2736	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12649	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2767	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7367	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12436	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12633	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12431	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12633	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12437	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2736	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12434	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12633	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2736	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2736	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12435	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2736	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12244	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11109	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12301	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11347	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12640	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2764	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulid1i 11159	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addid2i 11343	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7369	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12278	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2764	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12250	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulid1i 11159	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7367	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12703	. . . . . . . 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 12670	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12640	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12234	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mulid2i 11160	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addid2i 11343	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7369	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12305	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2764	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12432	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12734	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12238	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12690	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11347	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12680	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12670	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12671	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12253	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mulid2i 11160	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7367	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12710	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2764	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12746	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11164	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12683	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12684	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12633	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12425	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11417	. . . 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 12633	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2834	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12425	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11410	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 690	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2745	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12164	. 2 ⊢ 1 ∈ ℕ
81		2nn 12226	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12633	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 16949	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7368	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2767	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12750	. . . 4 ⊢ 8 < ;10
87		1lt10 12757	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12656	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12647	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5132	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17010	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17011	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16779	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2106 (class class class)co 7357 ℂcc 11049 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 < clt 11189 − cmin 11385 2c2 12208 3c3 12209 4c4 12210 5c5 12211 6c6 12212 7c7 12213 8c8 12214 9c9 12215 ℕ₀cn0 12413 cdc 12618 ↑cexp 13967 ℙcprime 16547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-gcd 16375 df-prm 16548 df-odz 16637 df-phi 16638 df-pc 16709

This theorem is referenced by: (None)

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