2503prm - Metamath Proof Explorer

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Theorem 2503prm 17330

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 17314	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12632	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12448	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12848	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12633	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12636	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12839	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12631	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12839	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12494	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2763	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12860	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2789	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7437	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12639	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12839	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12634	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12839	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12640	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2763	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12637	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12839	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2763	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2763	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12638	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2763	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12444	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11268	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12500	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11512	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12851	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2786	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11323	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11508	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7439	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12476	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2786	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12450	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11323	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7437	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12914	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2786	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12881	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12851	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12434	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11324	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11508	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7439	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12504	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2786	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12635	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12945	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12438	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12901	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11512	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12891	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12881	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12882	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12453	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11324	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7437	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12921	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2786	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12957	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11328	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12894	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12895	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12839	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12628	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11583	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2789	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2789	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12839	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2859	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12628	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11576	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 705	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2772	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12356	. 2 ⊢ 1 ∈ ℕ
81		2nn 12426	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12839	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17266	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7438	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2789	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12962	. . . 4 ⊢ 8 < ;10
87		1lt10 12969	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12867	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12858	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5134	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17328	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17329	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 17097	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1569 ∈ wcel 2146 (class class class)co 7427 ℂcc 11208 0cc0 11210 1c1 11211 + caddc 11213 · cmul 11215 < clt 11353 − cmin 11551 2c2 12407 3c3 12408 4c4 12409 5c5 12410 6c6 12411 7c7 12412 8c8 12413 9c9 12414 ℕ₀cn0 12616 cdc 12824 ↑cexp 14214 ℙcprime 16858

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 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 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8728 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9444 df-inf 9445 df-dju 9992 df-card 10030 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-7 12420 df-8 12421 df-9 12422 df-n0 12617 df-xnn0 12690 df-z 12704 df-dec 12825 df-uz 12976 df-q 13086 df-rp 13131 df-fz 13650 df-fzo 13799 df-fl 13942 df-mod 14020 df-seq 14155 df-exp 14215 df-hash 14485 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-dvds 16435 df-gcd 16677 df-prm 16859 df-odz 16954 df-phi 16955 df-pc 17027

This theorem is used by: (None)

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