2503prm - Metamath Proof Explorer

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Theorem 2503prm 16656

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 16640	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12071	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 11890	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12278	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12072	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12075	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12273	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12070	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12273	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 11937	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2736	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12289	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2762	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7201	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12078	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12273	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12073	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12273	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12079	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2736	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12076	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12273	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2736	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2736	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12077	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2736	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 11886	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 10752	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 11943	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 10989	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12280	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2759	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulid1i 10802	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addid2i 10985	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7203	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 11920	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2759	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 11892	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulid1i 10802	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7201	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12343	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2759	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12310	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12280	. . . . . . 7 ⊢ 6 = ;06
46		3cn 11876	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mulid2i 10803	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addid2i 10985	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7203	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 11947	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2759	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12074	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12374	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 11880	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12330	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 10989	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12320	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12310	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12311	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 11895	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mulid2i 10803	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7201	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12350	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2759	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12386	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 10807	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12323	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12324	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12273	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12067	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11059	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2762	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2762	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12273	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2827	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12067	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11052	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2745	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 11806	. 2 ⊢ 1 ∈ ℕ
81		2nn 11868	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12273	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 16593	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7202	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2762	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12390	. . . 4 ⊢ 8 < ;10
87		1lt10 12397	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12296	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12287	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5066	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 16654	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 16655	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16423	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1543 ∈ wcel 2112 (class class class)co 7191 ℂcc 10692 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 < clt 10832 − cmin 11027 2c2 11850 3c3 11851 4c4 11852 5c5 11853 6c6 11854 7c7 11855 8c8 11856 9c9 11857 ℕ₀cn0 12055 cdc 12258 ↑cexp 13600 ℙcprime 16191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-oadd 8184 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-xnn0 12128 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-dvds 15779 df-gcd 16017 df-prm 16192 df-odz 16281 df-phi 16282 df-pc 16353

This theorem is referenced by: (None)

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