2503prm - Metamath Proof Explorer

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Theorem 2503prm 16078

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 16062	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 11595	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 11413	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 11799	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 11596	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 11599	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 11794	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 11594	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 11794	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 11462	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2817	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 11810	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2842	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 6894	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 11602	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 11794	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 11597	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 11794	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 11603	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2817	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 11600	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 11794	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2817	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2817	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 11601	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2817	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 11407	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 10289	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 11467	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 10523	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 11801	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2839	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulid1i 10339	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addid2i 10519	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 6896	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 11445	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2839	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 11415	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulid1i 10339	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 6894	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 11864	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2839	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 11831	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 11801	. . . . . . 7 ⊢ 6 = ;06
46		3cn 11394	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mulid2i 10340	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addid2i 10519	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 6896	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 11471	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2839	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 11598	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 11895	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 11399	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 11851	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 10523	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 11841	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 11831	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 11832	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 11419	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mulid2i 10340	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 6894	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 11871	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2839	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 11907	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 10344	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 11844	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 11845	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 11794	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 11591	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 10592	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2842	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2842	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 11794	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2892	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 11591	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 10585	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 675	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2826	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 11328	. 2 ⊢ 1 ∈ ℕ
81		2nn 11386	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 11794	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 16018	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 6895	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2842	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 11911	. . . 4 ⊢ 8 < ;10
87		1lt10 11918	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 11817	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 11808	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 4882	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 16076	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 16077	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 15848	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1637 ∈ wcel 2157 (class class class)co 6884 ℂcc 10229 0cc0 10231 1c1 10232 + caddc 10234 · cmul 10236 < clt 10369 − cmin 10561 2c2 11368 3c3 11369 4c4 11370 5c5 11371 6c6 11372 7c7 11373 8c8 11374 9c9 11375 ℕ₀cn0 11579 cdc 11779 ↑cexp 13103 ℙcprime 15623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2069 ax-7 2105 ax-8 2159 ax-9 2166 ax-10 2186 ax-11 2202 ax-12 2215 ax-13 2422 ax-ext 2795 ax-rep 4977 ax-sep 4988 ax-nul 4996 ax-pow 5048 ax-pr 5109 ax-un 7189 ax-cnex 10287 ax-resscn 10288 ax-1cn 10289 ax-icn 10290 ax-addcl 10291 ax-addrcl 10292 ax-mulcl 10293 ax-mulrcl 10294 ax-mulcom 10295 ax-addass 10296 ax-mulass 10297 ax-distr 10298 ax-i2m1 10299 ax-1ne0 10300 ax-1rid 10301 ax-rnegex 10302 ax-rrecex 10303 ax-cnre 10304 ax-pre-lttri 10305 ax-pre-lttrn 10306 ax-pre-ltadd 10307 ax-pre-mulgt0 10308 ax-pre-sup 10309

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-ex 1860 df-nf 1864 df-sb 2062 df-mo 2635 df-eu 2642 df-clab 2804 df-cleq 2810 df-clel 2813 df-nfc 2948 df-ne 2990 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3404 df-sbc 3645 df-csb 3740 df-dif 3783 df-un 3785 df-in 3787 df-ss 3794 df-pss 3796 df-nul 4128 df-if 4291 df-pw 4364 df-sn 4382 df-pr 4384 df-tp 4386 df-op 4388 df-uni 4642 df-int 4681 df-iun 4725 df-br 4856 df-opab 4918 df-mpt 4935 df-tr 4958 df-id 5232 df-eprel 5237 df-po 5245 df-so 5246 df-fr 5283 df-we 5285 df-xp 5330 df-rel 5331 df-cnv 5332 df-co 5333 df-dm 5334 df-rn 5335 df-res 5336 df-ima 5337 df-pred 5907 df-ord 5953 df-on 5954 df-lim 5955 df-suc 5956 df-iota 6074 df-fun 6113 df-fn 6114 df-f 6115 df-f1 6116 df-fo 6117 df-f1o 6118 df-fv 6119 df-riota 6845 df-ov 6887 df-oprab 6888 df-mpt2 6889 df-om 7306 df-1st 7408 df-2nd 7409 df-wrecs 7652 df-recs 7714 df-rdg 7752 df-1o 7806 df-2o 7807 df-oadd 7810 df-er 7989 df-map 8104 df-en 8203 df-dom 8204 df-sdom 8205 df-fin 8206 df-sup 8597 df-inf 8598 df-card 9058 df-cda 9285 df-pnf 10371 df-mnf 10372 df-xr 10373 df-ltxr 10374 df-le 10375 df-sub 10563 df-neg 10564 df-div 10980 df-nn 11316 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11580 df-xnn0 11650 df-z 11664 df-dec 11780 df-uz 11925 df-q 12028 df-rp 12067 df-fz 12570 df-fzo 12710 df-fl 12837 df-mod 12913 df-seq 13045 df-exp 13104 df-hash 13358 df-cj 14082 df-re 14083 df-im 14084 df-sqrt 14218 df-abs 14219 df-dvds 15224 df-gcd 15456 df-prm 15624 df-odz 15707 df-phi 15708 df-pc 15779

This theorem is referenced by: (None)

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