2503prm - Metamath Proof Explorer

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Theorem 2503prm 17195

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 17179	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12515	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12331	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12730	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12516	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12519	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12721	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12514	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12721	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12377	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2763	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12742	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2789	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7420	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12522	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12721	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12517	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12721	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12523	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2763	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12520	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12721	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2763	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2763	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12521	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2763	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12327	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11153	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12383	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11397	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12733	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2786	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11208	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11393	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7422	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12359	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2786	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12333	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11208	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7420	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12796	. . . . . . . 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 12763	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12733	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12317	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11209	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11393	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7422	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12387	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2786	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12518	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12827	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12321	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12783	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11397	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12773	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12763	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12764	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12336	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11209	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7420	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12803	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2786	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12839	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11213	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12776	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12777	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12721	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12511	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11468	. . . 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 12721	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2859	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12511	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11461	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 704	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2772	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12239	. 2 ⊢ 1 ∈ ℕ
81		2nn 12309	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12721	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17131	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7421	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2789	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12844	. . . 4 ⊢ 8 < ;10
87		1lt10 12851	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12749	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12740	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5138	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17193	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17194	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16962	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1570 ∈ wcel 2143 (class class class)co 7410 ℂcc 11093 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 < clt 11238 − cmin 11436 2c2 12290 3c3 12291 4c4 12292 5c5 12293 6c6 12294 7c7 12295 8c8 12296 9c9 12297 ℕ₀cn0 12499 cdc 12706 ↑cexp 14093 ℙcprime 16724

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 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 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-dvds 16306 df-gcd 16548 df-prm 16725 df-odz 16819 df-phi 16820 df-pc 16892

This theorem is referenced by: (None)

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