2503prm - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 2503prm		Structured version Visualization version GIF version

Theorem 2503prm 17173

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 17157	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12539	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12358	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12750	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12540	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12543	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12745	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12538	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12745	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12405	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2734	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12761	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2765	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7440	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12546	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12745	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12541	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12745	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12547	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2734	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12544	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12745	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2734	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2734	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12545	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2734	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12354	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11210	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12411	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11450	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12752	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2762	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11262	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11446	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7442	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12388	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2762	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12360	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11262	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7440	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12815	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2762	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12782	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12752	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12344	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11263	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11446	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7442	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12415	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2762	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12542	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12846	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12348	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12802	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11450	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12792	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12782	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12783	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12363	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11263	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7440	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12822	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2762	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12858	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11267	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12795	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12796	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12745	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12535	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11521	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2765	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2765	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12745	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2834	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12535	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11514	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2743	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12274	. 2 ⊢ 1 ∈ ℕ
81		2nn 12336	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12745	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17110	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7441	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2765	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12862	. . . 4 ⊢ 8 < ;10
87		1lt10 12869	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12768	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12759	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5174	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17171	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17172	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16940	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1536 ∈ wcel 2105 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 − cmin 11489 2c2 12318 3c3 12319 4c4 12320 5c5 12321 6c6 12322 7c7 12323 8c8 12324 9c9 12325 ℕ₀cn0 12523 cdc 12730 ↑cexp 14098 ℙcprime 16704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-gcd 16528 df-prm 16705 df-odz 16798 df-phi 16799 df-pc 16870

This theorem is referenced by: (None)

W3C validator