2503prm - Metamath Proof Explorer

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Theorem 2503prm 17110

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 17094	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12458	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12281	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12669	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12459	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12462	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12664	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12457	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12664	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12323	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2729	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12680	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2755	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7397	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12465	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12664	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12460	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12664	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12466	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2729	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12463	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12664	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2729	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2729	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12464	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2729	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12277	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11126	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12329	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11366	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12671	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2752	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11178	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11362	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7399	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12306	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2752	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12283	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11178	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7397	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12734	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2752	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12701	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12671	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12267	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11179	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11362	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7399	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12333	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2752	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12461	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12765	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12271	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12721	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11366	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12711	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12701	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12702	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12286	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11179	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7397	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12741	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2752	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12777	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11183	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12714	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12715	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12664	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12454	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11437	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2755	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2755	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12664	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2824	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12454	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11430	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2738	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12197	. 2 ⊢ 1 ∈ ℕ
81		2nn 12259	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12664	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17047	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7398	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2755	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12781	. . . 4 ⊢ 8 < ;10
87		1lt10 12788	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12687	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12678	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5134	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17108	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17109	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16878	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 (class class class)co 7387 ℂcc 11066 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 − cmin 11405 2c2 12241 3c3 12242 4c4 12243 5c5 12244 6c6 12245 7c7 12246 8c8 12247 9c9 12248 ℕ₀cn0 12442 cdc 12649 ↑cexp 14026 ℙcprime 16641

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-gcd 16465 df-prm 16642 df-odz 16735 df-phi 16736 df-pc 16808

This theorem is referenced by: (None)

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