2503prm - Metamath Proof Explorer

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Theorem 2503prm 17200

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 17184	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12520	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12336	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12735	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12521	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12524	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12726	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12519	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12726	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12382	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2769	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12747	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2795	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7421	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12527	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12726	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12522	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12726	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12528	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2769	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12525	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12726	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2769	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2769	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12526	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2769	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12332	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11158	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12388	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11402	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12738	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2792	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11213	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11398	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7423	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12364	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2792	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12338	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11213	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7421	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12801	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2792	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12768	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12738	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12322	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11214	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11398	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7423	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12392	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2792	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12523	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12832	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12326	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12788	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11402	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12778	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12768	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12769	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12341	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11214	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7421	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12808	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2792	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12844	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11218	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12781	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12782	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12726	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12516	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11473	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2795	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2795	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12726	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2865	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12516	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11466	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 704	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2778	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12244	. 2 ⊢ 1 ∈ ℕ
81		2nn 12314	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12726	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17136	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7422	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2795	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12849	. . . 4 ⊢ 8 < ;10
87		1lt10 12856	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12754	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12745	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5142	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17198	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17199	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16967	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1567 ∈ wcel 2149 (class class class)co 7411 ℂcc 11098 0cc0 11100 1c1 11101 + caddc 11103 · cmul 11105 < clt 11243 − cmin 11441 2c2 12295 3c3 12296 4c4 12297 5c5 12298 6c6 12299 7c7 12300 8c8 12301 9c9 12302 ℕ₀cn0 12504 cdc 12711 ↑cexp 14097 ℙcprime 16729

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9402 df-inf 9403 df-dju 9887 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-xnn0 12578 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-fz 13536 df-fzo 13683 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-dvds 16311 df-gcd 16553 df-prm 16730 df-odz 16824 df-phi 16825 df-pc 16897

This theorem is referenced by: (None)

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