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 12453	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12276	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12664	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12454	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12457	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12659	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12452	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12659	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12318	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2736	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12675	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2762	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7377	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12460	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12659	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12455	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12659	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12461	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2736	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12458	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12659	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2736	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2736	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12459	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2736	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12272	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11096	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12324	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11338	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12666	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2759	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11149	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11334	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7379	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12301	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2759	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12278	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11149	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7377	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12729	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2759	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12696	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12666	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12262	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11150	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11334	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7379	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12328	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2759	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12456	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12760	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12266	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12716	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11338	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12706	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12696	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12697	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12281	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11150	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7377	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12736	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2759	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12772	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11154	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12709	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12710	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12659	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12449	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11409	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2762	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2762	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12659	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2832	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12449	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11402	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 693	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2745	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12185	. 2 ⊢ 1 ∈ ℕ
81		2nn 12254	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12659	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17047	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7378	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2762	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12776	. . . 4 ⊢ 8 < ;10
87		1lt10 12783	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12682	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12673	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5112	. 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 1542 ∈ wcel 2114 (class class class)co 7367 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 − cmin 11377 2c2 12236 3c3 12237 4c4 12238 5c5 12239 6c6 12240 7c7 12241 8c8 12242 9c9 12243 ℕ₀cn0 12437 cdc 12644 ↑cexp 14023 ℙcprime 16640

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-dvds 16222 df-gcd 16464 df-prm 16641 df-odz 16735 df-phi 16736 df-pc 16808

This theorem is referenced by: (None)

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