2503prm - Metamath Proof Explorer

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Theorem 2503prm 17187

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 17171	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12569	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12388	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12778	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12570	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12573	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12773	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12568	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12773	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12435	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2740	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12789	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2771	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7458	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12576	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12773	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12571	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12773	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12577	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2740	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12574	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12773	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2740	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2740	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12575	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2740	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12384	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11242	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12441	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11482	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12780	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2768	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11294	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11478	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7460	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12418	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2768	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12390	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11294	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7458	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12843	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2768	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12810	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12780	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12374	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11295	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11478	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7460	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12445	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2768	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12572	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12874	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12378	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12830	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11482	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12820	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12810	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12811	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12393	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11295	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7458	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12850	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2768	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12886	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11299	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12823	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12824	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12773	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12565	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11552	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2771	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2771	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12773	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2840	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12565	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11545	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 691	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2749	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12304	. 2 ⊢ 1 ∈ ℕ
81		2nn 12366	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12773	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17124	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7459	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2771	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12890	. . . 4 ⊢ 8 < ;10
87		1lt10 12897	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12796	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12787	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5193	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17185	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17186	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16954	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2108 (class class class)co 7448 ℂcc 11182 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 − cmin 11520 2c2 12348 3c3 12349 4c4 12350 5c5 12351 6c6 12352 7c7 12353 8c8 12354 9c9 12355 ℕ₀cn0 12553 cdc 12758 ↑cexp 14112 ℙcprime 16718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541 df-prm 16719 df-odz 16812 df-phi 16813 df-pc 16884

This theorem is referenced by: (None)

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