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 16476

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 16460	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 11916	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 11735	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12121	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 11917	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 11920	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12116	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 11915	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12116	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 11782	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2824	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12132	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2850	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7169	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 11923	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12116	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 11918	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12116	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 11924	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2824	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 11921	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12116	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2824	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2824	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 11922	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2824	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 11731	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 10598	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 11788	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 10835	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12123	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2847	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulid1i 10648	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addid2i 10831	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7171	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 11765	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2847	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 11737	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulid1i 10648	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7169	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12186	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2847	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12153	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12123	. . . . . . 7 ⊢ 6 = ;06
46		3cn 11721	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mulid2i 10649	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addid2i 10831	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7171	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 11792	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2847	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 11919	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12217	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 11725	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12173	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 10835	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12163	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12153	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12154	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 11740	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mulid2i 10649	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7169	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12193	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2847	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12229	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 10653	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12166	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12167	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12116	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 11912	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 10905	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2850	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2850	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12116	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2912	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 11912	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 10898	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 690	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2833	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 11652	. 2 ⊢ 1 ∈ ℕ
81		2nn 11713	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12116	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 16416	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7170	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2850	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12233	. . . 4 ⊢ 8 < ;10
87		1lt10 12240	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12139	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12130	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5096	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 16474	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 16475	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16246	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1536 ∈ wcel 2113 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 < clt 10678 − cmin 10873 2c2 11695 3c3 11696 4c4 11697 5c5 11698 6c6 11699 7c7 11700 8c8 11701 9c9 11702 ℕ₀cn0 11900 cdc 12101 ↑cexp 13432 ℙcprime 16018

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-dvds 15611 df-gcd 15847 df-prm 16019 df-odz 16105 df-phi 16106 df-pc 16177

This theorem is referenced by: (None)

W3C validator