2503prm - Metamath Proof Explorer

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Theorem 2503prm 16473

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 16457	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 11914	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 11733	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12119	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 11915	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 11918	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12114	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 11913	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12114	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 11780	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2821	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12130	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2847	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7166	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 11921	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12114	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 11916	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12114	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 11922	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2821	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 11919	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12114	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2821	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2821	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 11920	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2821	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 11729	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 10595	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 11786	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 10832	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12121	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2844	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulid1i 10645	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addid2i 10828	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7168	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 11763	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2844	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 11735	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulid1i 10645	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7166	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12184	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2844	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12151	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12121	. . . . . . 7 ⊢ 6 = ;06
46		3cn 11719	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mulid2i 10646	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addid2i 10828	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7168	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 11790	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2844	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 11917	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12215	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 11723	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12171	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 10832	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12161	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12151	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12152	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 11738	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mulid2i 10646	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7166	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12191	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2844	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12227	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 10650	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12164	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12165	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12114	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 11910	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 10902	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2847	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2847	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12114	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2909	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 11910	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 10895	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 690	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2830	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 11649	. 2 ⊢ 1 ∈ ℕ
81		2nn 11711	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12114	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 16413	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7167	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2847	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12231	. . . 4 ⊢ 8 < ;10
87		1lt10 12238	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12137	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12128	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5093	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 16471	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 16472	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16243	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 − cmin 10870 2c2 11693 3c3 11694 4c4 11695 5c5 11696 6c6 11697 7c7 11698 8c8 11699 9c9 11700 ℕ₀cn0 11898 cdc 12099 ↑cexp 13430 ℙcprime 16015

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 df-prm 16016 df-odz 16102 df-phi 16103 df-pc 16174

This theorem is referenced by: (None)

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