2503prm - Metamath Proof Explorer

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Theorem 2503prm 16319

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 16303	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 11718	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 11533	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 11925	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 11719	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 11722	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 11919	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 11717	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 11919	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 11582	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2772	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 11936	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2799	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 6980	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 11725	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 11919	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 11720	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 11919	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 11726	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2772	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 11723	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 11919	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2772	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2772	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 11724	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2772	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 11527	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 10385	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 11588	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 10624	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 11927	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2796	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulid1i 10436	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addid2i 10620	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 6982	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 11565	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2796	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 11535	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulid1i 10436	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 6980	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 11991	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2796	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 11957	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 11927	. . . . . . 7 ⊢ 6 = ;06
46		3cn 11514	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mulid2i 10437	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addid2i 10620	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 6982	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 11592	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2796	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 11721	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12022	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 11519	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 11978	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 10624	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 11967	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 11957	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 11958	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 11539	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mulid2i 10437	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 6980	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 11998	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2796	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12034	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 10441	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 11971	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 11972	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 11919	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 11713	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 10695	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2799	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2799	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 11919	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2856	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 11713	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 10688	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 679	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2781	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 11444	. 2 ⊢ 1 ∈ ℕ
81		2nn 11506	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 11919	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 16259	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 6981	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2799	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12038	. . . 4 ⊢ 8 < ;10
87		1lt10 12045	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 11943	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 11934	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 4950	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 16317	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 16318	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16089	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1507 ∈ wcel 2048 (class class class)co 6970 ℂcc 10325 0cc0 10327 1c1 10328 + caddc 10330 · cmul 10332 < clt 10466 − cmin 10662 2c2 11488 3c3 11489 4c4 11490 5c5 11491 6c6 11492 7c7 11493 8c8 11494 9c9 11495 ℕ₀cn0 11700 cdc 11904 ↑cexp 13237 ℙcprime 15861

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-sup 8693 df-inf 8694 df-dju 9116 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-xnn0 11773 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-fz 12702 df-fzo 12843 df-fl 12970 df-mod 13046 df-seq 13178 df-exp 13238 df-hash 13499 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-dvds 15458 df-gcd 15694 df-prm 15862 df-odz 15948 df-phi 15949 df-pc 16020

This theorem is referenced by: (None)

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