2503prm - Metamath Proof Explorer

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Theorem 2503prm 17051

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 17035	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12397	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12220	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12608	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12398	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12401	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12603	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12396	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12603	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12262	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2731	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12619	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2757	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7356	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12404	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12603	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12399	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12603	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12405	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2731	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12402	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12603	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2731	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2731	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12403	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2731	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12216	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11064	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12268	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11305	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12610	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2754	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11116	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11301	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7358	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12245	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2754	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12222	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11116	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7356	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12673	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2754	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12640	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12610	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12206	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11117	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11301	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7358	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12272	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2754	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12400	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12704	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12210	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12660	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11305	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12650	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12640	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12641	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12225	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11117	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7356	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12680	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2754	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12716	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11121	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12653	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12654	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12603	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12393	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11376	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2757	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2757	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12603	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2827	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12393	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11369	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2740	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12136	. 2 ⊢ 1 ∈ ℕ
81		2nn 12198	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12603	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 16988	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7357	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2757	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12720	. . . 4 ⊢ 8 < ;10
87		1lt10 12727	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12626	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12617	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5116	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17049	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17050	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16819	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2111 (class class class)co 7346 ℂcc 11004 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 < clt 11146 − cmin 11344 2c2 12180 3c3 12181 4c4 12182 5c5 12183 6c6 12184 7c7 12185 8c8 12186 9c9 12187 ℕ₀cn0 12381 cdc 12588 ↑cexp 13968 ℙcprime 16582

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-gcd 16406 df-prm 16583 df-odz 16676 df-phi 16677 df-pc 16749

This theorem is referenced by: (None)

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