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 17104

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 17088	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12447	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12270	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12658	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12448	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12451	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12653	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12446	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12653	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12312	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2737	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12669	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2763	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7371	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12454	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12653	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12449	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12653	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12455	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2737	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12452	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12653	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2737	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2737	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12453	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2737	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12266	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11090	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12318	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11332	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12660	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2760	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11143	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11328	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7373	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12295	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2760	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12272	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11143	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7371	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12723	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2760	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12690	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12660	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12256	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11144	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11328	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7373	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12322	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2760	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12450	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12754	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12260	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12710	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11332	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12700	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12690	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12691	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12275	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11144	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7371	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12730	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2760	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12766	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11148	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12703	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12704	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12653	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12443	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11403	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2763	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2763	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12653	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2833	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12443	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11396	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 693	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2746	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12179	. 2 ⊢ 1 ∈ ℕ
81		2nn 12248	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12653	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17041	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7372	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2763	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12770	. . . 4 ⊢ 8 < ;10
87		1lt10 12777	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12676	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12667	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5113	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17102	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17103	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16872	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 (class class class)co 7361 ℂcc 11030 0cc0 11032 1c1 11033 + caddc 11035 · cmul 11037 < clt 11173 − cmin 11371 2c2 12230 3c3 12231 4c4 12232 5c5 12233 6c6 12234 7c7 12235 8c8 12236 9c9 12237 ℕ₀cn0 12431 cdc 12638 ↑cexp 14017 ℙcprime 16634

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-xnn0 12505 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-hash 14287 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-dvds 16216 df-gcd 16458 df-prm 16635 df-odz 16729 df-phi 16730 df-pc 16802

This theorem is referenced by: (None)

W3C validator