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 17065

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 17049	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12415	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12238	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12625	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12416	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12419	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12620	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12414	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12620	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12280	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2734	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12636	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2760	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7366	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12422	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12620	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12417	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12620	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12423	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2734	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12420	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12620	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2734	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2734	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12421	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2734	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12234	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11082	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12286	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11323	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12627	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2757	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11134	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11319	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7368	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12263	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2757	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12240	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11134	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7366	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12690	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2757	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12657	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12627	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12224	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11135	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11319	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7368	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12290	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2757	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12418	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12721	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12228	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12677	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11323	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12667	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12657	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12658	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12243	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11135	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7366	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12697	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2757	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12733	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11139	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12670	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12671	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12620	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12411	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11394	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2760	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2760	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12620	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2830	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12411	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11387	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2743	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12154	. 2 ⊢ 1 ∈ ℕ
81		2nn 12216	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12620	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17002	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7367	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2760	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12737	. . . 4 ⊢ 8 < ;10
87		1lt10 12744	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12643	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12634	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5123	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17063	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17064	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16833	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2113 (class class class)co 7356 ℂcc 11022 0cc0 11024 1c1 11025 + caddc 11027 · cmul 11029 < clt 11164 − cmin 11362 2c2 12198 3c3 12199 4c4 12200 5c5 12201 6c6 12202 7c7 12203 8c8 12204 9c9 12205 ℕ₀cn0 12399 cdc 12605 ↑cexp 13982 ℙcprime 16596

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-inf 9344 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-xnn0 12473 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-dvds 16178 df-gcd 16420 df-prm 16597 df-odz 16690 df-phi 16691 df-pc 16763

This theorem is referenced by: (None)

W3C validator