2503prm - Metamath Proof Explorer

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Theorem 2503prm 17067

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 17051	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12417	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12240	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12627	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12418	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12421	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12622	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12416	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12622	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12282	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2736	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12638	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2762	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7368	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12424	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12622	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12419	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12622	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12425	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2736	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12422	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12622	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2736	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2736	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12423	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2736	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12236	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11084	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12288	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11325	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12629	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2759	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11136	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11321	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7370	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12265	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2759	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12242	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11136	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7368	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12692	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2759	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12659	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12629	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12226	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11137	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11321	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7370	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12292	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2759	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12420	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12723	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12230	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12679	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11325	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12669	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12659	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12660	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12245	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11137	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7368	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12699	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2759	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12735	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11141	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12672	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12673	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12622	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12413	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11396	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2762	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2762	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12622	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2832	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12413	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11389	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2745	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12156	. 2 ⊢ 1 ∈ ℕ
81		2nn 12218	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12622	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17004	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7369	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2762	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12739	. . . 4 ⊢ 8 < ;10
87		1lt10 12746	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12645	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12636	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5125	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17065	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17066	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16835	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2113 (class class class)co 7358 ℂcc 11024 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 < clt 11166 − cmin 11364 2c2 12200 3c3 12201 4c4 12202 5c5 12203 6c6 12204 7c7 12205 8c8 12206 9c9 12207 ℕ₀cn0 12401 cdc 12607 ↑cexp 13984 ℙcprime 16598

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-dvds 16180 df-gcd 16422 df-prm 16599 df-odz 16692 df-phi 16693 df-pc 16765

This theorem is referenced by: (None)

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