2503prm - Metamath Proof Explorer

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Theorem 2503prm 17101

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 17085	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12444	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12267	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12655	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12445	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12448	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12650	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12443	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12650	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12309	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2739	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12666	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2765	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7366	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12451	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12650	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12446	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12650	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12452	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2739	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12449	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12650	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2739	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2739	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12450	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2739	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12263	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11087	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12315	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11329	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12657	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2762	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11140	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11325	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7368	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12292	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2762	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12269	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11140	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7366	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12720	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2762	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12687	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12657	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12253	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11141	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11325	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7368	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12319	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2762	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12447	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12751	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12257	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12707	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11329	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12697	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12687	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12688	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12272	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11141	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7366	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12727	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2762	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12763	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11145	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12700	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12701	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12650	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12440	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11400	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2765	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2765	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12650	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2835	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12440	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11393	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 698	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2748	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12176	. 2 ⊢ 1 ∈ ℕ
81		2nn 12245	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12650	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17038	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7367	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2765	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12767	. . . 4 ⊢ 8 < ;10
87		1lt10 12774	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12673	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12664	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5099	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17099	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17100	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16869	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1547 ∈ wcel 2119 (class class class)co 7356 ℂcc 11027 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 − cmin 11368 2c2 12227 3c3 12228 4c4 12229 5c5 12230 6c6 12231 7c7 12232 8c8 12233 9c9 12234 ℕ₀cn0 12428 cdc 12635 ↑cexp 14014 ℙcprime 16631

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 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 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16213 df-gcd 16455 df-prm 16632 df-odz 16726 df-phi 16727 df-pc 16799

This theorem is referenced by: (None)

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