2503prm - Metamath Proof Explorer

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Theorem 2503prm 17362

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 17346	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12662	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12478	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12878	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12663	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12666	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12869	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12661	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12869	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12524	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2763	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12890	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2789	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7461	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12669	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12869	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12664	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12869	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12670	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2763	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12667	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12869	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2763	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2763	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12668	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2763	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12474	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11297	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12530	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11541	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12881	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2786	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11352	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11537	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7463	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12506	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2786	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12480	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11352	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7461	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12944	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2786	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12911	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12881	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12464	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11353	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11537	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7463	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12534	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2786	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12665	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12975	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12468	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12931	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11541	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12921	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12911	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12912	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12483	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11353	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7461	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12951	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2786	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12987	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11357	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12924	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12925	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12869	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12658	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11612	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2789	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2789	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12869	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2860	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12658	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11605	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 705	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2772	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12386	. 2 ⊢ 1 ∈ ℕ
81		2nn 12456	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12869	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17298	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7462	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2789	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12992	. . . 4 ⊢ 8 < ;10
87		1lt10 12999	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12897	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12888	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5138	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17360	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17361	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 17129	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2147 (class class class)co 7451 ℂcc 11237 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 < clt 11382 − cmin 11580 2c2 12437 3c3 12438 4c4 12439 5c5 12440 6c6 12441 7c7 12442 8c8 12443 9c9 12444 ℕ₀cn0 12646 cdc 12854 ↑cexp 14244 ℙcprime 16890

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-inf 9472 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-dvds 16467 df-gcd 16709 df-prm 16891 df-odz 16986 df-phi 16987 df-pc 17059

This theorem is used by: (None)

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