2503prm - Metamath Proof Explorer

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Theorem 2503prm 17318

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 17302	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12622	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12438	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12838	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12623	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12626	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12829	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12621	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12829	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12484	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2761	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12850	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2787	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7430	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12629	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12829	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12624	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12829	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12630	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2761	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12627	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12829	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2761	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2761	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12628	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2761	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12434	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11258	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12490	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11502	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12841	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2784	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11313	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11498	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7432	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12466	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2784	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12440	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11313	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7430	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12904	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2784	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12871	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12841	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12424	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11314	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11498	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7432	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12494	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2784	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12625	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12935	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12428	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12891	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11502	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12881	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12871	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12872	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12443	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11314	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7430	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12911	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2784	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12947	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11318	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12884	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12885	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12829	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12618	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11573	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2787	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2787	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12829	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2857	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12618	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11566	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 705	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2770	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12346	. 2 ⊢ 1 ∈ ℕ
81		2nn 12416	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12829	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17254	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7431	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2787	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12952	. . . 4 ⊢ 8 < ;10
87		1lt10 12959	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12857	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12848	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5132	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17316	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17317	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 17085	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2145 (class class class)co 7420 ℂcc 11198 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 < clt 11343 − cmin 11541 2c2 12397 3c3 12398 4c4 12399 5c5 12400 6c6 12401 7c7 12402 8c8 12403 9c9 12404 ℕ₀cn0 12606 cdc 12814 ↑cexp 14204 ℙcprime 16846

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9434 df-inf 9435 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-xnn0 12680 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-dvds 16423 df-gcd 16665 df-prm 16847 df-odz 16942 df-phi 16943 df-pc 17015

This theorem is used by: (None)

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