2503prm - Metamath Proof Explorer

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Theorem 2503prm 17137

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 17121	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12534	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12353	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12743	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12535	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12538	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12738	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12533	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12738	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12400	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2726	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12754	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2757	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7426	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12541	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12738	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12536	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12738	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12542	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2726	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12539	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12738	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2726	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2726	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12540	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2726	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12349	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11207	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12406	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11447	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12745	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2754	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11259	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11443	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7428	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12383	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2754	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12355	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11259	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7426	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12808	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2754	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12775	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12745	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12339	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11260	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11443	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7428	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12410	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2754	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12537	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12839	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12343	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12795	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11447	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12785	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12775	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12776	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12358	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11260	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7426	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12815	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2754	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12851	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11264	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12788	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12789	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12738	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12530	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11517	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2757	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2757	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12738	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2822	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12530	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11510	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 690	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2735	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12269	. 2 ⊢ 1 ∈ ℕ
81		2nn 12331	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12738	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17074	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7427	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2757	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12855	. . . 4 ⊢ 8 < ;10
87		1lt10 12862	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12761	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12752	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5172	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17135	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17136	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16904	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7416 ℂcc 11147 0cc0 11149 1c1 11150 + caddc 11152 · cmul 11154 < clt 11289 − cmin 11485 2c2 12313 3c3 12314 4c4 12315 5c5 12316 6c6 12317 7c7 12318 8c8 12319 9c9 12320 ℕ₀cn0 12518 cdc 12723 ↑cexp 14075 ℙcprime 16667

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-oadd 8492 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-dju 9937 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-xnn0 12591 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-fz 13533 df-fzo 13676 df-fl 13806 df-mod 13884 df-seq 14016 df-exp 14076 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-dvds 16252 df-gcd 16490 df-prm 16668 df-odz 16762 df-phi 16763 df-pc 16834

This theorem is referenced by: (None)

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