2503prm - Metamath Proof Explorer

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Theorem 2503prm 17117

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 17101	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12465	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12288	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12676	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12466	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12469	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12671	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12464	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12671	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12330	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2730	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12687	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2756	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7400	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12472	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12671	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12467	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12671	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12473	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2730	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12470	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12671	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2730	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2730	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12471	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2730	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12284	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11133	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12336	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11373	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12678	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2753	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11185	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11369	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7402	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12313	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2753	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12290	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11185	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7400	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12741	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2753	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12708	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12678	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12274	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11186	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11369	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7402	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12340	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2753	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12468	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12772	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12278	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12728	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11373	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12718	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12708	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12709	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12293	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11186	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7400	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12748	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2753	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12784	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11190	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12721	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12722	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12671	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12461	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11444	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2756	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2756	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12671	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2825	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12461	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11437	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 692	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2739	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12204	. 2 ⊢ 1 ∈ ℕ
81		2nn 12266	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12671	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17054	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7401	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2756	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12788	. . . 4 ⊢ 8 < ;10
87		1lt10 12795	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12694	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12685	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5137	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17115	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17116	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16885	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 (class class class)co 7390 ℂcc 11073 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 < clt 11215 − cmin 11412 2c2 12248 3c3 12249 4c4 12250 5c5 12251 6c6 12252 7c7 12253 8c8 12254 9c9 12255 ℕ₀cn0 12449 cdc 12656 ↑cexp 14033 ℙcprime 16648

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-xnn0 12523 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16230 df-gcd 16472 df-prm 16649 df-odz 16742 df-phi 16743 df-pc 16815

This theorem is referenced by: (None)

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