2503prm - Metamath Proof Explorer

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Theorem 2503prm 17235

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 17219	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12547	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12363	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12763	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12548	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12551	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12754	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12546	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12754	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12409	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2760	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12775	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2786	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7424	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12554	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12754	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12549	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12754	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12555	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2760	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12552	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12754	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2760	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2760	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12553	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2760	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12359	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11185	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12415	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11429	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12766	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2783	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11240	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11425	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7426	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12391	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2783	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12365	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11240	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7424	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12829	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2783	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12796	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12766	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12349	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11241	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11425	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7426	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12419	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2783	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12550	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12860	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12353	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12816	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11429	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12806	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12796	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12797	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12368	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11241	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7424	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12836	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2783	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12872	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11245	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12809	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12810	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12754	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12543	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11500	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2786	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2786	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12754	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2856	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12543	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11493	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 705	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2769	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12271	. 2 ⊢ 1 ∈ ℕ
81		2nn 12341	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12754	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17171	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7425	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2786	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12877	. . . 4 ⊢ 8 < ;10
87		1lt10 12884	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12782	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12773	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5132	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17233	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17234	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 17002	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2145 (class class class)co 7414 ℂcc 11125 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 < clt 11270 − cmin 11468 2c2 12322 3c3 12323 4c4 12324 5c5 12325 6c6 12326 7c7 12327 8c8 12328 9c9 12329 ℕ₀cn0 12531 cdc 12739 ↑cexp 14128 ℙcprime 16764

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 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-oadd 8462 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-inf 9416 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-xnn0 12605 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13046 df-fz 13565 df-fzo 13713 df-fl 13856 df-mod 13934 df-seq 14069 df-exp 14129 df-hash 14398 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-dvds 16346 df-gcd 16588 df-prm 16765 df-odz 16859 df-phi 16860 df-pc 16932

This theorem is used by: (None)

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