2503prm - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 2503prm		Structured version Visualization version GIF version

Theorem 2503prm 17176

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 17160	. 2 ⊢ ;;139 ∈ ℙ
2		1nn0 12497	. . 3 ⊢ 1 ∈ ℕ₀
3		8nn 12313	. . 3 ⊢ 8 ∈ ℕ
4	2, 3	decnncl 12712	. 2 ⊢ ;18 ∈ ℕ
5		2503prm.1	. . . . 5 ⊢ 𝑁 = ;;;2503
6		2nn0 12498	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
7		5nn0 12501	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
8	6, 7	deccl 12703	. . . . . . 7 ⊢ ;25 ∈ ℕ₀
9		0nn0 12496	. . . . . . 7 ⊢ 0 ∈ ℕ₀
10	8, 9	deccl 12703	. . . . . 6 ⊢ ;;250 ∈ ℕ₀
11		2p1e3 12359	. . . . . 6 ⊢ (2 + 1) = 3
12		eqid 2762	. . . . . 6 ⊢ ;;;2502 = ;;;2502
13	10, 6, 11, 12	decsuc 12724	. . . . 5 ⊢ (;;;2502 + 1) = ;;;2503
14	5, 13	eqtr4i 2788	. . . 4 ⊢ 𝑁 = (;;;2502 + 1)
15	14	oveq1i 7406	. . 3 ⊢ (𝑁 − 1) = ((;;;2502 + 1) − 1)
16		8nn0 12504	. . . . . 6 ⊢ 8 ∈ ℕ₀
17	2, 16	deccl 12703	. . . . 5 ⊢ ;18 ∈ ℕ₀
18		3nn0 12499	. . . . . 6 ⊢ 3 ∈ ℕ₀
19	2, 18	deccl 12703	. . . . 5 ⊢ ;13 ∈ ℕ₀
20		9nn0 12505	. . . . 5 ⊢ 9 ∈ ℕ₀
21		eqid 2762	. . . . 5 ⊢ ;;139 = ;;139
22		6nn0 12502	. . . . . 6 ⊢ 6 ∈ ℕ₀
23	2, 22	deccl 12703	. . . . 5 ⊢ ;16 ∈ ℕ₀
24		eqid 2762	. . . . . 6 ⊢ ;13 = ;13
25		eqid 2762	. . . . . 6 ⊢ ;16 = ;16
26		7nn0 12503	. . . . . . 7 ⊢ 7 ∈ ℕ₀
27		eqid 2762	. . . . . . 7 ⊢ ;18 = ;18
28		6cn 12309	. . . . . . . . 9 ⊢ 6 ∈ ℂ
29		ax-1cn 11131	. . . . . . . . 9 ⊢ 1 ∈ ℂ
30		6p1e7 12365	. . . . . . . . 9 ⊢ (6 + 1) = 7
31	28, 29, 30	addcomli 11375	. . . . . . . 8 ⊢ (1 + 6) = 7
32	26	dec0h 12715	. . . . . . . 8 ⊢ 7 = ;07
33	31, 32	eqtri 2785	. . . . . . 7 ⊢ (1 + 6) = ;07
34	29	mulridi 11186	. . . . . . . . 9 ⊢ (1 · 1) = 1
35	29	addlidi 11371	. . . . . . . . 9 ⊢ (0 + 1) = 1
36	34, 35	oveq12i 7408	. . . . . . . 8 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
37		1p1e2 12341	. . . . . . . 8 ⊢ (1 + 1) = 2
38	36, 37	eqtri 2785	. . . . . . 7 ⊢ ((1 · 1) + (0 + 1)) = 2
39		8cn 12315	. . . . . . . . . 10 ⊢ 8 ∈ ℂ
40	39	mulridi 11186	. . . . . . . . 9 ⊢ (8 · 1) = 8
41	40	oveq1i 7406	. . . . . . . 8 ⊢ ((8 · 1) + 7) = (8 + 7)
42		8p7e15 12778	. . . . . . . 8 ⊢ (8 + 7) = ;15
43	41, 42	eqtri 2785	. . . . . . 7 ⊢ ((8 · 1) + 7) = ;15
44	2, 16, 9, 26, 27, 33, 2, 7, 2, 38, 43	decmac 12745	. . . . . 6 ⊢ ((;18 · 1) + (1 + 6)) = ;25
45	22	dec0h 12715	. . . . . . 7 ⊢ 6 = ;06
46		3cn 12299	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
47	46	mullidi 11187	. . . . . . . . 9 ⊢ (1 · 3) = 3
48	46	addlidi 11371	. . . . . . . . 9 ⊢ (0 + 3) = 3
49	47, 48	oveq12i 7408	. . . . . . . 8 ⊢ ((1 · 3) + (0 + 3)) = (3 + 3)
50		3p3e6 12369	. . . . . . . 8 ⊢ (3 + 3) = 6
51	49, 50	eqtri 2785	. . . . . . 7 ⊢ ((1 · 3) + (0 + 3)) = 6
52		4nn0 12500	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
53		8t3e24 12809	. . . . . . . 8 ⊢ (8 · 3) = ;24
54		4cn 12303	. . . . . . . . 9 ⊢ 4 ∈ ℂ
55		6p4e10 12765	. . . . . . . . 9 ⊢ (6 + 4) = ;10
56	28, 54, 55	addcomli 11375	. . . . . . . 8 ⊢ (4 + 6) = ;10
57	6, 52, 22, 53, 11, 56	decaddci2 12755	. . . . . . 7 ⊢ ((8 · 3) + 6) = ;30
58	2, 16, 9, 22, 27, 45, 18, 9, 18, 51, 57	decmac 12745	. . . . . 6 ⊢ ((;18 · 3) + 6) = ;60
59	2, 18, 2, 22, 24, 25, 17, 9, 22, 44, 58	decma2c 12746	. . . . 5 ⊢ ((;18 · ;13) + ;16) = ;;250
60		9cn 12318	. . . . . . . . 9 ⊢ 9 ∈ ℂ
61	60	mullidi 11187	. . . . . . . 8 ⊢ (1 · 9) = 9
62	61	oveq1i 7406	. . . . . . 7 ⊢ ((1 · 9) + 7) = (9 + 7)
63		9p7e16 12785	. . . . . . 7 ⊢ (9 + 7) = ;16
64	62, 63	eqtri 2785	. . . . . 6 ⊢ ((1 · 9) + 7) = ;16
65		9t8e72 12821	. . . . . . 7 ⊢ (9 · 8) = ;72
66	60, 39, 65	mulcomli 11191	. . . . . 6 ⊢ (8 · 9) = ;72
67	20, 2, 16, 27, 6, 26, 64, 66	decmul1c 12758	. . . . 5 ⊢ (;18 · 9) = ;;162
68	17, 19, 20, 21, 6, 23, 59, 67	decmul2c 12759	. . . 4 ⊢ (;18 · ;;139) = ;;;2502
69	10, 6	deccl 12703	. . . . . 6 ⊢ ;;;2502 ∈ ℕ₀
70	69	nn0cni 12493	. . . . 5 ⊢ ;;;2502 ∈ ℂ
71	70, 29	pncan3oi 11446	. . . 4 ⊢ ((;;;2502 + 1) − 1) = ;;;2502
72	68, 71	eqtr4i 2788	. . 3 ⊢ (;18 · ;;139) = ((;;;2502 + 1) − 1)
73	15, 72	eqtr4i 2788	. 2 ⊢ (𝑁 − 1) = (;18 · ;;139)
74	10, 18	deccl 12703	. . . . . 6 ⊢ ;;;2503 ∈ ℕ₀
75	5, 74	eqeltri 2858	. . . . 5 ⊢ 𝑁 ∈ ℕ₀
76	75	nn0cni 12493	. . . 4 ⊢ 𝑁 ∈ ℂ
77		npcan 11439	. . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
78	76, 29, 77	mp2an 702	. . 3 ⊢ ((𝑁 − 1) + 1) = 𝑁
79	78	eqcomi 2771	. 2 ⊢ 𝑁 = ((𝑁 − 1) + 1)
80		1nn 12221	. 2 ⊢ 1 ∈ ℕ
81		2nn 12291	. 2 ⊢ 2 ∈ ℕ
82	19, 20	deccl 12703	. . . . 5 ⊢ ;;139 ∈ ℕ₀
83	82	numexp1 17112	. . . 4 ⊢ (;;139↑1) = ;;139
84	83	oveq2i 7407	. . 3 ⊢ (;18 · (;;139↑1)) = (;18 · ;;139)
85	73, 84	eqtr4i 2788	. 2 ⊢ (𝑁 − 1) = (;18 · (;;139↑1))
86		8lt10 12826	. . . 4 ⊢ 8 < ;10
87		1lt10 12833	. . . . 5 ⊢ 1 < ;10
88	80, 18, 2, 87	declti 12731	. . . 4 ⊢ 1 < ;13
89	2, 19, 16, 20, 86, 88	decltc 12722	. . 3 ⊢ ;18 < ;;139
90	89, 83	breqtrri 5127	. 2 ⊢ ;18 < (;;139↑1)
91	5	2503lem2 17174	. 2 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
92	5	2503lem3 17175	. 2 ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
93	1, 4, 73, 79, 4, 80, 81, 85, 90, 91, 92	pockthi 16943	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff setvar class

Syntax hints: = wceq 1560 ∈ wcel 2142 (class class class)co 7396 ℂcc 11071 0cc0 11073 1c1 11074 + caddc 11076 · cmul 11078 < clt 11216 − cmin 11414 2c2 12272 3c3 12273 4c4 12274 5c5 12275 6c6 12276 7c7 12277 8c8 12278 9c9 12279 ℕ₀cn0 12481 cdc 12688 ↑cexp 14074 ℙcprime 16705

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9388 df-inf 9389 df-dju 9859 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-xnn0 12555 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-fz 13513 df-fzo 13660 df-fl 13802 df-mod 13880 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-dvds 16287 df-gcd 16529 df-prm 16706 df-odz 16800 df-phi 16801 df-pc 16873

This theorem is referenced by: (None)

W3C validator