4001lem3 - Metamath Proof Explorer

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Theorem 4001lem3 16468

Description: Lemma for 4001prm 16470. Calculate a power mod. In decimal, we calculate 2↑1000 = 2↑800 · 2↑200≡2311 · 902 = 521𝑁 + 1 and finally 2↑(𝑁 − 1) = (2↑1000)↑4≡1↑4 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)

**Hypothesis**
Ref	Expression
4001prm.1	⊢ 𝑁 = ;;;4001

**Assertion**
Ref	Expression
4001lem3	⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

**Proof of Theorem 4001lem3**
Step	Hyp	Ref	Expression
1		4001prm.1	. . 3 ⊢ 𝑁 = ;;;4001
2		4nn0 11904	. . . . . 6 ⊢ 4 ∈ ℕ₀
3		0nn0 11900	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 12101	. . . . 5 ⊢ ;40 ∈ ℕ₀
5	4, 3	deccl 12101	. . . 4 ⊢ ;;400 ∈ ℕ₀
6		1nn 11636	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 12106	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2886	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 11698	. 2 ⊢ 2 ∈ ℕ
10		2nn0 11902	. . . . 5 ⊢ 2 ∈ ℕ₀
11	10, 3	deccl 12101	. . . 4 ⊢ ;20 ∈ ℕ₀
12	11, 3	deccl 12101	. . 3 ⊢ ;;200 ∈ ℕ₀
13	12, 3	deccl 12101	. 2 ⊢ ;;;2000 ∈ ℕ₀
14		0z 11980	. 2 ⊢ 0 ∈ ℤ
15		1nn0 11901	. 2 ⊢ 1 ∈ ℕ₀
16		10nn0 12104	. . . . 5 ⊢ ;10 ∈ ℕ₀
17	16, 3	deccl 12101	. . . 4 ⊢ ;;100 ∈ ℕ₀
18	17, 3	deccl 12101	. . 3 ⊢ ;;;1000 ∈ ℕ₀
19		8nn0 11908	. . . . . 6 ⊢ 8 ∈ ℕ₀
20	19, 3	deccl 12101	. . . . 5 ⊢ ;80 ∈ ℕ₀
21	20, 3	deccl 12101	. . . 4 ⊢ ;;800 ∈ ℕ₀
22		5nn0 11905	. . . . . . 7 ⊢ 5 ∈ ℕ₀
23	22, 10	deccl 12101	. . . . . 6 ⊢ ;52 ∈ ℕ₀
24	23, 15	deccl 12101	. . . . 5 ⊢ ;;521 ∈ ℕ₀
25	24	nn0zi 11995	. . . 4 ⊢ ;;521 ∈ ℤ
26		3nn0 11903	. . . . . . 7 ⊢ 3 ∈ ℕ₀
27	10, 26	deccl 12101	. . . . . 6 ⊢ ;23 ∈ ℕ₀
28	27, 15	deccl 12101	. . . . 5 ⊢ ;;231 ∈ ℕ₀
29	28, 15	deccl 12101	. . . 4 ⊢ ;;;2311 ∈ ℕ₀
30		9nn0 11909	. . . . . 6 ⊢ 9 ∈ ℕ₀
31	30, 3	deccl 12101	. . . . 5 ⊢ ;90 ∈ ℕ₀
32	31, 10	deccl 12101	. . . 4 ⊢ ;;902 ∈ ℕ₀
33	1	4001lem2 16467	. . . 4 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
34	1	4001lem1 16466	. . . 4 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
35		eqid 2798	. . . . 5 ⊢ ;;800 = ;;800
36		eqid 2798	. . . . 5 ⊢ ;;200 = ;;200
37		eqid 2798	. . . . . 6 ⊢ ;80 = ;80
38		eqid 2798	. . . . . 6 ⊢ ;20 = ;20
39		8p2e10 12166	. . . . . 6 ⊢ (8 + 2) = ;10
40		00id 10804	. . . . . 6 ⊢ (0 + 0) = 0
41	19, 3, 10, 3, 37, 38, 39, 40	decadd 12140	. . . . 5 ⊢ (;80 + ;20) = ;;100
42	20, 3, 11, 3, 35, 36, 41, 40	decadd 12140	. . . 4 ⊢ (;;800 + ;;200) = ;;;1000
43	15	dec0h 12108	. . . . . 6 ⊢ 1 = ;01
44		eqid 2798	. . . . . . 7 ⊢ ;;400 = ;;400
45	23	nn0cni 11897	. . . . . . . 8 ⊢ ;52 ∈ ℂ
46	45	addid2i 10817	. . . . . . 7 ⊢ (0 + ;52) = ;52
47		eqid 2798	. . . . . . . 8 ⊢ ;40 = ;40
48		5cn 11713	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
49	48	addid1i 10816	. . . . . . . . 9 ⊢ (5 + 0) = 5
50	22	dec0h 12108	. . . . . . . . 9 ⊢ 5 = ;05
51	49, 50	eqtri 2821	. . . . . . . 8 ⊢ (5 + 0) = ;05
52	40, 3	eqeltri 2886	. . . . . . . . 9 ⊢ (0 + 0) ∈ ℕ₀
53		eqid 2798	. . . . . . . . 9 ⊢ ;;521 = ;;521
54		eqid 2798	. . . . . . . . . 10 ⊢ ;52 = ;52
55		5t4e20 12188	. . . . . . . . . 10 ⊢ (5 · 4) = ;20
56		4cn 11710	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
57		2cn 11700	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
58		4t2e8 11793	. . . . . . . . . . 11 ⊢ (4 · 2) = 8
59	56, 57, 58	mulcomli 10639	. . . . . . . . . 10 ⊢ (2 · 4) = 8
60	2, 22, 10, 54, 55, 59	decmul1 12150	. . . . . . . . 9 ⊢ (;52 · 4) = ;;208
61	56	mulid2i 10635	. . . . . . . . . . 11 ⊢ (1 · 4) = 4
62	61, 40	oveq12i 7147	. . . . . . . . . 10 ⊢ ((1 · 4) + (0 + 0)) = (4 + 0)
63	56	addid1i 10816	. . . . . . . . . 10 ⊢ (4 + 0) = 4
64	62, 63	eqtri 2821	. . . . . . . . 9 ⊢ ((1 · 4) + (0 + 0)) = 4
65	23, 15, 52, 53, 2, 60, 64	decrmanc 12143	. . . . . . . 8 ⊢ ((;;521 · 4) + (0 + 0)) = ;;;2084
66	24	nn0cni 11897	. . . . . . . . . . 11 ⊢ ;;521 ∈ ℂ
67	66	mul01i 10819	. . . . . . . . . 10 ⊢ (;;521 · 0) = 0
68	67	oveq1i 7145	. . . . . . . . 9 ⊢ ((;;521 · 0) + 5) = (0 + 5)
69	48	addid2i 10817	. . . . . . . . 9 ⊢ (0 + 5) = 5
70	68, 69, 50	3eqtri 2825	. . . . . . . 8 ⊢ ((;;521 · 0) + 5) = ;05
71	2, 3, 3, 22, 47, 51, 24, 22, 3, 65, 70	decma2c 12139	. . . . . . 7 ⊢ ((;;521 · ;40) + (5 + 0)) = ;;;;20845
72	67	oveq1i 7145	. . . . . . . 8 ⊢ ((;;521 · 0) + 2) = (0 + 2)
73	57	addid2i 10817	. . . . . . . 8 ⊢ (0 + 2) = 2
74	10	dec0h 12108	. . . . . . . 8 ⊢ 2 = ;02
75	72, 73, 74	3eqtri 2825	. . . . . . 7 ⊢ ((;;521 · 0) + 2) = ;02
76	4, 3, 22, 10, 44, 46, 24, 10, 3, 71, 75	decma2c 12139	. . . . . 6 ⊢ ((;;521 · ;;400) + (0 + ;52)) = ;;;;;208452
77	45	mulid1i 10634	. . . . . . 7 ⊢ (;52 · 1) = ;52
78		ax-1cn 10584	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
79	78	mulid2i 10635	. . . . . . . . 9 ⊢ (1 · 1) = 1
80	79	oveq1i 7145	. . . . . . . 8 ⊢ ((1 · 1) + 1) = (1 + 1)
81		1p1e2 11750	. . . . . . . 8 ⊢ (1 + 1) = 2
82	80, 81	eqtri 2821	. . . . . . 7 ⊢ ((1 · 1) + 1) = 2
83	23, 15, 15, 53, 15, 77, 82	decrmanc 12143	. . . . . 6 ⊢ ((;;521 · 1) + 1) = ;;522
84	5, 15, 3, 15, 1, 43, 24, 10, 23, 76, 83	decma2c 12139	. . . . 5 ⊢ ((;;521 · 𝑁) + 1) = ;;;;;;2084522
85		eqid 2798	. . . . . 6 ⊢ ;;902 = ;;902
86		6nn0 11906	. . . . . . . 8 ⊢ 6 ∈ ℕ₀
87	2, 86	deccl 12101	. . . . . . 7 ⊢ ;46 ∈ ℕ₀
88	87, 10	deccl 12101	. . . . . 6 ⊢ ;;462 ∈ ℕ₀
89		eqid 2798	. . . . . . 7 ⊢ ;90 = ;90
90		eqid 2798	. . . . . . 7 ⊢ ;;462 = ;;462
91		eqid 2798	. . . . . . . 8 ⊢ ;;;2311 = ;;;2311
92	87	nn0cni 11897	. . . . . . . . 9 ⊢ ;46 ∈ ℂ
93	92	addid1i 10816	. . . . . . . 8 ⊢ (;46 + 0) = ;46
94		4p1e5 11771	. . . . . . . . . 10 ⊢ (4 + 1) = 5
95	94, 22	eqeltri 2886	. . . . . . . . 9 ⊢ (4 + 1) ∈ ℕ₀
96		eqid 2798	. . . . . . . . 9 ⊢ ;;231 = ;;231
97		eqid 2798	. . . . . . . . . 10 ⊢ ;23 = ;23
98		9cn 11725	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
99		9t2e18 12208	. . . . . . . . . . . 12 ⊢ (9 · 2) = ;18
100	98, 57, 99	mulcomli 10639	. . . . . . . . . . 11 ⊢ (2 · 9) = ;18
101	15, 19, 10, 100, 81, 39	decaddci2 12148	. . . . . . . . . 10 ⊢ ((2 · 9) + 2) = ;20
102		7nn0 11907	. . . . . . . . . . 11 ⊢ 7 ∈ ℕ₀
103		7p1e8 11774	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
104		3cn 11706	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
105		9t3e27 12209	. . . . . . . . . . . 12 ⊢ (9 · 3) = ;27
106	98, 104, 105	mulcomli 10639	. . . . . . . . . . 11 ⊢ (3 · 9) = ;27
107	10, 102, 103, 106	decsuc 12117	. . . . . . . . . 10 ⊢ ((3 · 9) + 1) = ;28
108	10, 26, 15, 97, 30, 19, 10, 101, 107	decrmac 12144	. . . . . . . . 9 ⊢ ((;23 · 9) + 1) = ;;208
109	98	mulid2i 10635	. . . . . . . . . . 11 ⊢ (1 · 9) = 9
110	109, 94	oveq12i 7147	. . . . . . . . . 10 ⊢ ((1 · 9) + (4 + 1)) = (9 + 5)
111		9p5e14 12176	. . . . . . . . . 10 ⊢ (9 + 5) = ;14
112	110, 111	eqtri 2821	. . . . . . . . 9 ⊢ ((1 · 9) + (4 + 1)) = ;14
113	27, 15, 95, 96, 30, 2, 15, 108, 112	decrmac 12144	. . . . . . . 8 ⊢ ((;;231 · 9) + (4 + 1)) = ;;;2084
114	109	oveq1i 7145	. . . . . . . . 9 ⊢ ((1 · 9) + 6) = (9 + 6)
115		9p6e15 12177	. . . . . . . . 9 ⊢ (9 + 6) = ;15
116	114, 115	eqtri 2821	. . . . . . . 8 ⊢ ((1 · 9) + 6) = ;15
117	28, 15, 2, 86, 91, 93, 30, 22, 15, 113, 116	decmac 12138	. . . . . . 7 ⊢ ((;;;2311 · 9) + (;46 + 0)) = ;;;;20845
118	29	nn0cni 11897	. . . . . . . . . 10 ⊢ ;;;2311 ∈ ℂ
119	118	mul01i 10819	. . . . . . . . 9 ⊢ (;;;2311 · 0) = 0
120	119	oveq1i 7145	. . . . . . . 8 ⊢ ((;;;2311 · 0) + 2) = (0 + 2)
121	120, 73, 74	3eqtri 2825	. . . . . . 7 ⊢ ((;;;2311 · 0) + 2) = ;02
122	30, 3, 87, 10, 89, 90, 29, 10, 3, 117, 121	decma2c 12139	. . . . . 6 ⊢ ((;;;2311 · ;90) + ;;462) = ;;;;;208452
123		2t2e4 11789	. . . . . . . . 9 ⊢ (2 · 2) = 4
124		3t2e6 11791	. . . . . . . . 9 ⊢ (3 · 2) = 6
125	10, 10, 26, 97, 123, 124	decmul1 12150	. . . . . . . 8 ⊢ (;23 · 2) = ;46
126	57	mulid2i 10635	. . . . . . . 8 ⊢ (1 · 2) = 2
127	10, 27, 15, 96, 125, 126	decmul1 12150	. . . . . . 7 ⊢ (;;231 · 2) = ;;462
128	10, 28, 15, 91, 127, 126	decmul1 12150	. . . . . 6 ⊢ (;;;2311 · 2) = ;;;4622
129	29, 31, 10, 85, 10, 88, 122, 128	decmul2c 12152	. . . . 5 ⊢ (;;;2311 · ;;902) = ;;;;;;2084522
130	84, 129	eqtr4i 2824	. . . 4 ⊢ ((;;521 · 𝑁) + 1) = (;;;2311 · ;;902)
131	8, 9, 21, 25, 29, 15, 12, 32, 33, 34, 42, 130	modxai 16394	. . 3 ⊢ ((2↑;;;1000) mod 𝑁) = (1 mod 𝑁)
132	18	nn0cni 11897	. . . 4 ⊢ ;;;1000 ∈ ℂ
133		eqid 2798	. . . . 5 ⊢ ;;;1000 = ;;;1000
134		eqid 2798	. . . . . 6 ⊢ ;;100 = ;;100
135	10	dec0u 12107	. . . . . 6 ⊢ (;10 · 2) = ;20
136	57	mul02i 10818	. . . . . 6 ⊢ (0 · 2) = 0
137	10, 16, 3, 134, 135, 136	decmul1 12150	. . . . 5 ⊢ (;;100 · 2) = ;;200
138	10, 17, 3, 133, 137, 136	decmul1 12150	. . . 4 ⊢ (;;;1000 · 2) = ;;;2000
139	132, 57, 138	mulcomli 10639	. . 3 ⊢ (2 · ;;;1000) = ;;;2000
140	8	nncni 11635	. . . . . 6 ⊢ 𝑁 ∈ ℂ
141	140	mul02i 10818	. . . . 5 ⊢ (0 · 𝑁) = 0
142	141	oveq1i 7145	. . . 4 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
143	78	addid2i 10817	. . . . 5 ⊢ (0 + 1) = 1
144	79, 143	eqtr4i 2824	. . . 4 ⊢ (1 · 1) = (0 + 1)
145	142, 144	eqtr4i 2824	. . 3 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
146	8, 9, 18, 14, 15, 15, 131, 139, 145	mod2xi 16395	. 2 ⊢ ((2↑;;;2000) mod 𝑁) = (1 mod 𝑁)
147	13	nn0cni 11897	. . . 4 ⊢ ;;;2000 ∈ ℂ
148		eqid 2798	. . . . 5 ⊢ ;;;2000 = ;;;2000
149	10, 10, 3, 38, 123, 136	decmul1 12150	. . . . . 6 ⊢ (;20 · 2) = ;40
150	10, 11, 3, 36, 149, 136	decmul1 12150	. . . . 5 ⊢ (;;200 · 2) = ;;400
151	10, 12, 3, 148, 150, 136	decmul1 12150	. . . 4 ⊢ (;;;2000 · 2) = ;;;4000
152	147, 57, 151	mulcomli 10639	. . 3 ⊢ (2 · ;;;2000) = ;;;4000
153	5, 3	deccl 12101	. . . . 5 ⊢ ;;;4000 ∈ ℕ₀
154	153	nn0cni 11897	. . . 4 ⊢ ;;;4000 ∈ ℂ
155		eqid 2798	. . . . . 6 ⊢ ;;;4000 = ;;;4000
156	5, 3, 143, 155	decsuc 12117	. . . . 5 ⊢ (;;;4000 + 1) = ;;;4001
157	1, 156	eqtr4i 2824	. . . 4 ⊢ 𝑁 = (;;;4000 + 1)
158	154, 78, 157	mvrraddi 10892	. . 3 ⊢ (𝑁 − 1) = ;;;4000
159	152, 158	eqtr4i 2824	. 2 ⊢ (2 · ;;;2000) = (𝑁 − 1)
160	8, 9, 13, 14, 15, 15, 146, 159, 145	mod2xi 16395	1 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 ℕcn 11625 2c2 11680 3c3 11681 4c4 11682 5c5 11683 6c6 11684 7c7 11685 8c8 11686 9c9 11687 ℕ₀cn0 11885 cdc 12086 mod cmo 13232 ↑cexp 13425

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426

This theorem is referenced by: 4001prm 16470

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