4001lem3 - Metamath Proof Explorer

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Theorem 4001lem3 17121

Description: Lemma for 4001prm 17123. 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 12531	. . . . . 6 ⊢ 4 ∈ ℕ₀
3		0nn0 12527	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 12732	. . . . 5 ⊢ ;40 ∈ ℕ₀
5	4, 3	deccl 12732	. . . 4 ⊢ ;;400 ∈ ℕ₀
6		1nn 12263	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 12737	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2825	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 12325	. 2 ⊢ 2 ∈ ℕ
10		2nn0 12529	. . . . 5 ⊢ 2 ∈ ℕ₀
11	10, 3	deccl 12732	. . . 4 ⊢ ;20 ∈ ℕ₀
12	11, 3	deccl 12732	. . 3 ⊢ ;;200 ∈ ℕ₀
13	12, 3	deccl 12732	. 2 ⊢ ;;;2000 ∈ ℕ₀
14		0z 12609	. 2 ⊢ 0 ∈ ℤ
15		1nn0 12528	. 2 ⊢ 1 ∈ ℕ₀
16		10nn0 12735	. . . . 5 ⊢ ;10 ∈ ℕ₀
17	16, 3	deccl 12732	. . . 4 ⊢ ;;100 ∈ ℕ₀
18	17, 3	deccl 12732	. . 3 ⊢ ;;;1000 ∈ ℕ₀
19		8nn0 12535	. . . . . 6 ⊢ 8 ∈ ℕ₀
20	19, 3	deccl 12732	. . . . 5 ⊢ ;80 ∈ ℕ₀
21	20, 3	deccl 12732	. . . 4 ⊢ ;;800 ∈ ℕ₀
22		5nn0 12532	. . . . . . 7 ⊢ 5 ∈ ℕ₀
23	22, 10	deccl 12732	. . . . . 6 ⊢ ;52 ∈ ℕ₀
24	23, 15	deccl 12732	. . . . 5 ⊢ ;;521 ∈ ℕ₀
25	24	nn0zi 12627	. . . 4 ⊢ ;;521 ∈ ℤ
26		3nn0 12530	. . . . . . 7 ⊢ 3 ∈ ℕ₀
27	10, 26	deccl 12732	. . . . . 6 ⊢ ;23 ∈ ℕ₀
28	27, 15	deccl 12732	. . . . 5 ⊢ ;;231 ∈ ℕ₀
29	28, 15	deccl 12732	. . . 4 ⊢ ;;;2311 ∈ ℕ₀
30		9nn0 12536	. . . . . 6 ⊢ 9 ∈ ℕ₀
31	30, 3	deccl 12732	. . . . 5 ⊢ ;90 ∈ ℕ₀
32	31, 10	deccl 12732	. . . 4 ⊢ ;;902 ∈ ℕ₀
33	1	4001lem2 17120	. . . 4 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
34	1	4001lem1 17119	. . . 4 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
35		eqid 2728	. . . . 5 ⊢ ;;800 = ;;800
36		eqid 2728	. . . . 5 ⊢ ;;200 = ;;200
37		eqid 2728	. . . . . 6 ⊢ ;80 = ;80
38		eqid 2728	. . . . . 6 ⊢ ;20 = ;20
39		8p2e10 12797	. . . . . 6 ⊢ (8 + 2) = ;10
40		00id 11429	. . . . . 6 ⊢ (0 + 0) = 0
41	19, 3, 10, 3, 37, 38, 39, 40	decadd 12771	. . . . 5 ⊢ (;80 + ;20) = ;;100
42	20, 3, 11, 3, 35, 36, 41, 40	decadd 12771	. . . 4 ⊢ (;;800 + ;;200) = ;;;1000
43	15	dec0h 12739	. . . . . 6 ⊢ 1 = ;01
44		eqid 2728	. . . . . . 7 ⊢ ;;400 = ;;400
45	23	nn0cni 12524	. . . . . . . 8 ⊢ ;52 ∈ ℂ
46	45	addlidi 11442	. . . . . . 7 ⊢ (0 + ;52) = ;52
47		eqid 2728	. . . . . . . 8 ⊢ ;40 = ;40
48		5cn 12340	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
49	48	addridi 11441	. . . . . . . . 9 ⊢ (5 + 0) = 5
50	22	dec0h 12739	. . . . . . . . 9 ⊢ 5 = ;05
51	49, 50	eqtri 2756	. . . . . . . 8 ⊢ (5 + 0) = ;05
52	40, 3	eqeltri 2825	. . . . . . . . 9 ⊢ (0 + 0) ∈ ℕ₀
53		eqid 2728	. . . . . . . . 9 ⊢ ;;521 = ;;521
54		eqid 2728	. . . . . . . . . 10 ⊢ ;52 = ;52
55		5t4e20 12819	. . . . . . . . . 10 ⊢ (5 · 4) = ;20
56		4cn 12337	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
57		2cn 12327	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
58		4t2e8 12420	. . . . . . . . . . 11 ⊢ (4 · 2) = 8
59	56, 57, 58	mulcomli 11263	. . . . . . . . . 10 ⊢ (2 · 4) = 8
60	2, 22, 10, 54, 55, 59	decmul1 12781	. . . . . . . . 9 ⊢ (;52 · 4) = ;;208
61	56	mullidi 11259	. . . . . . . . . . 11 ⊢ (1 · 4) = 4
62	61, 40	oveq12i 7438	. . . . . . . . . 10 ⊢ ((1 · 4) + (0 + 0)) = (4 + 0)
63	56	addridi 11441	. . . . . . . . . 10 ⊢ (4 + 0) = 4
64	62, 63	eqtri 2756	. . . . . . . . 9 ⊢ ((1 · 4) + (0 + 0)) = 4
65	23, 15, 52, 53, 2, 60, 64	decrmanc 12774	. . . . . . . 8 ⊢ ((;;521 · 4) + (0 + 0)) = ;;;2084
66	24	nn0cni 12524	. . . . . . . . . . 11 ⊢ ;;521 ∈ ℂ
67	66	mul01i 11444	. . . . . . . . . 10 ⊢ (;;521 · 0) = 0
68	67	oveq1i 7436	. . . . . . . . 9 ⊢ ((;;521 · 0) + 5) = (0 + 5)
69	48	addlidi 11442	. . . . . . . . 9 ⊢ (0 + 5) = 5
70	68, 69, 50	3eqtri 2760	. . . . . . . 8 ⊢ ((;;521 · 0) + 5) = ;05
71	2, 3, 3, 22, 47, 51, 24, 22, 3, 65, 70	decma2c 12770	. . . . . . 7 ⊢ ((;;521 · ;40) + (5 + 0)) = ;;;;20845
72	67	oveq1i 7436	. . . . . . . 8 ⊢ ((;;521 · 0) + 2) = (0 + 2)
73	57	addlidi 11442	. . . . . . . 8 ⊢ (0 + 2) = 2
74	10	dec0h 12739	. . . . . . . 8 ⊢ 2 = ;02
75	72, 73, 74	3eqtri 2760	. . . . . . 7 ⊢ ((;;521 · 0) + 2) = ;02
76	4, 3, 22, 10, 44, 46, 24, 10, 3, 71, 75	decma2c 12770	. . . . . 6 ⊢ ((;;521 · ;;400) + (0 + ;52)) = ;;;;;208452
77	45	mulridi 11258	. . . . . . 7 ⊢ (;52 · 1) = ;52
78		ax-1cn 11206	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
79	78	mullidi 11259	. . . . . . . . 9 ⊢ (1 · 1) = 1
80	79	oveq1i 7436	. . . . . . . 8 ⊢ ((1 · 1) + 1) = (1 + 1)
81		1p1e2 12377	. . . . . . . 8 ⊢ (1 + 1) = 2
82	80, 81	eqtri 2756	. . . . . . 7 ⊢ ((1 · 1) + 1) = 2
83	23, 15, 15, 53, 15, 77, 82	decrmanc 12774	. . . . . 6 ⊢ ((;;521 · 1) + 1) = ;;522
84	5, 15, 3, 15, 1, 43, 24, 10, 23, 76, 83	decma2c 12770	. . . . 5 ⊢ ((;;521 · 𝑁) + 1) = ;;;;;;2084522
85		eqid 2728	. . . . . 6 ⊢ ;;902 = ;;902
86		6nn0 12533	. . . . . . . 8 ⊢ 6 ∈ ℕ₀
87	2, 86	deccl 12732	. . . . . . 7 ⊢ ;46 ∈ ℕ₀
88	87, 10	deccl 12732	. . . . . 6 ⊢ ;;462 ∈ ℕ₀
89		eqid 2728	. . . . . . 7 ⊢ ;90 = ;90
90		eqid 2728	. . . . . . 7 ⊢ ;;462 = ;;462
91		eqid 2728	. . . . . . . 8 ⊢ ;;;2311 = ;;;2311
92	87	nn0cni 12524	. . . . . . . . 9 ⊢ ;46 ∈ ℂ
93	92	addridi 11441	. . . . . . . 8 ⊢ (;46 + 0) = ;46
94		4p1e5 12398	. . . . . . . . . 10 ⊢ (4 + 1) = 5
95	94, 22	eqeltri 2825	. . . . . . . . 9 ⊢ (4 + 1) ∈ ℕ₀
96		eqid 2728	. . . . . . . . 9 ⊢ ;;231 = ;;231
97		eqid 2728	. . . . . . . . . 10 ⊢ ;23 = ;23
98		9cn 12352	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
99		9t2e18 12839	. . . . . . . . . . . 12 ⊢ (9 · 2) = ;18
100	98, 57, 99	mulcomli 11263	. . . . . . . . . . 11 ⊢ (2 · 9) = ;18
101	15, 19, 10, 100, 81, 39	decaddci2 12779	. . . . . . . . . 10 ⊢ ((2 · 9) + 2) = ;20
102		7nn0 12534	. . . . . . . . . . 11 ⊢ 7 ∈ ℕ₀
103		7p1e8 12401	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
104		3cn 12333	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
105		9t3e27 12840	. . . . . . . . . . . 12 ⊢ (9 · 3) = ;27
106	98, 104, 105	mulcomli 11263	. . . . . . . . . . 11 ⊢ (3 · 9) = ;27
107	10, 102, 103, 106	decsuc 12748	. . . . . . . . . 10 ⊢ ((3 · 9) + 1) = ;28
108	10, 26, 15, 97, 30, 19, 10, 101, 107	decrmac 12775	. . . . . . . . 9 ⊢ ((;23 · 9) + 1) = ;;208
109	98	mullidi 11259	. . . . . . . . . . 11 ⊢ (1 · 9) = 9
110	109, 94	oveq12i 7438	. . . . . . . . . 10 ⊢ ((1 · 9) + (4 + 1)) = (9 + 5)
111		9p5e14 12807	. . . . . . . . . 10 ⊢ (9 + 5) = ;14
112	110, 111	eqtri 2756	. . . . . . . . 9 ⊢ ((1 · 9) + (4 + 1)) = ;14
113	27, 15, 95, 96, 30, 2, 15, 108, 112	decrmac 12775	. . . . . . . 8 ⊢ ((;;231 · 9) + (4 + 1)) = ;;;2084
114	109	oveq1i 7436	. . . . . . . . 9 ⊢ ((1 · 9) + 6) = (9 + 6)
115		9p6e15 12808	. . . . . . . . 9 ⊢ (9 + 6) = ;15
116	114, 115	eqtri 2756	. . . . . . . 8 ⊢ ((1 · 9) + 6) = ;15
117	28, 15, 2, 86, 91, 93, 30, 22, 15, 113, 116	decmac 12769	. . . . . . 7 ⊢ ((;;;2311 · 9) + (;46 + 0)) = ;;;;20845
118	29	nn0cni 12524	. . . . . . . . . 10 ⊢ ;;;2311 ∈ ℂ
119	118	mul01i 11444	. . . . . . . . 9 ⊢ (;;;2311 · 0) = 0
120	119	oveq1i 7436	. . . . . . . 8 ⊢ ((;;;2311 · 0) + 2) = (0 + 2)
121	120, 73, 74	3eqtri 2760	. . . . . . 7 ⊢ ((;;;2311 · 0) + 2) = ;02
122	30, 3, 87, 10, 89, 90, 29, 10, 3, 117, 121	decma2c 12770	. . . . . 6 ⊢ ((;;;2311 · ;90) + ;;462) = ;;;;;208452
123		2t2e4 12416	. . . . . . . . 9 ⊢ (2 · 2) = 4
124		3t2e6 12418	. . . . . . . . 9 ⊢ (3 · 2) = 6
125	10, 10, 26, 97, 123, 124	decmul1 12781	. . . . . . . 8 ⊢ (;23 · 2) = ;46
126	57	mullidi 11259	. . . . . . . 8 ⊢ (1 · 2) = 2
127	10, 27, 15, 96, 125, 126	decmul1 12781	. . . . . . 7 ⊢ (;;231 · 2) = ;;462
128	10, 28, 15, 91, 127, 126	decmul1 12781	. . . . . 6 ⊢ (;;;2311 · 2) = ;;;4622
129	29, 31, 10, 85, 10, 88, 122, 128	decmul2c 12783	. . . . 5 ⊢ (;;;2311 · ;;902) = ;;;;;;2084522
130	84, 129	eqtr4i 2759	. . . 4 ⊢ ((;;521 · 𝑁) + 1) = (;;;2311 · ;;902)
131	8, 9, 21, 25, 29, 15, 12, 32, 33, 34, 42, 130	modxai 17046	. . 3 ⊢ ((2↑;;;1000) mod 𝑁) = (1 mod 𝑁)
132	18	nn0cni 12524	. . . 4 ⊢ ;;;1000 ∈ ℂ
133		eqid 2728	. . . . 5 ⊢ ;;;1000 = ;;;1000
134		eqid 2728	. . . . . 6 ⊢ ;;100 = ;;100
135	10	dec0u 12738	. . . . . 6 ⊢ (;10 · 2) = ;20
136	57	mul02i 11443	. . . . . 6 ⊢ (0 · 2) = 0
137	10, 16, 3, 134, 135, 136	decmul1 12781	. . . . 5 ⊢ (;;100 · 2) = ;;200
138	10, 17, 3, 133, 137, 136	decmul1 12781	. . . 4 ⊢ (;;;1000 · 2) = ;;;2000
139	132, 57, 138	mulcomli 11263	. . 3 ⊢ (2 · ;;;1000) = ;;;2000
140	8	nncni 12262	. . . . . 6 ⊢ 𝑁 ∈ ℂ
141	140	mul02i 11443	. . . . 5 ⊢ (0 · 𝑁) = 0
142	141	oveq1i 7436	. . . 4 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
143	78	addlidi 11442	. . . . 5 ⊢ (0 + 1) = 1
144	79, 143	eqtr4i 2759	. . . 4 ⊢ (1 · 1) = (0 + 1)
145	142, 144	eqtr4i 2759	. . 3 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
146	8, 9, 18, 14, 15, 15, 131, 139, 145	mod2xi 17047	. 2 ⊢ ((2↑;;;2000) mod 𝑁) = (1 mod 𝑁)
147	13	nn0cni 12524	. . . 4 ⊢ ;;;2000 ∈ ℂ
148		eqid 2728	. . . . 5 ⊢ ;;;2000 = ;;;2000
149	10, 10, 3, 38, 123, 136	decmul1 12781	. . . . . 6 ⊢ (;20 · 2) = ;40
150	10, 11, 3, 36, 149, 136	decmul1 12781	. . . . 5 ⊢ (;;200 · 2) = ;;400
151	10, 12, 3, 148, 150, 136	decmul1 12781	. . . 4 ⊢ (;;;2000 · 2) = ;;;4000
152	147, 57, 151	mulcomli 11263	. . 3 ⊢ (2 · ;;;2000) = ;;;4000
153	5, 3	deccl 12732	. . . . 5 ⊢ ;;;4000 ∈ ℕ₀
154	153	nn0cni 12524	. . . 4 ⊢ ;;;4000 ∈ ℂ
155		eqid 2728	. . . . . 6 ⊢ ;;;4000 = ;;;4000
156	5, 3, 143, 155	decsuc 12748	. . . . 5 ⊢ (;;;4000 + 1) = ;;;4001
157	1, 156	eqtr4i 2759	. . . 4 ⊢ 𝑁 = (;;;4000 + 1)
158	154, 78, 157	mvrraddi 11517	. . 3 ⊢ (𝑁 − 1) = ;;;4000
159	152, 158	eqtr4i 2759	. 2 ⊢ (2 · ;;;2000) = (𝑁 − 1)
160	8, 9, 13, 14, 15, 15, 146, 159, 145	mod2xi 17047	1 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 (class class class)co 7426 0cc0 11148 1c1 11149 + caddc 11151 · cmul 11153 − cmin 11484 ℕcn 12252 2c2 12307 3c3 12308 4c4 12309 5c5 12310 6c6 12311 7c7 12312 8c8 12313 9c9 12314 ℕ₀cn0 12512 cdc 12717 mod cmo 13876 ↑cexp 14068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-sup 9475 df-inf 9476 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-rp 13017 df-fl 13799 df-mod 13877 df-seq 14009 df-exp 14069

This theorem is referenced by: 4001prm 17123

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