4001lem3 - Metamath Proof Explorer

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Theorem 4001lem3 17071

Description: Lemma for 4001prm 17073. 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 12421	. . . . . 6 ⊢ 4 ∈ ℕ₀
3		0nn0 12417	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 12623	. . . . 5 ⊢ ;40 ∈ ℕ₀
5	4, 3	deccl 12623	. . . 4 ⊢ ;;400 ∈ ℕ₀
6		1nn 12157	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 12628	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2833	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 12219	. 2 ⊢ 2 ∈ ℕ
10		2nn0 12419	. . . . 5 ⊢ 2 ∈ ℕ₀
11	10, 3	deccl 12623	. . . 4 ⊢ ;20 ∈ ℕ₀
12	11, 3	deccl 12623	. . 3 ⊢ ;;200 ∈ ℕ₀
13	12, 3	deccl 12623	. 2 ⊢ ;;;2000 ∈ ℕ₀
14		0z 12500	. 2 ⊢ 0 ∈ ℤ
15		1nn0 12418	. 2 ⊢ 1 ∈ ℕ₀
16		10nn0 12626	. . . . 5 ⊢ ;10 ∈ ℕ₀
17	16, 3	deccl 12623	. . . 4 ⊢ ;;100 ∈ ℕ₀
18	17, 3	deccl 12623	. . 3 ⊢ ;;;1000 ∈ ℕ₀
19		8nn0 12425	. . . . . 6 ⊢ 8 ∈ ℕ₀
20	19, 3	deccl 12623	. . . . 5 ⊢ ;80 ∈ ℕ₀
21	20, 3	deccl 12623	. . . 4 ⊢ ;;800 ∈ ℕ₀
22		5nn0 12422	. . . . . . 7 ⊢ 5 ∈ ℕ₀
23	22, 10	deccl 12623	. . . . . 6 ⊢ ;52 ∈ ℕ₀
24	23, 15	deccl 12623	. . . . 5 ⊢ ;;521 ∈ ℕ₀
25	24	nn0zi 12517	. . . 4 ⊢ ;;521 ∈ ℤ
26		3nn0 12420	. . . . . . 7 ⊢ 3 ∈ ℕ₀
27	10, 26	deccl 12623	. . . . . 6 ⊢ ;23 ∈ ℕ₀
28	27, 15	deccl 12623	. . . . 5 ⊢ ;;231 ∈ ℕ₀
29	28, 15	deccl 12623	. . . 4 ⊢ ;;;2311 ∈ ℕ₀
30		9nn0 12426	. . . . . 6 ⊢ 9 ∈ ℕ₀
31	30, 3	deccl 12623	. . . . 5 ⊢ ;90 ∈ ℕ₀
32	31, 10	deccl 12623	. . . 4 ⊢ ;;902 ∈ ℕ₀
33	1	4001lem2 17070	. . . 4 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
34	1	4001lem1 17069	. . . 4 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
35		eqid 2737	. . . . 5 ⊢ ;;800 = ;;800
36		eqid 2737	. . . . 5 ⊢ ;;200 = ;;200
37		eqid 2737	. . . . . 6 ⊢ ;80 = ;80
38		eqid 2737	. . . . . 6 ⊢ ;20 = ;20
39		8p2e10 12688	. . . . . 6 ⊢ (8 + 2) = ;10
40		00id 11309	. . . . . 6 ⊢ (0 + 0) = 0
41	19, 3, 10, 3, 37, 38, 39, 40	decadd 12662	. . . . 5 ⊢ (;80 + ;20) = ;;100
42	20, 3, 11, 3, 35, 36, 41, 40	decadd 12662	. . . 4 ⊢ (;;800 + ;;200) = ;;;1000
43	15	dec0h 12630	. . . . . 6 ⊢ 1 = ;01
44		eqid 2737	. . . . . . 7 ⊢ ;;400 = ;;400
45	23	nn0cni 12414	. . . . . . . 8 ⊢ ;52 ∈ ℂ
46	45	addlidi 11322	. . . . . . 7 ⊢ (0 + ;52) = ;52
47		eqid 2737	. . . . . . . 8 ⊢ ;40 = ;40
48		5cn 12234	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
49	48	addridi 11321	. . . . . . . . 9 ⊢ (5 + 0) = 5
50	22	dec0h 12630	. . . . . . . . 9 ⊢ 5 = ;05
51	49, 50	eqtri 2760	. . . . . . . 8 ⊢ (5 + 0) = ;05
52	40, 3	eqeltri 2833	. . . . . . . . 9 ⊢ (0 + 0) ∈ ℕ₀
53		eqid 2737	. . . . . . . . 9 ⊢ ;;521 = ;;521
54		eqid 2737	. . . . . . . . . 10 ⊢ ;52 = ;52
55		5t4e20 12710	. . . . . . . . . 10 ⊢ (5 · 4) = ;20
56		4cn 12231	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
57		2cn 12221	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
58		4t2e8 12309	. . . . . . . . . . 11 ⊢ (4 · 2) = 8
59	56, 57, 58	mulcomli 11142	. . . . . . . . . 10 ⊢ (2 · 4) = 8
60	2, 22, 10, 54, 55, 59	decmul1 12672	. . . . . . . . 9 ⊢ (;52 · 4) = ;;208
61	56	mullidi 11138	. . . . . . . . . . 11 ⊢ (1 · 4) = 4
62	61, 40	oveq12i 7370	. . . . . . . . . 10 ⊢ ((1 · 4) + (0 + 0)) = (4 + 0)
63	56	addridi 11321	. . . . . . . . . 10 ⊢ (4 + 0) = 4
64	62, 63	eqtri 2760	. . . . . . . . 9 ⊢ ((1 · 4) + (0 + 0)) = 4
65	23, 15, 52, 53, 2, 60, 64	decrmanc 12665	. . . . . . . 8 ⊢ ((;;521 · 4) + (0 + 0)) = ;;;2084
66	24	nn0cni 12414	. . . . . . . . . . 11 ⊢ ;;521 ∈ ℂ
67	66	mul01i 11324	. . . . . . . . . 10 ⊢ (;;521 · 0) = 0
68	67	oveq1i 7368	. . . . . . . . 9 ⊢ ((;;521 · 0) + 5) = (0 + 5)
69	48	addlidi 11322	. . . . . . . . 9 ⊢ (0 + 5) = 5
70	68, 69, 50	3eqtri 2764	. . . . . . . 8 ⊢ ((;;521 · 0) + 5) = ;05
71	2, 3, 3, 22, 47, 51, 24, 22, 3, 65, 70	decma2c 12661	. . . . . . 7 ⊢ ((;;521 · ;40) + (5 + 0)) = ;;;;20845
72	67	oveq1i 7368	. . . . . . . 8 ⊢ ((;;521 · 0) + 2) = (0 + 2)
73	57	addlidi 11322	. . . . . . . 8 ⊢ (0 + 2) = 2
74	10	dec0h 12630	. . . . . . . 8 ⊢ 2 = ;02
75	72, 73, 74	3eqtri 2764	. . . . . . 7 ⊢ ((;;521 · 0) + 2) = ;02
76	4, 3, 22, 10, 44, 46, 24, 10, 3, 71, 75	decma2c 12661	. . . . . 6 ⊢ ((;;521 · ;;400) + (0 + ;52)) = ;;;;;208452
77	45	mulridi 11137	. . . . . . 7 ⊢ (;52 · 1) = ;52
78		ax-1cn 11085	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
79	78	mullidi 11138	. . . . . . . . 9 ⊢ (1 · 1) = 1
80	79	oveq1i 7368	. . . . . . . 8 ⊢ ((1 · 1) + 1) = (1 + 1)
81		1p1e2 12266	. . . . . . . 8 ⊢ (1 + 1) = 2
82	80, 81	eqtri 2760	. . . . . . 7 ⊢ ((1 · 1) + 1) = 2
83	23, 15, 15, 53, 15, 77, 82	decrmanc 12665	. . . . . 6 ⊢ ((;;521 · 1) + 1) = ;;522
84	5, 15, 3, 15, 1, 43, 24, 10, 23, 76, 83	decma2c 12661	. . . . 5 ⊢ ((;;521 · 𝑁) + 1) = ;;;;;;2084522
85		eqid 2737	. . . . . 6 ⊢ ;;902 = ;;902
86		6nn0 12423	. . . . . . . 8 ⊢ 6 ∈ ℕ₀
87	2, 86	deccl 12623	. . . . . . 7 ⊢ ;46 ∈ ℕ₀
88	87, 10	deccl 12623	. . . . . 6 ⊢ ;;462 ∈ ℕ₀
89		eqid 2737	. . . . . . 7 ⊢ ;90 = ;90
90		eqid 2737	. . . . . . 7 ⊢ ;;462 = ;;462
91		eqid 2737	. . . . . . . 8 ⊢ ;;;2311 = ;;;2311
92	87	nn0cni 12414	. . . . . . . . 9 ⊢ ;46 ∈ ℂ
93	92	addridi 11321	. . . . . . . 8 ⊢ (;46 + 0) = ;46
94		4p1e5 12287	. . . . . . . . . 10 ⊢ (4 + 1) = 5
95	94, 22	eqeltri 2833	. . . . . . . . 9 ⊢ (4 + 1) ∈ ℕ₀
96		eqid 2737	. . . . . . . . 9 ⊢ ;;231 = ;;231
97		eqid 2737	. . . . . . . . . 10 ⊢ ;23 = ;23
98		9cn 12246	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
99		9t2e18 12730	. . . . . . . . . . . 12 ⊢ (9 · 2) = ;18
100	98, 57, 99	mulcomli 11142	. . . . . . . . . . 11 ⊢ (2 · 9) = ;18
101	15, 19, 10, 100, 81, 39	decaddci2 12670	. . . . . . . . . 10 ⊢ ((2 · 9) + 2) = ;20
102		7nn0 12424	. . . . . . . . . . 11 ⊢ 7 ∈ ℕ₀
103		7p1e8 12290	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
104		3cn 12227	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
105		9t3e27 12731	. . . . . . . . . . . 12 ⊢ (9 · 3) = ;27
106	98, 104, 105	mulcomli 11142	. . . . . . . . . . 11 ⊢ (3 · 9) = ;27
107	10, 102, 103, 106	decsuc 12639	. . . . . . . . . 10 ⊢ ((3 · 9) + 1) = ;28
108	10, 26, 15, 97, 30, 19, 10, 101, 107	decrmac 12666	. . . . . . . . 9 ⊢ ((;23 · 9) + 1) = ;;208
109	98	mullidi 11138	. . . . . . . . . . 11 ⊢ (1 · 9) = 9
110	109, 94	oveq12i 7370	. . . . . . . . . 10 ⊢ ((1 · 9) + (4 + 1)) = (9 + 5)
111		9p5e14 12698	. . . . . . . . . 10 ⊢ (9 + 5) = ;14
112	110, 111	eqtri 2760	. . . . . . . . 9 ⊢ ((1 · 9) + (4 + 1)) = ;14
113	27, 15, 95, 96, 30, 2, 15, 108, 112	decrmac 12666	. . . . . . . 8 ⊢ ((;;231 · 9) + (4 + 1)) = ;;;2084
114	109	oveq1i 7368	. . . . . . . . 9 ⊢ ((1 · 9) + 6) = (9 + 6)
115		9p6e15 12699	. . . . . . . . 9 ⊢ (9 + 6) = ;15
116	114, 115	eqtri 2760	. . . . . . . 8 ⊢ ((1 · 9) + 6) = ;15
117	28, 15, 2, 86, 91, 93, 30, 22, 15, 113, 116	decmac 12660	. . . . . . 7 ⊢ ((;;;2311 · 9) + (;46 + 0)) = ;;;;20845
118	29	nn0cni 12414	. . . . . . . . . 10 ⊢ ;;;2311 ∈ ℂ
119	118	mul01i 11324	. . . . . . . . 9 ⊢ (;;;2311 · 0) = 0
120	119	oveq1i 7368	. . . . . . . 8 ⊢ ((;;;2311 · 0) + 2) = (0 + 2)
121	120, 73, 74	3eqtri 2764	. . . . . . 7 ⊢ ((;;;2311 · 0) + 2) = ;02
122	30, 3, 87, 10, 89, 90, 29, 10, 3, 117, 121	decma2c 12661	. . . . . 6 ⊢ ((;;;2311 · ;90) + ;;462) = ;;;;;208452
123		2t2e4 12305	. . . . . . . . 9 ⊢ (2 · 2) = 4
124		3t2e6 12307	. . . . . . . . 9 ⊢ (3 · 2) = 6
125	10, 10, 26, 97, 123, 124	decmul1 12672	. . . . . . . 8 ⊢ (;23 · 2) = ;46
126	57	mullidi 11138	. . . . . . . 8 ⊢ (1 · 2) = 2
127	10, 27, 15, 96, 125, 126	decmul1 12672	. . . . . . 7 ⊢ (;;231 · 2) = ;;462
128	10, 28, 15, 91, 127, 126	decmul1 12672	. . . . . 6 ⊢ (;;;2311 · 2) = ;;;4622
129	29, 31, 10, 85, 10, 88, 122, 128	decmul2c 12674	. . . . 5 ⊢ (;;;2311 · ;;902) = ;;;;;;2084522
130	84, 129	eqtr4i 2763	. . . 4 ⊢ ((;;521 · 𝑁) + 1) = (;;;2311 · ;;902)
131	8, 9, 21, 25, 29, 15, 12, 32, 33, 34, 42, 130	modxai 16997	. . 3 ⊢ ((2↑;;;1000) mod 𝑁) = (1 mod 𝑁)
132	18	nn0cni 12414	. . . 4 ⊢ ;;;1000 ∈ ℂ
133		eqid 2737	. . . . 5 ⊢ ;;;1000 = ;;;1000
134		eqid 2737	. . . . . 6 ⊢ ;;100 = ;;100
135	10	dec0u 12629	. . . . . 6 ⊢ (;10 · 2) = ;20
136	57	mul02i 11323	. . . . . 6 ⊢ (0 · 2) = 0
137	10, 16, 3, 134, 135, 136	decmul1 12672	. . . . 5 ⊢ (;;100 · 2) = ;;200
138	10, 17, 3, 133, 137, 136	decmul1 12672	. . . 4 ⊢ (;;;1000 · 2) = ;;;2000
139	132, 57, 138	mulcomli 11142	. . 3 ⊢ (2 · ;;;1000) = ;;;2000
140	8	nncni 12156	. . . . . 6 ⊢ 𝑁 ∈ ℂ
141	140	mul02i 11323	. . . . 5 ⊢ (0 · 𝑁) = 0
142	141	oveq1i 7368	. . . 4 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
143	78	addlidi 11322	. . . . 5 ⊢ (0 + 1) = 1
144	79, 143	eqtr4i 2763	. . . 4 ⊢ (1 · 1) = (0 + 1)
145	142, 144	eqtr4i 2763	. . 3 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
146	8, 9, 18, 14, 15, 15, 131, 139, 145	mod2xi 16998	. 2 ⊢ ((2↑;;;2000) mod 𝑁) = (1 mod 𝑁)
147	13	nn0cni 12414	. . . 4 ⊢ ;;;2000 ∈ ℂ
148		eqid 2737	. . . . 5 ⊢ ;;;2000 = ;;;2000
149	10, 10, 3, 38, 123, 136	decmul1 12672	. . . . . 6 ⊢ (;20 · 2) = ;40
150	10, 11, 3, 36, 149, 136	decmul1 12672	. . . . 5 ⊢ (;;200 · 2) = ;;400
151	10, 12, 3, 148, 150, 136	decmul1 12672	. . . 4 ⊢ (;;;2000 · 2) = ;;;4000
152	147, 57, 151	mulcomli 11142	. . 3 ⊢ (2 · ;;;2000) = ;;;4000
153	5, 3	deccl 12623	. . . . 5 ⊢ ;;;4000 ∈ ℕ₀
154	153	nn0cni 12414	. . . 4 ⊢ ;;;4000 ∈ ℂ
155		eqid 2737	. . . . . 6 ⊢ ;;;4000 = ;;;4000
156	5, 3, 143, 155	decsuc 12639	. . . . 5 ⊢ (;;;4000 + 1) = ;;;4001
157	1, 156	eqtr4i 2763	. . . 4 ⊢ 𝑁 = (;;;4000 + 1)
158	154, 78, 157	mvrraddi 11398	. . 3 ⊢ (𝑁 − 1) = ;;;4000
159	152, 158	eqtr4i 2763	. 2 ⊢ (2 · ;;;2000) = (𝑁 − 1)
160	8, 9, 13, 14, 15, 15, 146, 159, 145	mod2xi 16998	1 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 (class class class)co 7358 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 − cmin 11365 ℕcn 12146 2c2 12201 3c3 12202 4c4 12203 5c5 12204 6c6 12205 7c7 12206 8c8 12207 9c9 12208 ℕ₀cn0 12402 cdc 12608 mod cmo 13790 ↑cexp 13985

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9346 df-inf 9347 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12609 df-uz 12753 df-rp 12907 df-fl 13713 df-mod 13791 df-seq 13926 df-exp 13986

This theorem is referenced by: 4001prm 17073

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