4001lem3 - Metamath Proof Explorer

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Theorem 4001lem3 16215

Description: Lemma for 4001prm 16217. 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 11639	. . . . . 6 ⊢ 4 ∈ ℕ₀
3		0nn0 11635	. . . . . 6 ⊢ 0 ∈ ℕ₀
4	2, 3	deccl 11836	. . . . 5 ⊢ ;40 ∈ ℕ₀
5	4, 3	deccl 11836	. . . 4 ⊢ ;;400 ∈ ℕ₀
6		1nn 11363	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 11842	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2902	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 11424	. 2 ⊢ 2 ∈ ℕ
10		2nn0 11637	. . . . 5 ⊢ 2 ∈ ℕ₀
11	10, 3	deccl 11836	. . . 4 ⊢ ;20 ∈ ℕ₀
12	11, 3	deccl 11836	. . 3 ⊢ ;;200 ∈ ℕ₀
13	12, 3	deccl 11836	. 2 ⊢ ;;;2000 ∈ ℕ₀
14		0z 11715	. 2 ⊢ 0 ∈ ℤ
15		1nn0 11636	. 2 ⊢ 1 ∈ ℕ₀
16		10nn0 11839	. . . . 5 ⊢ ;10 ∈ ℕ₀
17	16, 3	deccl 11836	. . . 4 ⊢ ;;100 ∈ ℕ₀
18	17, 3	deccl 11836	. . 3 ⊢ ;;;1000 ∈ ℕ₀
19		8nn0 11643	. . . . . 6 ⊢ 8 ∈ ℕ₀
20	19, 3	deccl 11836	. . . . 5 ⊢ ;80 ∈ ℕ₀
21	20, 3	deccl 11836	. . . 4 ⊢ ;;800 ∈ ℕ₀
22		5nn0 11640	. . . . . . 7 ⊢ 5 ∈ ℕ₀
23	22, 10	deccl 11836	. . . . . 6 ⊢ ;52 ∈ ℕ₀
24	23, 15	deccl 11836	. . . . 5 ⊢ ;;521 ∈ ℕ₀
25	24	nn0zi 11730	. . . 4 ⊢ ;;521 ∈ ℤ
26		3nn0 11638	. . . . . . 7 ⊢ 3 ∈ ℕ₀
27	10, 26	deccl 11836	. . . . . 6 ⊢ ;23 ∈ ℕ₀
28	27, 15	deccl 11836	. . . . 5 ⊢ ;;231 ∈ ℕ₀
29	28, 15	deccl 11836	. . . 4 ⊢ ;;;2311 ∈ ℕ₀
30		9nn0 11644	. . . . . 6 ⊢ 9 ∈ ℕ₀
31	30, 3	deccl 11836	. . . . 5 ⊢ ;90 ∈ ℕ₀
32	31, 10	deccl 11836	. . . 4 ⊢ ;;902 ∈ ℕ₀
33	1	4001lem2 16214	. . . 4 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
34	1	4001lem1 16213	. . . 4 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
35		eqid 2825	. . . . 5 ⊢ ;;800 = ;;800
36		eqid 2825	. . . . 5 ⊢ ;;200 = ;;200
37		eqid 2825	. . . . . 6 ⊢ ;80 = ;80
38		eqid 2825	. . . . . 6 ⊢ ;20 = ;20
39		8p2e10 11903	. . . . . 6 ⊢ (8 + 2) = ;10
40		00id 10530	. . . . . 6 ⊢ (0 + 0) = 0
41	19, 3, 10, 3, 37, 38, 39, 40	decadd 11876	. . . . 5 ⊢ (;80 + ;20) = ;;100
42	20, 3, 11, 3, 35, 36, 41, 40	decadd 11876	. . . 4 ⊢ (;;800 + ;;200) = ;;;1000
43	15	dec0h 11844	. . . . . 6 ⊢ 1 = ;01
44		eqid 2825	. . . . . . 7 ⊢ ;;400 = ;;400
45	23	nn0cni 11631	. . . . . . . 8 ⊢ ;52 ∈ ℂ
46	45	addid2i 10543	. . . . . . 7 ⊢ (0 + ;52) = ;52
47		eqid 2825	. . . . . . . 8 ⊢ ;40 = ;40
48		5cn 11441	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
49	48	addid1i 10542	. . . . . . . . 9 ⊢ (5 + 0) = 5
50	22	dec0h 11844	. . . . . . . . 9 ⊢ 5 = ;05
51	49, 50	eqtri 2849	. . . . . . . 8 ⊢ (5 + 0) = ;05
52	40, 3	eqeltri 2902	. . . . . . . . 9 ⊢ (0 + 0) ∈ ℕ₀
53		eqid 2825	. . . . . . . . 9 ⊢ ;;521 = ;;521
54		eqid 2825	. . . . . . . . . 10 ⊢ ;52 = ;52
55		5t4e20 11925	. . . . . . . . . 10 ⊢ (5 · 4) = ;20
56		4cn 11437	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
57		2cn 11426	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
58		4t2e8 11526	. . . . . . . . . . 11 ⊢ (4 · 2) = 8
59	56, 57, 58	mulcomli 10366	. . . . . . . . . 10 ⊢ (2 · 4) = 8
60	2, 22, 10, 54, 55, 59	decmul1 11886	. . . . . . . . 9 ⊢ (;52 · 4) = ;;208
61	56	mulid2i 10362	. . . . . . . . . . 11 ⊢ (1 · 4) = 4
62	61, 40	oveq12i 6917	. . . . . . . . . 10 ⊢ ((1 · 4) + (0 + 0)) = (4 + 0)
63	56	addid1i 10542	. . . . . . . . . 10 ⊢ (4 + 0) = 4
64	62, 63	eqtri 2849	. . . . . . . . 9 ⊢ ((1 · 4) + (0 + 0)) = 4
65	23, 15, 52, 53, 2, 60, 64	decrmanc 11879	. . . . . . . 8 ⊢ ((;;521 · 4) + (0 + 0)) = ;;;2084
66	24	nn0cni 11631	. . . . . . . . . . 11 ⊢ ;;521 ∈ ℂ
67	66	mul01i 10545	. . . . . . . . . 10 ⊢ (;;521 · 0) = 0
68	67	oveq1i 6915	. . . . . . . . 9 ⊢ ((;;521 · 0) + 5) = (0 + 5)
69	48	addid2i 10543	. . . . . . . . 9 ⊢ (0 + 5) = 5
70	68, 69, 50	3eqtri 2853	. . . . . . . 8 ⊢ ((;;521 · 0) + 5) = ;05
71	2, 3, 3, 22, 47, 51, 24, 22, 3, 65, 70	decma2c 11875	. . . . . . 7 ⊢ ((;;521 · ;40) + (5 + 0)) = ;;;;20845
72	67	oveq1i 6915	. . . . . . . 8 ⊢ ((;;521 · 0) + 2) = (0 + 2)
73	57	addid2i 10543	. . . . . . . 8 ⊢ (0 + 2) = 2
74	10	dec0h 11844	. . . . . . . 8 ⊢ 2 = ;02
75	72, 73, 74	3eqtri 2853	. . . . . . 7 ⊢ ((;;521 · 0) + 2) = ;02
76	4, 3, 22, 10, 44, 46, 24, 10, 3, 71, 75	decma2c 11875	. . . . . 6 ⊢ ((;;521 · ;;400) + (0 + ;52)) = ;;;;;208452
77	45	mulid1i 10361	. . . . . . 7 ⊢ (;52 · 1) = ;52
78		ax-1cn 10310	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
79	78	mulid2i 10362	. . . . . . . . 9 ⊢ (1 · 1) = 1
80	79	oveq1i 6915	. . . . . . . 8 ⊢ ((1 · 1) + 1) = (1 + 1)
81		1p1e2 11483	. . . . . . . 8 ⊢ (1 + 1) = 2
82	80, 81	eqtri 2849	. . . . . . 7 ⊢ ((1 · 1) + 1) = 2
83	23, 15, 15, 53, 15, 77, 82	decrmanc 11879	. . . . . 6 ⊢ ((;;521 · 1) + 1) = ;;522
84	5, 15, 3, 15, 1, 43, 24, 10, 23, 76, 83	decma2c 11875	. . . . 5 ⊢ ((;;521 · 𝑁) + 1) = ;;;;;;2084522
85		eqid 2825	. . . . . 6 ⊢ ;;902 = ;;902
86		6nn0 11641	. . . . . . . 8 ⊢ 6 ∈ ℕ₀
87	2, 86	deccl 11836	. . . . . . 7 ⊢ ;46 ∈ ℕ₀
88	87, 10	deccl 11836	. . . . . 6 ⊢ ;;462 ∈ ℕ₀
89		eqid 2825	. . . . . . 7 ⊢ ;90 = ;90
90		eqid 2825	. . . . . . 7 ⊢ ;;462 = ;;462
91		eqid 2825	. . . . . . . 8 ⊢ ;;;2311 = ;;;2311
92	87	nn0cni 11631	. . . . . . . . 9 ⊢ ;46 ∈ ℂ
93	92	addid1i 10542	. . . . . . . 8 ⊢ (;46 + 0) = ;46
94		4p1e5 11504	. . . . . . . . . 10 ⊢ (4 + 1) = 5
95	94, 22	eqeltri 2902	. . . . . . . . 9 ⊢ (4 + 1) ∈ ℕ₀
96		eqid 2825	. . . . . . . . 9 ⊢ ;;231 = ;;231
97		eqid 2825	. . . . . . . . . 10 ⊢ ;23 = ;23
98		9cn 11457	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
99		9t2e18 11945	. . . . . . . . . . . 12 ⊢ (9 · 2) = ;18
100	98, 57, 99	mulcomli 10366	. . . . . . . . . . 11 ⊢ (2 · 9) = ;18
101	15, 19, 10, 100, 81, 39	decaddci2 11884	. . . . . . . . . 10 ⊢ ((2 · 9) + 2) = ;20
102		7nn0 11642	. . . . . . . . . . 11 ⊢ 7 ∈ ℕ₀
103		7p1e8 11507	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
104		3cn 11432	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
105		9t3e27 11946	. . . . . . . . . . . 12 ⊢ (9 · 3) = ;27
106	98, 104, 105	mulcomli 10366	. . . . . . . . . . 11 ⊢ (3 · 9) = ;27
107	10, 102, 103, 106	decsuc 11853	. . . . . . . . . 10 ⊢ ((3 · 9) + 1) = ;28
108	10, 26, 15, 97, 30, 19, 10, 101, 107	decrmac 11880	. . . . . . . . 9 ⊢ ((;23 · 9) + 1) = ;;208
109	98	mulid2i 10362	. . . . . . . . . . 11 ⊢ (1 · 9) = 9
110	109, 94	oveq12i 6917	. . . . . . . . . 10 ⊢ ((1 · 9) + (4 + 1)) = (9 + 5)
111		9p5e14 11913	. . . . . . . . . 10 ⊢ (9 + 5) = ;14
112	110, 111	eqtri 2849	. . . . . . . . 9 ⊢ ((1 · 9) + (4 + 1)) = ;14
113	27, 15, 95, 96, 30, 2, 15, 108, 112	decrmac 11880	. . . . . . . 8 ⊢ ((;;231 · 9) + (4 + 1)) = ;;;2084
114	109	oveq1i 6915	. . . . . . . . 9 ⊢ ((1 · 9) + 6) = (9 + 6)
115		9p6e15 11914	. . . . . . . . 9 ⊢ (9 + 6) = ;15
116	114, 115	eqtri 2849	. . . . . . . 8 ⊢ ((1 · 9) + 6) = ;15
117	28, 15, 2, 86, 91, 93, 30, 22, 15, 113, 116	decmac 11874	. . . . . . 7 ⊢ ((;;;2311 · 9) + (;46 + 0)) = ;;;;20845
118	29	nn0cni 11631	. . . . . . . . . 10 ⊢ ;;;2311 ∈ ℂ
119	118	mul01i 10545	. . . . . . . . 9 ⊢ (;;;2311 · 0) = 0
120	119	oveq1i 6915	. . . . . . . 8 ⊢ ((;;;2311 · 0) + 2) = (0 + 2)
121	120, 73, 74	3eqtri 2853	. . . . . . 7 ⊢ ((;;;2311 · 0) + 2) = ;02
122	30, 3, 87, 10, 89, 90, 29, 10, 3, 117, 121	decma2c 11875	. . . . . 6 ⊢ ((;;;2311 · ;90) + ;;462) = ;;;;;208452
123		2t2e4 11522	. . . . . . . . 9 ⊢ (2 · 2) = 4
124		3t2e6 11524	. . . . . . . . 9 ⊢ (3 · 2) = 6
125	10, 10, 26, 97, 123, 124	decmul1 11886	. . . . . . . 8 ⊢ (;23 · 2) = ;46
126	57	mulid2i 10362	. . . . . . . 8 ⊢ (1 · 2) = 2
127	10, 27, 15, 96, 125, 126	decmul1 11886	. . . . . . 7 ⊢ (;;231 · 2) = ;;462
128	10, 28, 15, 91, 127, 126	decmul1 11886	. . . . . 6 ⊢ (;;;2311 · 2) = ;;;4622
129	29, 31, 10, 85, 10, 88, 122, 128	decmul2c 11889	. . . . 5 ⊢ (;;;2311 · ;;902) = ;;;;;;2084522
130	84, 129	eqtr4i 2852	. . . 4 ⊢ ((;;521 · 𝑁) + 1) = (;;;2311 · ;;902)
131	8, 9, 21, 25, 29, 15, 12, 32, 33, 34, 42, 130	modxai 16143	. . 3 ⊢ ((2↑;;;1000) mod 𝑁) = (1 mod 𝑁)
132	18	nn0cni 11631	. . . 4 ⊢ ;;;1000 ∈ ℂ
133		eqid 2825	. . . . 5 ⊢ ;;;1000 = ;;;1000
134		eqid 2825	. . . . . 6 ⊢ ;;100 = ;;100
135	10	dec0u 11843	. . . . . 6 ⊢ (;10 · 2) = ;20
136	57	mul02i 10544	. . . . . 6 ⊢ (0 · 2) = 0
137	10, 16, 3, 134, 135, 136	decmul1 11886	. . . . 5 ⊢ (;;100 · 2) = ;;200
138	10, 17, 3, 133, 137, 136	decmul1 11886	. . . 4 ⊢ (;;;1000 · 2) = ;;;2000
139	132, 57, 138	mulcomli 10366	. . 3 ⊢ (2 · ;;;1000) = ;;;2000
140	8	nncni 11361	. . . . . 6 ⊢ 𝑁 ∈ ℂ
141	140	mul02i 10544	. . . . 5 ⊢ (0 · 𝑁) = 0
142	141	oveq1i 6915	. . . 4 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
143	78	addid2i 10543	. . . . 5 ⊢ (0 + 1) = 1
144	79, 143	eqtr4i 2852	. . . 4 ⊢ (1 · 1) = (0 + 1)
145	142, 144	eqtr4i 2852	. . 3 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
146	8, 9, 18, 14, 15, 15, 131, 139, 145	mod2xi 16144	. 2 ⊢ ((2↑;;;2000) mod 𝑁) = (1 mod 𝑁)
147	13	nn0cni 11631	. . . 4 ⊢ ;;;2000 ∈ ℂ
148		eqid 2825	. . . . 5 ⊢ ;;;2000 = ;;;2000
149	10, 10, 3, 38, 123, 136	decmul1 11886	. . . . . 6 ⊢ (;20 · 2) = ;40
150	10, 11, 3, 36, 149, 136	decmul1 11886	. . . . 5 ⊢ (;;200 · 2) = ;;400
151	10, 12, 3, 148, 150, 136	decmul1 11886	. . . 4 ⊢ (;;;2000 · 2) = ;;;4000
152	147, 57, 151	mulcomli 10366	. . 3 ⊢ (2 · ;;;2000) = ;;;4000
153	5, 3	deccl 11836	. . . . 5 ⊢ ;;;4000 ∈ ℕ₀
154	153	nn0cni 11631	. . . 4 ⊢ ;;;4000 ∈ ℂ
155		eqid 2825	. . . . . 6 ⊢ ;;;4000 = ;;;4000
156	5, 3, 143, 155	decsuc 11853	. . . . 5 ⊢ (;;;4000 + 1) = ;;;4001
157	1, 156	eqtr4i 2852	. . . 4 ⊢ 𝑁 = (;;;4000 + 1)
158	154, 78, 157	mvrraddi 10619	. . 3 ⊢ (𝑁 − 1) = ;;;4000
159	152, 158	eqtr4i 2852	. 2 ⊢ (2 · ;;;2000) = (𝑁 − 1)
160	8, 9, 13, 14, 15, 15, 146, 159, 145	mod2xi 16144	1 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

Colors of variables: wff setvar class

Syntax hints: = wceq 1658 (class class class)co 6905 0cc0 10252 1c1 10253 + caddc 10255 · cmul 10257 − cmin 10585 ℕcn 11350 2c2 11406 3c3 11407 4c4 11408 5c5 11409 6c6 11410 7c7 11411 8c8 11412 9c9 11413 ℕ₀cn0 11618 cdc 11821 mod cmo 12963 ↑cexp 13154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-rp 12113 df-fl 12888 df-mod 12964 df-seq 13096 df-exp 13155

This theorem is referenced by: 4001prm 16217

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