Theorem 4001lem3 15850

Description: Lemma for 4001prm 15852. 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 11311	. . . . . 6 ⊢ 4 ∈ ℕ0
3		0nn0 11307	. . . . . 6 ⊢ 0 ∈ ℕ0
4	2, 3	deccl 11512	. . . . 5 ⊢ ;40 ∈ ℕ0
5	4, 3	deccl 11512	. . . 4 ⊢ ;;400 ∈ ℕ0
6		1nn 11031	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 11518	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2697	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 11185	. 2 ⊢ 2 ∈ ℕ
10		2nn0 11309	. . . . 5 ⊢ 2 ∈ ℕ0
11	10, 3	deccl 11512	. . . 4 ⊢ ;20 ∈ ℕ0
12	11, 3	deccl 11512	. . 3 ⊢ ;;200 ∈ ℕ0
13	12, 3	deccl 11512	. 2 ⊢ ;;;2000 ∈ ℕ0
14		0z 11388	. 2 ⊢ 0 ∈ ℤ
15		1nn0 11308	. 2 ⊢ 1 ∈ ℕ0
16		10nn0 11516	. . . . 5 ⊢ ;10 ∈ ℕ0
17	16, 3	deccl 11512	. . . 4 ⊢ ;;100 ∈ ℕ0
18	17, 3	deccl 11512	. . 3 ⊢ ;;;1000 ∈ ℕ0
19		8nn0 11315	. . . . . 6 ⊢ 8 ∈ ℕ0
20	19, 3	deccl 11512	. . . . 5 ⊢ ;80 ∈ ℕ0
21	20, 3	deccl 11512	. . . 4 ⊢ ;;800 ∈ ℕ0
22		5nn0 11312	. . . . . . 7 ⊢ 5 ∈ ℕ0
23	22, 10	deccl 11512	. . . . . 6 ⊢ ;52 ∈ ℕ0
24	23, 15	deccl 11512	. . . . 5 ⊢ ;;521 ∈ ℕ0
25	24	nn0zi 11402	. . . 4 ⊢ ;;521 ∈ ℤ
26		3nn0 11310	. . . . . . 7 ⊢ 3 ∈ ℕ0
27	10, 26	deccl 11512	. . . . . 6 ⊢ ;23 ∈ ℕ0
28	27, 15	deccl 11512	. . . . 5 ⊢ ;;231 ∈ ℕ0
29	28, 15	deccl 11512	. . . 4 ⊢ ;;;2311 ∈ ℕ0
30		9nn0 11316	. . . . . 6 ⊢ 9 ∈ ℕ0
31	30, 3	deccl 11512	. . . . 5 ⊢ ;90 ∈ ℕ0
32	31, 10	deccl 11512	. . . 4 ⊢ ;;902 ∈ ℕ0
33	1	4001lem2 15849	. . . 4 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
34	1	4001lem1 15848	. . . 4 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
35		eqid 2622	. . . . 5 ⊢ ;;800 = ;;800
36		eqid 2622	. . . . 5 ⊢ ;;200 = ;;200
37		eqid 2622	. . . . . 6 ⊢ ;80 = ;80
38		eqid 2622	. . . . . 6 ⊢ ;20 = ;20
39		8p2e10 11610	. . . . . 6 ⊢ (8 + 2) = ;10
40		00id 10211	. . . . . 6 ⊢ (0 + 0) = 0
41	19, 3, 10, 3, 37, 38, 39, 40	decadd 11570	. . . . 5 ⊢ (;80 + ;20) = ;;100
42	20, 3, 11, 3, 35, 36, 41, 40	decadd 11570	. . . 4 ⊢ (;;800 + ;;200) = ;;;1000
43	15	dec0h 11522	. . . . . 6 ⊢ 1 = ;01
44		eqid 2622	. . . . . . 7 ⊢ ;;400 = ;;400
45	23	nn0cni 11304	. . . . . . . 8 ⊢ ;52 ∈ ℂ
46	45	addid2i 10224	. . . . . . 7 ⊢ (0 + ;52) = ;52
47		eqid 2622	. . . . . . . 8 ⊢ ;40 = ;40
48		5cn 11100	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
49	48	addid1i 10223	. . . . . . . . 9 ⊢ (5 + 0) = 5
50	22	dec0h 11522	. . . . . . . . 9 ⊢ 5 = ;05
51	49, 50	eqtri 2644	. . . . . . . 8 ⊢ (5 + 0) = ;05
52	40, 3	eqeltri 2697	. . . . . . . . 9 ⊢ (0 + 0) ∈ ℕ0
53		eqid 2622	. . . . . . . . 9 ⊢ ;;521 = ;;521
54		eqid 2622	. . . . . . . . . 10 ⊢ ;52 = ;52
55		5t4e20 11637	. . . . . . . . . 10 ⊢ (5 · 4) = ;20
56		4cn 11098	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
57		2cn 11091	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
58		4t2e8 11181	. . . . . . . . . . 11 ⊢ (4 · 2) = 8
59	56, 57, 58	mulcomli 10047	. . . . . . . . . 10 ⊢ (2 · 4) = 8
60	2, 22, 10, 54, 19, 55, 59	decmul1 11585	. . . . . . . . 9 ⊢ (;52 · 4) = ;;208
61	56	mulid2i 10043	. . . . . . . . . . 11 ⊢ (1 · 4) = 4
62	61, 40	oveq12i 6662	. . . . . . . . . 10 ⊢ ((1 · 4) + (0 + 0)) = (4 + 0)
63	56	addid1i 10223	. . . . . . . . . 10 ⊢ (4 + 0) = 4
64	62, 63	eqtri 2644	. . . . . . . . 9 ⊢ ((1 · 4) + (0 + 0)) = 4
65	23, 15, 52, 53, 2, 60, 64	decrmanc 11576	. . . . . . . 8 ⊢ ((;;521 · 4) + (0 + 0)) = ;;;2084
66	24	nn0cni 11304	. . . . . . . . . . 11 ⊢ ;;521 ∈ ℂ
67	66	mul01i 10226	. . . . . . . . . 10 ⊢ (;;521 · 0) = 0
68	67	oveq1i 6660	. . . . . . . . 9 ⊢ ((;;521 · 0) + 5) = (0 + 5)
69	48	addid2i 10224	. . . . . . . . 9 ⊢ (0 + 5) = 5
70	68, 69, 50	3eqtri 2648	. . . . . . . 8 ⊢ ((;;521 · 0) + 5) = ;05
71	2, 3, 3, 22, 47, 51, 24, 22, 3, 65, 70	decma2c 11568	. . . . . . 7 ⊢ ((;;521 · ;40) + (5 + 0)) = ;;;;20845
72	67	oveq1i 6660	. . . . . . . 8 ⊢ ((;;521 · 0) + 2) = (0 + 2)
73	57	addid2i 10224	. . . . . . . 8 ⊢ (0 + 2) = 2
74	10	dec0h 11522	. . . . . . . 8 ⊢ 2 = ;02
75	72, 73, 74	3eqtri 2648	. . . . . . 7 ⊢ ((;;521 · 0) + 2) = ;02
76	4, 3, 22, 10, 44, 46, 24, 10, 3, 71, 75	decma2c 11568	. . . . . 6 ⊢ ((;;521 · ;;400) + (0 + ;52)) = ;;;;;208452
77	45	mulid1i 10042	. . . . . . 7 ⊢ (;52 · 1) = ;52
78		ax-1cn 9994	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
79	78	mulid2i 10043	. . . . . . . . 9 ⊢ (1 · 1) = 1
80	79	oveq1i 6660	. . . . . . . 8 ⊢ ((1 · 1) + 1) = (1 + 1)
81		1p1e2 11134	. . . . . . . 8 ⊢ (1 + 1) = 2
82	80, 81	eqtri 2644	. . . . . . 7 ⊢ ((1 · 1) + 1) = 2
83	23, 15, 15, 53, 15, 77, 82	decrmanc 11576	. . . . . 6 ⊢ ((;;521 · 1) + 1) = ;;522
84	5, 15, 3, 15, 1, 43, 24, 10, 23, 76, 83	decma2c 11568	. . . . 5 ⊢ ((;;521 · 𝑁) + 1) = ;;;;;;2084522
85		eqid 2622	. . . . . 6 ⊢ ;;902 = ;;902
86		6nn0 11313	. . . . . . . 8 ⊢ 6 ∈ ℕ0
87	2, 86	deccl 11512	. . . . . . 7 ⊢ ;46 ∈ ℕ0
88	87, 10	deccl 11512	. . . . . 6 ⊢ ;;462 ∈ ℕ0
89		eqid 2622	. . . . . . 7 ⊢ ;90 = ;90
90		eqid 2622	. . . . . . 7 ⊢ ;;462 = ;;462
91		eqid 2622	. . . . . . . 8 ⊢ ;;;2311 = ;;;2311
92	87	nn0cni 11304	. . . . . . . . 9 ⊢ ;46 ∈ ℂ
93	92	addid1i 10223	. . . . . . . 8 ⊢ (;46 + 0) = ;46
94		4p1e5 11154	. . . . . . . . . 10 ⊢ (4 + 1) = 5
95	94, 22	eqeltri 2697	. . . . . . . . 9 ⊢ (4 + 1) ∈ ℕ0
96		eqid 2622	. . . . . . . . 9 ⊢ ;;231 = ;;231
97		eqid 2622	. . . . . . . . . 10 ⊢ ;23 = ;23
98		9cn 11108	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
99		9t2e18 11663	. . . . . . . . . . . 12 ⊢ (9 · 2) = ;18
100	98, 57, 99	mulcomli 10047	. . . . . . . . . . 11 ⊢ (2 · 9) = ;18
101	15, 19, 10, 100, 81, 39	decaddci2 11581	. . . . . . . . . 10 ⊢ ((2 · 9) + 2) = ;20
102		7nn0 11314	. . . . . . . . . . 11 ⊢ 7 ∈ ℕ0
103		7p1e8 11157	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
104		3cn 11095	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
105		9t3e27 11664	. . . . . . . . . . . 12 ⊢ (9 · 3) = ;27
106	98, 104, 105	mulcomli 10047	. . . . . . . . . . 11 ⊢ (3 · 9) = ;27
107	10, 102, 103, 106	decsuc 11535	. . . . . . . . . 10 ⊢ ((3 · 9) + 1) = ;28
108	10, 26, 15, 97, 30, 19, 10, 101, 107	decrmac 11577	. . . . . . . . 9 ⊢ ((;23 · 9) + 1) = ;;208
109	98	mulid2i 10043	. . . . . . . . . . 11 ⊢ (1 · 9) = 9
110	109, 94	oveq12i 6662	. . . . . . . . . 10 ⊢ ((1 · 9) + (4 + 1)) = (9 + 5)
111		9p5e14 11623	. . . . . . . . . 10 ⊢ (9 + 5) = ;14
112	110, 111	eqtri 2644	. . . . . . . . 9 ⊢ ((1 · 9) + (4 + 1)) = ;14
113	27, 15, 95, 96, 30, 2, 15, 108, 112	decrmac 11577	. . . . . . . 8 ⊢ ((;;231 · 9) + (4 + 1)) = ;;;2084
114	109	oveq1i 6660	. . . . . . . . 9 ⊢ ((1 · 9) + 6) = (9 + 6)
115		9p6e15 11624	. . . . . . . . 9 ⊢ (9 + 6) = ;15
116	114, 115	eqtri 2644	. . . . . . . 8 ⊢ ((1 · 9) + 6) = ;15
117	28, 15, 2, 86, 91, 93, 30, 22, 15, 113, 116	decmac 11566	. . . . . . 7 ⊢ ((;;;2311 · 9) + (;46 + 0)) = ;;;;20845
118	29	nn0cni 11304	. . . . . . . . . 10 ⊢ ;;;2311 ∈ ℂ
119	118	mul01i 10226	. . . . . . . . 9 ⊢ (;;;2311 · 0) = 0
120	119	oveq1i 6660	. . . . . . . 8 ⊢ ((;;;2311 · 0) + 2) = (0 + 2)
121	120, 73, 74	3eqtri 2648	. . . . . . 7 ⊢ ((;;;2311 · 0) + 2) = ;02
122	30, 3, 87, 10, 89, 90, 29, 10, 3, 117, 121	decma2c 11568	. . . . . 6 ⊢ ((;;;2311 · ;90) + ;;462) = ;;;;;208452
123		2t2e4 11177	. . . . . . . . 9 ⊢ (2 · 2) = 4
124		3t2e6 11179	. . . . . . . . 9 ⊢ (3 · 2) = 6
125	10, 10, 26, 97, 86, 123, 124	decmul1 11585	. . . . . . . 8 ⊢ (;23 · 2) = ;46
126	57	mulid2i 10043	. . . . . . . 8 ⊢ (1 · 2) = 2
127	10, 27, 15, 96, 10, 125, 126	decmul1 11585	. . . . . . 7 ⊢ (;;231 · 2) = ;;462
128	10, 28, 15, 91, 10, 127, 126	decmul1 11585	. . . . . 6 ⊢ (;;;2311 · 2) = ;;;4622
129	29, 31, 10, 85, 10, 88, 122, 128	decmul2c 11589	. . . . 5 ⊢ (;;;2311 · ;;902) = ;;;;;;2084522
130	84, 129	eqtr4i 2647	. . . 4 ⊢ ((;;521 · 𝑁) + 1) = (;;;2311 · ;;902)
131	8, 9, 21, 25, 29, 15, 12, 32, 33, 34, 42, 130	modxai 15772	. . 3 ⊢ ((2↑;;;1000) mod 𝑁) = (1 mod 𝑁)
132	18	nn0cni 11304	. . . 4 ⊢ ;;;1000 ∈ ℂ
133		eqid 2622	. . . . 5 ⊢ ;;;1000 = ;;;1000
134		eqid 2622	. . . . . 6 ⊢ ;;100 = ;;100
135	10	dec0u 11520	. . . . . 6 ⊢ (;10 · 2) = ;20
136	57	mul02i 10225	. . . . . 6 ⊢ (0 · 2) = 0
137	10, 16, 3, 134, 3, 135, 136	decmul1 11585	. . . . 5 ⊢ (;;100 · 2) = ;;200
138	10, 17, 3, 133, 3, 137, 136	decmul1 11585	. . . 4 ⊢ (;;;1000 · 2) = ;;;2000
139	132, 57, 138	mulcomli 10047	. . 3 ⊢ (2 · ;;;1000) = ;;;2000
140	8	nncni 11030	. . . . . 6 ⊢ 𝑁 ∈ ℂ
141	140	mul02i 10225	. . . . 5 ⊢ (0 · 𝑁) = 0
142	141	oveq1i 6660	. . . 4 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
143	78	addid2i 10224	. . . . 5 ⊢ (0 + 1) = 1
144	79, 143	eqtr4i 2647	. . . 4 ⊢ (1 · 1) = (0 + 1)
145	142, 144	eqtr4i 2647	. . 3 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
146	8, 9, 18, 14, 15, 15, 131, 139, 145	mod2xi 15773	. 2 ⊢ ((2↑;;;2000) mod 𝑁) = (1 mod 𝑁)
147	13	nn0cni 11304	. . . 4 ⊢ ;;;2000 ∈ ℂ
148		eqid 2622	. . . . 5 ⊢ ;;;2000 = ;;;2000
149	10, 10, 3, 38, 3, 123, 136	decmul1 11585	. . . . . 6 ⊢ (;20 · 2) = ;40
150	10, 11, 3, 36, 3, 149, 136	decmul1 11585	. . . . 5 ⊢ (;;200 · 2) = ;;400
151	10, 12, 3, 148, 3, 150, 136	decmul1 11585	. . . 4 ⊢ (;;;2000 · 2) = ;;;4000
152	147, 57, 151	mulcomli 10047	. . 3 ⊢ (2 · ;;;2000) = ;;;4000
153	5, 3	deccl 11512	. . . . 5 ⊢ ;;;4000 ∈ ℕ0
154	153	nn0cni 11304	. . . 4 ⊢ ;;;4000 ∈ ℂ
155		eqid 2622	. . . . . 6 ⊢ ;;;4000 = ;;;4000
156	5, 3, 143, 155	decsuc 11535	. . . . 5 ⊢ (;;;4000 + 1) = ;;;4001
157	1, 156	eqtr4i 2647	. . . 4 ⊢ 𝑁 = (;;;4000 + 1)
158	154, 78, 157	mvrraddi 10298	. . 3 ⊢ (𝑁 − 1) = ;;;4000
159	152, 158	eqtr4i 2647	. 2 ⊢ (2 · ;;;2000) = (𝑁 − 1)
160	8, 9, 13, 14, 15, 15, 146, 159, 145	mod2xi 15773	1 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

Syntax hints:   = wceq 1483  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   − cmin 10266  ℕcn 11020  2c2 11070  3c3 11071  4c4 11072  5c5 11073  6c6 11074  7c7 11075  8c8 11076  9c9 11077  ℕ₀cn0 11292  ;cdc 11493   mod cmo 12668  ↑cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861

This theorem is referenced by:  4001prm  15852