Theorem 4001lem3 17321

Description: Lemma for 4001prm 17323. 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 12625	. . . . . 6 ⊢ 4 ∈ ℕ0
3		0nn0 12621	. . . . . 6 ⊢ 0 ∈ ℕ0
4	2, 3	deccl 12829	. . . . 5 ⊢ ;40 ∈ ℕ0
5	4, 3	deccl 12829	. . . 4 ⊢ ;;400 ∈ ℕ0
6		1nn 12346	. . . 4 ⊢ 1 ∈ ℕ
7	5, 6	decnncl 12838	. . 3 ⊢ ;;;4001 ∈ ℕ
8	1, 7	eqeltri 2857	. 2 ⊢ 𝑁 ∈ ℕ
9		2nn 12416	. 2 ⊢ 2 ∈ ℕ
10		2nn0 12623	. . . . 5 ⊢ 2 ∈ ℕ0
11	10, 3	deccl 12829	. . . 4 ⊢ ;20 ∈ ℕ0
12	11, 3	deccl 12829	. . 3 ⊢ ;;200 ∈ ℕ0
13	12, 3	deccl 12829	. 2 ⊢ ;;;2000 ∈ ℕ0
14		0z 12704	. 2 ⊢ 0 ∈ ℤ
15		1nn0 12622	. 2 ⊢ 1 ∈ ℕ0
16		10nn0 12836	. . . . 5 ⊢ ;10 ∈ ℕ0
17	16, 3	deccl 12829	. . . 4 ⊢ ;;100 ∈ ℕ0
18	17, 3	deccl 12829	. . 3 ⊢ ;;;1000 ∈ ℕ0
19		8nn0 12629	. . . . . 6 ⊢ 8 ∈ ℕ0
20	19, 3	deccl 12829	. . . . 5 ⊢ ;80 ∈ ℕ0
21	20, 3	deccl 12829	. . . 4 ⊢ ;;800 ∈ ℕ0
22		5nn0 12626	. . . . . . 7 ⊢ 5 ∈ ℕ0
23	22, 10	deccl 12829	. . . . . 6 ⊢ ;52 ∈ ℕ0
24	23, 15	deccl 12829	. . . . 5 ⊢ ;;521 ∈ ℕ0
25	24	nn0zi 12721	. . . 4 ⊢ ;;521 ∈ ℤ
26		3nn0 12624	. . . . . . 7 ⊢ 3 ∈ ℕ0
27	10, 26	deccl 12829	. . . . . 6 ⊢ ;23 ∈ ℕ0
28	27, 15	deccl 12829	. . . . 5 ⊢ ;;231 ∈ ℕ0
29	28, 15	deccl 12829	. . . 4 ⊢ ;;;2311 ∈ ℕ0
30		9nn0 12630	. . . . . 6 ⊢ 9 ∈ ℕ0
31	30, 3	deccl 12829	. . . . 5 ⊢ ;90 ∈ ℕ0
32	31, 10	deccl 12829	. . . 4 ⊢ ;;902 ∈ ℕ0
33	1	4001lem2 17320	. . . 4 ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
34	1	4001lem1 17319	. . . 4 ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
35		eqid 2761	. . . . 5 ⊢ ;;800 = ;;800
36		eqid 2761	. . . . 5 ⊢ ;;200 = ;;200
37		eqid 2761	. . . . . 6 ⊢ ;80 = ;80
38		eqid 2761	. . . . . 6 ⊢ ;20 = ;20
39		8p2e10 12899	. . . . . 6 ⊢ (8 + 2) = ;10
40		00id 11485	. . . . . 6 ⊢ (0 + 0) = 0
41	19, 3, 10, 3, 37, 38, 39, 40	decadd 12873	. . . . 5 ⊢ (;80 + ;20) = ;;100
42	20, 3, 11, 3, 35, 36, 41, 40	decadd 12873	. . . 4 ⊢ (;;800 + ;;200) = ;;;1000
43	15	dec0h 12841	. . . . . 6 ⊢ 1 = ;01
44		eqid 2761	. . . . . . 7 ⊢ ;;400 = ;;400
45	23	nn0cni 12618	. . . . . . . 8 ⊢ ;52 ∈ ℂ
46	45	addlidi 11498	. . . . . . 7 ⊢ (0 + ;52) = ;52
47		eqid 2761	. . . . . . . 8 ⊢ ;40 = ;40
48		5cn 12431	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
49	48	addridi 11497	. . . . . . . . 9 ⊢ (5 + 0) = 5
50	22	dec0h 12841	. . . . . . . . 9 ⊢ 5 = ;05
51	49, 50	eqtri 2784	. . . . . . . 8 ⊢ (5 + 0) = ;05
52	40, 3	eqeltri 2857	. . . . . . . . 9 ⊢ (0 + 0) ∈ ℕ0
53		eqid 2761	. . . . . . . . 9 ⊢ ;;521 = ;;521
54		eqid 2761	. . . . . . . . . 10 ⊢ ;52 = ;52
55		5t4e20 12921	. . . . . . . . . 10 ⊢ (5 · 4) = ;20
56		2t4e8 12512	. . . . . . . . . 10 ⊢ (2 · 4) = 8
57	2, 22, 10, 54, 55, 56	decmul1 12883	. . . . . . . . 9 ⊢ (;52 · 4) = ;;208
58		4cn 12428	. . . . . . . . . . . 12 ⊢ 4 ∈ ℂ
59	58	mullidi 11314	. . . . . . . . . . 11 ⊢ (1 · 4) = 4
60	59, 40	oveq12i 7432	. . . . . . . . . 10 ⊢ ((1 · 4) + (0 + 0)) = (4 + 0)
61	58	addridi 11497	. . . . . . . . . 10 ⊢ (4 + 0) = 4
62	60, 61	eqtri 2784	. . . . . . . . 9 ⊢ ((1 · 4) + (0 + 0)) = 4
63	23, 15, 52, 53, 2, 57, 62	decrmanc 12876	. . . . . . . 8 ⊢ ((;;521 · 4) + (0 + 0)) = ;;;2084
64	24	nn0cni 12618	. . . . . . . . . . 11 ⊢ ;;521 ∈ ℂ
65	64	mul01i 11500	. . . . . . . . . 10 ⊢ (;;521 · 0) = 0
66	65	oveq1i 7430	. . . . . . . . 9 ⊢ ((;;521 · 0) + 5) = (0 + 5)
67	48	addlidi 11498	. . . . . . . . 9 ⊢ (0 + 5) = 5
68	66, 67, 50	3eqtri 2788	. . . . . . . 8 ⊢ ((;;521 · 0) + 5) = ;05
69	2, 3, 3, 22, 47, 51, 24, 22, 3, 63, 68	decma2c 12872	. . . . . . 7 ⊢ ((;;521 · ;40) + (5 + 0)) = ;;;;20845
70	65	oveq1i 7430	. . . . . . . 8 ⊢ ((;;521 · 0) + 2) = (0 + 2)
71		2cn 12418	. . . . . . . . 9 ⊢ 2 ∈ ℂ
72	71	addlidi 11498	. . . . . . . 8 ⊢ (0 + 2) = 2
73	10	dec0h 12841	. . . . . . . 8 ⊢ 2 = ;02
74	70, 72, 73	3eqtri 2788	. . . . . . 7 ⊢ ((;;521 · 0) + 2) = ;02
75	4, 3, 22, 10, 44, 46, 24, 10, 3, 69, 74	decma2c 12872	. . . . . 6 ⊢ ((;;521 · ;;400) + (0 + ;52)) = ;;;;;208452
76	45	mulridi 11313	. . . . . . 7 ⊢ (;52 · 1) = ;52
77		ax-1cn 11258	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
78	77	mullidi 11314	. . . . . . . . 9 ⊢ (1 · 1) = 1
79	78	oveq1i 7430	. . . . . . . 8 ⊢ ((1 · 1) + 1) = (1 + 1)
80		1p1e2 12466	. . . . . . . 8 ⊢ (1 + 1) = 2
81	79, 80	eqtri 2784	. . . . . . 7 ⊢ ((1 · 1) + 1) = 2
82	23, 15, 15, 53, 15, 76, 81	decrmanc 12876	. . . . . 6 ⊢ ((;;521 · 1) + 1) = ;;522
83	5, 15, 3, 15, 1, 43, 24, 10, 23, 75, 82	decma2c 12872	. . . . 5 ⊢ ((;;521 · 𝑁) + 1) = ;;;;;;2084522
84		eqid 2761	. . . . . 6 ⊢ ;;902 = ;;902
85		6nn0 12627	. . . . . . . 8 ⊢ 6 ∈ ℕ0
86	2, 85	deccl 12829	. . . . . . 7 ⊢ ;46 ∈ ℕ0
87	86, 10	deccl 12829	. . . . . 6 ⊢ ;;462 ∈ ℕ0
88		eqid 2761	. . . . . . 7 ⊢ ;90 = ;90
89		eqid 2761	. . . . . . 7 ⊢ ;;462 = ;;462
90		eqid 2761	. . . . . . . 8 ⊢ ;;;2311 = ;;;2311
91	86	nn0cni 12618	. . . . . . . . 9 ⊢ ;46 ∈ ℂ
92	91	addridi 11497	. . . . . . . 8 ⊢ (;46 + 0) = ;46
93		4p1e5 12488	. . . . . . . . . 10 ⊢ (4 + 1) = 5
94	93, 22	eqeltri 2857	. . . . . . . . 9 ⊢ (4 + 1) ∈ ℕ0
95		eqid 2761	. . . . . . . . 9 ⊢ ;;231 = ;;231
96		eqid 2761	. . . . . . . . . 10 ⊢ ;23 = ;23
97		9cn 12443	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
98		9t2e18 12941	. . . . . . . . . . . 12 ⊢ (9 · 2) = ;18
99	97, 71, 98	mulcomli 11318	. . . . . . . . . . 11 ⊢ (2 · 9) = ;18
100	15, 19, 10, 99, 80, 39	decaddci2 12881	. . . . . . . . . 10 ⊢ ((2 · 9) + 2) = ;20
101		7nn0 12628	. . . . . . . . . . 11 ⊢ 7 ∈ ℕ0
102		7p1e8 12491	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
103		3cn 12424	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
104		9t3e27 12942	. . . . . . . . . . . 12 ⊢ (9 · 3) = ;27
105	97, 103, 104	mulcomli 11318	. . . . . . . . . . 11 ⊢ (3 · 9) = ;27
106	10, 101, 102, 105	decsuc 12850	. . . . . . . . . 10 ⊢ ((3 · 9) + 1) = ;28
107	10, 26, 15, 96, 30, 19, 10, 100, 106	decrmac 12877	. . . . . . . . 9 ⊢ ((;23 · 9) + 1) = ;;208
108	97	mullidi 11314	. . . . . . . . . . 11 ⊢ (1 · 9) = 9
109	108, 93	oveq12i 7432	. . . . . . . . . 10 ⊢ ((1 · 9) + (4 + 1)) = (9 + 5)
110		9p5e14 12909	. . . . . . . . . 10 ⊢ (9 + 5) = ;14
111	109, 110	eqtri 2784	. . . . . . . . 9 ⊢ ((1 · 9) + (4 + 1)) = ;14
112	27, 15, 94, 95, 30, 2, 15, 107, 111	decrmac 12877	. . . . . . . 8 ⊢ ((;;231 · 9) + (4 + 1)) = ;;;2084
113	108	oveq1i 7430	. . . . . . . . 9 ⊢ ((1 · 9) + 6) = (9 + 6)
114		9p6e15 12910	. . . . . . . . 9 ⊢ (9 + 6) = ;15
115	113, 114	eqtri 2784	. . . . . . . 8 ⊢ ((1 · 9) + 6) = ;15
116	28, 15, 2, 85, 90, 92, 30, 22, 15, 112, 115	decmac 12871	. . . . . . 7 ⊢ ((;;;2311 · 9) + (;46 + 0)) = ;;;;20845
117	29	nn0cni 12618	. . . . . . . . . 10 ⊢ ;;;2311 ∈ ℂ
118	117	mul01i 11500	. . . . . . . . 9 ⊢ (;;;2311 · 0) = 0
119	118	oveq1i 7430	. . . . . . . 8 ⊢ ((;;;2311 · 0) + 2) = (0 + 2)
120	119, 72, 73	3eqtri 2788	. . . . . . 7 ⊢ ((;;;2311 · 0) + 2) = ;02
121	30, 3, 86, 10, 88, 89, 29, 10, 3, 116, 120	decma2c 12872	. . . . . 6 ⊢ ((;;;2311 · ;90) + ;;462) = ;;;;;208452
122		2t2e4 12506	. . . . . . . . 9 ⊢ (2 · 2) = 4
123		3t2e6 12508	. . . . . . . . 9 ⊢ (3 · 2) = 6
124	10, 10, 26, 96, 122, 123	decmul1 12883	. . . . . . . 8 ⊢ (;23 · 2) = ;46
125	71	mullidi 11314	. . . . . . . 8 ⊢ (1 · 2) = 2
126	10, 27, 15, 95, 124, 125	decmul1 12883	. . . . . . 7 ⊢ (;;231 · 2) = ;;462
127	10, 28, 15, 90, 126, 125	decmul1 12883	. . . . . 6 ⊢ (;;;2311 · 2) = ;;;4622
128	29, 31, 10, 84, 10, 87, 121, 127	decmul2c 12885	. . . . 5 ⊢ (;;;2311 · ;;902) = ;;;;;;2084522
129	83, 128	eqtr4i 2787	. . . 4 ⊢ ((;;521 · 𝑁) + 1) = (;;;2311 · ;;902)
130	8, 9, 21, 25, 29, 15, 12, 32, 33, 34, 42, 129	modxai 17246	. . 3 ⊢ ((2↑;;;1000) mod 𝑁) = (1 mod 𝑁)
131	18	nn0cni 12618	. . . 4 ⊢ ;;;1000 ∈ ℂ
132		eqid 2761	. . . . 5 ⊢ ;;;1000 = ;;;1000
133		eqid 2761	. . . . . 6 ⊢ ;;100 = ;;100
134	10	dec0u 12840	. . . . . 6 ⊢ (;10 · 2) = ;20
135	71	mul02i 11499	. . . . . 6 ⊢ (0 · 2) = 0
136	10, 16, 3, 133, 134, 135	decmul1 12883	. . . . 5 ⊢ (;;100 · 2) = ;;200
137	10, 17, 3, 132, 136, 135	decmul1 12883	. . . 4 ⊢ (;;;1000 · 2) = ;;;2000
138	131, 71, 137	mulcomli 11318	. . 3 ⊢ (2 · ;;;1000) = ;;;2000
139	8	nncni 12345	. . . . . 6 ⊢ 𝑁 ∈ ℂ
140	139	mul02i 11499	. . . . 5 ⊢ (0 · 𝑁) = 0
141	140	oveq1i 7430	. . . 4 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
142	77	addlidi 11498	. . . . 5 ⊢ (0 + 1) = 1
143	78, 142	eqtr4i 2787	. . . 4 ⊢ (1 · 1) = (0 + 1)
144	141, 143	eqtr4i 2787	. . 3 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
145	8, 9, 18, 14, 15, 15, 130, 138, 144	mod2xi 17247	. 2 ⊢ ((2↑;;;2000) mod 𝑁) = (1 mod 𝑁)
146	13	nn0cni 12618	. . . 4 ⊢ ;;;2000 ∈ ℂ
147		eqid 2761	. . . . 5 ⊢ ;;;2000 = ;;;2000
148	10, 10, 3, 38, 122, 135	decmul1 12883	. . . . . 6 ⊢ (;20 · 2) = ;40
149	10, 11, 3, 36, 148, 135	decmul1 12883	. . . . 5 ⊢ (;;200 · 2) = ;;400
150	10, 12, 3, 147, 149, 135	decmul1 12883	. . . 4 ⊢ (;;;2000 · 2) = ;;;4000
151	146, 71, 150	mulcomli 11318	. . 3 ⊢ (2 · ;;;2000) = ;;;4000
152	5, 3	deccl 12829	. . . . 5 ⊢ ;;;4000 ∈ ℕ0
153	152	nn0cni 12618	. . . 4 ⊢ ;;;4000 ∈ ℂ
154		eqid 2761	. . . . . 6 ⊢ ;;;4000 = ;;;4000
155	5, 3, 142, 154	decsuc 12850	. . . . 5 ⊢ (;;;4000 + 1) = ;;;4001
156	1, 155	eqtr4i 2787	. . . 4 ⊢ 𝑁 = (;;;4000 + 1)
157	153, 77, 156	mvrraddi 11574	. . 3 ⊢ (𝑁 − 1) = ;;;4000
158	151, 157	eqtr4i 2787	. 2 ⊢ (2 · ;;;2000) = (𝑁 − 1)
159	8, 9, 13, 14, 15, 15, 145, 158, 144	mod2xi 17247	1 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

This proof depends on syntax axioms:   = wceq 1570  (class class class)co 7420  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205   − cmin 11541  ℕcn 12335  2c2 12397  3c3 12398  4c4 12399  5c5 12400  6c6 12401  7c7 12402  8c8 12403  9c9 12404  ℕ₀cn0 12606  ;cdc 12814   mod cmo 14009  ↑cexp 14204

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-rp 13121  df-fl 13932  df-mod 14010  df-seq 14145  df-exp 14205

This theorem is used by:  4001prm  17323