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Theorem 4001lem3 13141
Description: Lemma for 4001prm 13143. Calculate a power mod. In decimal, we calculate  2 ^ 1 0 0 0  =  2 ^ 8 0 0  x.  2 ^ 2 0 0  ==  2 3 1 1  x.  9 0 2  =  5 2 1 N  +  1 and finally  2 ^ ( N  -  1 )  =  ( 2 ^ 1 0 0 0 ) ^ 4  ==  1 ^ 4  =  1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
Hypothesis
Ref Expression
4001prm.1  |-  N  = ;;; 4 0 0 1
Assertion
Ref Expression
4001lem3  |-  ( ( 2 ^ ( N  -  1 ) )  mod  N )  =  ( 1  mod  N
)

Proof of Theorem 4001lem3
StepHypRef Expression
1 4001prm.1 . . 3  |-  N  = ;;; 4 0 0 1
2 4nn0 9984 . . . . . 6  |-  4  e.  NN0
3 0nn0 9980 . . . . . 6  |-  0  e.  NN0
42, 3deccl 10138 . . . . 5  |- ; 4 0  e.  NN0
54, 3deccl 10138 . . . 4  |- ;; 4 0 0  e.  NN0
6 1nn 9757 . . . 4  |-  1  e.  NN
75, 6decnncl 10137 . . 3  |- ;;; 4 0 0 1  e.  NN
81, 7eqeltri 2353 . 2  |-  N  e.  NN
9 2nn 9877 . 2  |-  2  e.  NN
10 2nn0 9982 . . . . 5  |-  2  e.  NN0
1110, 3deccl 10138 . . . 4  |- ; 2 0  e.  NN0
1211, 3deccl 10138 . . 3  |- ;; 2 0 0  e.  NN0
1312, 3deccl 10138 . 2  |- ;;; 2 0 0 0  e.  NN0
14 0z 10035 . 2  |-  0  e.  ZZ
15 1nn0 9981 . 2  |-  1  e.  NN0
16 10nn0 9990 . . . . 5  |-  10  e.  NN0
1716, 3deccl 10138 . . . 4  |- ; 10 0  e.  NN0
1817, 3deccl 10138 . . 3  |- ;; 10 0 0  e.  NN0
19 8nn0 9988 . . . . . 6  |-  8  e.  NN0
2019, 3deccl 10138 . . . . 5  |- ; 8 0  e.  NN0
2120, 3deccl 10138 . . . 4  |- ;; 8 0 0  e.  NN0
22 5nn0 9985 . . . . . . 7  |-  5  e.  NN0
2322, 10deccl 10138 . . . . . 6  |- ; 5 2  e.  NN0
2423, 15deccl 10138 . . . . 5  |- ;; 5 2 1  e.  NN0
2524nn0zi 10048 . . . 4  |- ;; 5 2 1  e.  ZZ
26 3nn0 9983 . . . . . . 7  |-  3  e.  NN0
2710, 26deccl 10138 . . . . . 6  |- ; 2 3  e.  NN0
2827, 15deccl 10138 . . . . 5  |- ;; 2 3 1  e.  NN0
2928, 15deccl 10138 . . . 4  |- ;;; 2 3 1 1  e.  NN0
30 9nn0 9989 . . . . . 6  |-  9  e.  NN0
3130, 3deccl 10138 . . . . 5  |- ; 9 0  e.  NN0
3231, 10deccl 10138 . . . 4  |- ;; 9 0 2  e.  NN0
3314001lem2 13140 . . . 4  |-  ( ( 2 ^;; 8 0 0 )  mod 
N )  =  (;;; 2 3 1 1  mod 
N )
3414001lem1 13139 . . . 4  |-  ( ( 2 ^;; 2 0 0 )  mod 
N )  =  (;; 9 0 2  mod 
N )
35 eqid 2283 . . . . 5  |- ;; 8 0 0  = ;; 8 0 0
36 eqid 2283 . . . . 5  |- ;; 2 0 0  = ;; 2 0 0
37 eqid 2283 . . . . . 6  |- ; 8 0  = ; 8 0
38 eqid 2283 . . . . . 6  |- ; 2 0  = ; 2 0
39 8p2e10 9869 . . . . . 6  |-  ( 8  +  2 )  =  10
40 00id 8987 . . . . . 6  |-  ( 0  +  0 )  =  0
4119, 3, 10, 3, 37, 38, 39, 40decadd 10165 . . . . 5  |-  (; 8 0  + ; 2 0 )  = ; 10 0
4220, 3, 11, 3, 35, 36, 41, 40decadd 10165 . . . 4  |-  (;; 8 0 0  + ;; 2 0 0 )  = ;; 10 0 0
4315dec0h 10140 . . . . . 6  |-  1  = ; 0 1
44 eqid 2283 . . . . . . 7  |- ;; 4 0 0  = ;; 4 0 0
4523nn0cni 9977 . . . . . . . 8  |- ; 5 2  e.  CC
4645addid2i 9000 . . . . . . 7  |-  ( 0  + ; 5 2 )  = ; 5
2
47 eqid 2283 . . . . . . . 8  |- ; 4 0  = ; 4 0
48 5nn 9880 . . . . . . . . . . 11  |-  5  e.  NN
4948nncni 9756 . . . . . . . . . 10  |-  5  e.  CC
5049addid1i 8999 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
5122dec0h 10140 . . . . . . . . 9  |-  5  = ; 0 5
5250, 51eqtri 2303 . . . . . . . 8  |-  ( 5  +  0 )  = ; 0
5
53 eqid 2283 . . . . . . . . 9  |- ;; 5 2 1  = ;; 5 2 1
543dec0h 10140 . . . . . . . . . 10  |-  0  = ; 0 0
5540, 54eqtri 2303 . . . . . . . . 9  |-  ( 0  +  0 )  = ; 0
0
56 eqid 2283 . . . . . . . . . 10  |- ; 5 2  = ; 5 2
57 5t4e20 10199 . . . . . . . . . . . 12  |-  ( 5  x.  4 )  = ; 2
0
5857oveq1i 5868 . . . . . . . . . . 11  |-  ( ( 5  x.  4 )  +  0 )  =  (; 2 0  +  0 )
5911nn0cni 9977 . . . . . . . . . . . 12  |- ; 2 0  e.  CC
6059addid1i 8999 . . . . . . . . . . 11  |-  (; 2 0  +  0 )  = ; 2 0
6158, 60eqtri 2303 . . . . . . . . . 10  |-  ( ( 5  x.  4 )  +  0 )  = ; 2
0
62 4cn 9820 . . . . . . . . . . . . 13  |-  4  e.  CC
63 2cn 9816 . . . . . . . . . . . . 13  |-  2  e.  CC
64 4t2e8 9874 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
6562, 63, 64mulcomli 8844 . . . . . . . . . . . 12  |-  ( 2  x.  4 )  =  8
6665oveq1i 5868 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  0 )  =  ( 8  +  0 )
67 8nn 9883 . . . . . . . . . . . . 13  |-  8  e.  NN
6867nncni 9756 . . . . . . . . . . . 12  |-  8  e.  CC
6968addid1i 8999 . . . . . . . . . . 11  |-  ( 8  +  0 )  =  8
7066, 69eqtri 2303 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  0 )  =  8
7122, 10, 3, 3, 56, 55, 2, 61, 70decma 10162 . . . . . . . . 9  |-  ( (; 5
2  x.  4 )  +  ( 0  +  0 ) )  = ;; 2 0 8
7262mulid2i 8840 . . . . . . . . . . 11  |-  ( 1  x.  4 )  =  4
7372oveq1i 5868 . . . . . . . . . 10  |-  ( ( 1  x.  4 )  +  0 )  =  ( 4  +  0 )
7462addid1i 8999 . . . . . . . . . 10  |-  ( 4  +  0 )  =  4
752dec0h 10140 . . . . . . . . . 10  |-  4  = ; 0 4
7673, 74, 753eqtri 2307 . . . . . . . . 9  |-  ( ( 1  x.  4 )  +  0 )  = ; 0
4
7723, 15, 3, 3, 53, 55, 2, 2, 3, 71, 76decmac 10163 . . . . . . . 8  |-  ( (;; 5 2 1  x.  4 )  +  ( 0  +  0 ) )  = ;;; 2 0 8 4
7824nn0cni 9977 . . . . . . . . . . 11  |- ;; 5 2 1  e.  CC
7978mul01i 9002 . . . . . . . . . 10  |-  (;; 5 2 1  x.  0 )  =  0
8079oveq1i 5868 . . . . . . . . 9  |-  ( (;; 5 2 1  x.  0 )  +  5 )  =  ( 0  +  5 )
8149addid2i 9000 . . . . . . . . 9  |-  ( 0  +  5 )  =  5
8280, 81, 513eqtri 2307 . . . . . . . 8  |-  ( (;; 5 2 1  x.  0 )  +  5 )  = ; 0 5
832, 3, 3, 22, 47, 52, 24, 22, 3, 77, 82decma2c 10164 . . . . . . 7  |-  ( (;; 5 2 1  x. ; 4
0 )  +  ( 5  +  0 ) )  = ;;;; 2 0 8 4 5
8479oveq1i 5868 . . . . . . . 8  |-  ( (;; 5 2 1  x.  0 )  +  2 )  =  ( 0  +  2 )
8563addid2i 9000 . . . . . . . 8  |-  ( 0  +  2 )  =  2
8610dec0h 10140 . . . . . . . 8  |-  2  = ; 0 2
8784, 85, 863eqtri 2307 . . . . . . 7  |-  ( (;; 5 2 1  x.  0 )  +  2 )  = ; 0 2
884, 3, 22, 10, 44, 46, 24, 10, 3, 83, 87decma2c 10164 . . . . . 6  |-  ( (;; 5 2 1  x. ;; 4 0 0 )  +  ( 0  + ; 5
2 ) )  = ;;;;; 2 0 8 4 5 2
8949mulid1i 8839 . . . . . . . . . 10  |-  ( 5  x.  1 )  =  5
9089, 40oveq12i 5870 . . . . . . . . 9  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  ( 5  +  0 )
9190, 50eqtri 2303 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  5
9263mulid1i 8839 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
9392oveq1i 5868 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  0 )  =  ( 2  +  0 )
9463addid1i 8999 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
9593, 94, 863eqtri 2307 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  0 )  = ; 0
2
9622, 10, 3, 3, 56, 55, 15, 10, 3, 91, 95decmac 10163 . . . . . . 7  |-  ( (; 5
2  x.  1 )  +  ( 0  +  0 ) )  = ; 5
2
97 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
9897mulid2i 8840 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
9998oveq1i 5868 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  1 )  =  ( 1  +  1 )
100 1p1e2 9840 . . . . . . . 8  |-  ( 1  +  1 )  =  2
10199, 100, 863eqtri 2307 . . . . . . 7  |-  ( ( 1  x.  1 )  +  1 )  = ; 0
2
10223, 15, 3, 15, 53, 43, 15, 10, 3, 96, 101decmac 10163 . . . . . 6  |-  ( (;; 5 2 1  x.  1 )  +  1 )  = ;; 5 2 2
1035, 15, 3, 15, 1, 43, 24, 10, 23, 88, 102decma2c 10164 . . . . 5  |-  ( (;; 5 2 1  x.  N )  +  1 )  = ;;;;;; 2 0 8 4 5 2 2
104 eqid 2283 . . . . . 6  |- ;; 9 0 2  = ;; 9 0 2
105 6nn0 9986 . . . . . . . 8  |-  6  e.  NN0
1062, 105deccl 10138 . . . . . . 7  |- ; 4 6  e.  NN0
107106, 10deccl 10138 . . . . . 6  |- ;; 4 6 2  e.  NN0
108 eqid 2283 . . . . . . 7  |- ; 9 0  = ; 9 0
109 eqid 2283 . . . . . . 7  |- ;; 4 6 2  = ;; 4 6 2
110 eqid 2283 . . . . . . . 8  |- ;;; 2 3 1 1  = ;;; 2 3 1 1
111106nn0cni 9977 . . . . . . . . 9  |- ; 4 6  e.  CC
112111addid1i 8999 . . . . . . . 8  |-  (; 4 6  +  0 )  = ; 4 6
113 eqid 2283 . . . . . . . . 9  |- ;; 2 3 1  = ;; 2 3 1
114 4p1e5 9849 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
115114, 51eqtri 2303 . . . . . . . . 9  |-  ( 4  +  1 )  = ; 0
5
116 eqid 2283 . . . . . . . . . 10  |- ; 2 3  = ; 2 3
11797addid2i 9000 . . . . . . . . . . 11  |-  ( 0  +  1 )  =  1
118117, 43eqtri 2303 . . . . . . . . . 10  |-  ( 0  +  1 )  = ; 0
1
11985oveq2i 5869 . . . . . . . . . . 11  |-  ( ( 2  x.  9 )  +  ( 0  +  2 ) )  =  ( ( 2  x.  9 )  +  2 )
120 9nn 9884 . . . . . . . . . . . . . 14  |-  9  e.  NN
121120nncni 9756 . . . . . . . . . . . . 13  |-  9  e.  CC
122 9t2e18 10219 . . . . . . . . . . . . 13  |-  ( 9  x.  2 )  = ; 1
8
123121, 63, 122mulcomli 8844 . . . . . . . . . . . 12  |-  ( 2  x.  9 )  = ; 1
8
12415, 19, 10, 123, 100, 39decaddci2 10170 . . . . . . . . . . 11  |-  ( ( 2  x.  9 )  +  2 )  = ; 2
0
125119, 124eqtri 2303 . . . . . . . . . 10  |-  ( ( 2  x.  9 )  +  ( 0  +  2 ) )  = ; 2
0
126 7nn0 9987 . . . . . . . . . . 11  |-  7  e.  NN0
127 7p1e8 9852 . . . . . . . . . . 11  |-  ( 7  +  1 )  =  8
128 3cn 9818 . . . . . . . . . . . 12  |-  3  e.  CC
129 9t3e27 10220 . . . . . . . . . . . 12  |-  ( 9  x.  3 )  = ; 2
7
130121, 128, 129mulcomli 8844 . . . . . . . . . . 11  |-  ( 3  x.  9 )  = ; 2
7
13110, 126, 127, 130decsuc 10147 . . . . . . . . . 10  |-  ( ( 3  x.  9 )  +  1 )  = ; 2
8
13210, 26, 3, 15, 116, 118, 30, 19, 10, 125, 131decmac 10163 . . . . . . . . 9  |-  ( (; 2
3  x.  9 )  +  ( 0  +  1 ) )  = ;; 2 0 8
133121mulid2i 8840 . . . . . . . . . . 11  |-  ( 1  x.  9 )  =  9
134133oveq1i 5868 . . . . . . . . . 10  |-  ( ( 1  x.  9 )  +  5 )  =  ( 9  +  5 )
135 9p5e14 10189 . . . . . . . . . 10  |-  ( 9  +  5 )  = ; 1
4
136134, 135eqtri 2303 . . . . . . . . 9  |-  ( ( 1  x.  9 )  +  5 )  = ; 1
4
13727, 15, 3, 22, 113, 115, 30, 2, 15, 132, 136decmac 10163 . . . . . . . 8  |-  ( (;; 2 3 1  x.  9 )  +  ( 4  +  1 ) )  = ;;; 2 0 8 4
138133oveq1i 5868 . . . . . . . . 9  |-  ( ( 1  x.  9 )  +  6 )  =  ( 9  +  6 )
139 9p6e15 10190 . . . . . . . . 9  |-  ( 9  +  6 )  = ; 1
5
140138, 139eqtri 2303 . . . . . . . 8  |-  ( ( 1  x.  9 )  +  6 )  = ; 1
5
14128, 15, 2, 105, 110, 112, 30, 22, 15, 137, 140decmac 10163 . . . . . . 7  |-  ( (;;; 2 3 1 1  x.  9 )  +  (; 4
6  +  0 ) )  = ;;;; 2 0 8 4 5
14229nn0cni 9977 . . . . . . . . . 10  |- ;;; 2 3 1 1  e.  CC
143142mul01i 9002 . . . . . . . . 9  |-  (;;; 2 3 1 1  x.  0 )  =  0
144143oveq1i 5868 . . . . . . . 8  |-  ( (;;; 2 3 1 1  x.  0 )  +  2 )  =  ( 0  +  2 )
145144, 85, 863eqtri 2307 . . . . . . 7  |-  ( (;;; 2 3 1 1  x.  0 )  +  2 )  = ; 0 2
14630, 3, 106, 10, 108, 109, 29, 10, 3, 141, 145decma2c 10164 . . . . . 6  |-  ( (;;; 2 3 1 1  x. ; 9
0 )  + ;; 4 6 2 )  = ;;;;; 2 0 8 4 5 2
147 2t2e4 9871 . . . . . . . . . . . . . . 15  |-  ( 2  x.  2 )  =  4
148147oveq1i 5868 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  0 )  =  ( 4  +  0 )
149148, 74eqtri 2303 . . . . . . . . . . . . 13  |-  ( ( 2  x.  2 )  +  0 )  =  4
150 3t2e6 9872 . . . . . . . . . . . . . 14  |-  ( 3  x.  2 )  =  6
151105dec0h 10140 . . . . . . . . . . . . . 14  |-  6  = ; 0 6
152150, 151eqtri 2303 . . . . . . . . . . . . 13  |-  ( 3  x.  2 )  = ; 0
6
15310, 10, 26, 116, 105, 3, 149, 152decmul1c 10171 . . . . . . . . . . . 12  |-  (; 2 3  x.  2 )  = ; 4 6
154153oveq1i 5868 . . . . . . . . . . 11  |-  ( (; 2
3  x.  2 )  +  0 )  =  (; 4 6  +  0 )
155154, 112eqtri 2303 . . . . . . . . . 10  |-  ( (; 2
3  x.  2 )  +  0 )  = ; 4
6
15663mulid2i 8840 . . . . . . . . . . 11  |-  ( 1  x.  2 )  =  2
157156, 86eqtri 2303 . . . . . . . . . 10  |-  ( 1  x.  2 )  = ; 0
2
15810, 27, 15, 113, 10, 3, 155, 157decmul1c 10171 . . . . . . . . 9  |-  (;; 2 3 1  x.  2 )  = ;; 4 6 2
159158oveq1i 5868 . . . . . . . 8  |-  ( (;; 2 3 1  x.  2 )  +  0 )  =  (;; 4 6 2  +  0 )
160107nn0cni 9977 . . . . . . . . 9  |- ;; 4 6 2  e.  CC
161160addid1i 8999 . . . . . . . 8  |-  (;; 4 6 2  +  0 )  = ;; 4 6 2
162159, 161eqtri 2303 . . . . . . 7  |-  ( (;; 2 3 1  x.  2 )  +  0 )  = ;; 4 6 2
16310, 28, 15, 110, 10, 3, 162, 157decmul1c 10171 . . . . . 6  |-  (;;; 2 3 1 1  x.  2 )  = ;;; 4 6 2 2
16429, 31, 10, 104, 10, 107, 146, 163decmul2c 10172 . . . . 5  |-  (;;; 2 3 1 1  x. ;; 9 0 2 )  = ;;;;;; 2 0 8 4 5 2 2
165103, 164eqtr4i 2306 . . . 4  |-  ( (;; 5 2 1  x.  N )  +  1 )  =  (;;; 2 3 1 1  x. ;; 9 0 2 )
1668, 9, 21, 25, 29, 15, 12, 32, 33, 34, 42, 165modxai 13083 . . 3  |-  ( ( 2 ^;; 10 0 0 )  mod 
N )  =  ( 1  mod  N )
167 eqid 2283 . . . 4  |- ;; 10 0 0  = ;; 10 0 0
168 eqid 2283 . . . . . . 7  |- ; 10 0  = ; 10 0
169 dec10 10154 . . . . . . . . . 10  |-  10  = ; 1 0
17093, 94eqtri 2303 . . . . . . . . . 10  |-  ( ( 2  x.  1 )  +  0 )  =  2
17163mul01i 9002 . . . . . . . . . . 11  |-  ( 2  x.  0 )  =  0
172171, 54eqtri 2303 . . . . . . . . . 10  |-  ( 2  x.  0 )  = ; 0
0
17310, 15, 3, 169, 3, 3, 170, 172decmul2c 10172 . . . . . . . . 9  |-  ( 2  x.  10 )  = ; 2
0
174173oveq1i 5868 . . . . . . . 8  |-  ( ( 2  x.  10 )  +  0 )  =  (; 2 0  +  0 )
175174, 60eqtri 2303 . . . . . . 7  |-  ( ( 2  x.  10 )  +  0 )  = ; 2
0
17610, 16, 3, 168, 3, 3, 175, 172decmul2c 10172 . . . . . 6  |-  ( 2  x. ; 10 0 )  = ;; 2 0 0
177176oveq1i 5868 . . . . 5  |-  ( ( 2  x. ; 10 0 )  +  0 )  =  (;; 2 0 0  +  0 )
17812nn0cni 9977 . . . . . 6  |- ;; 2 0 0  e.  CC
179178addid1i 8999 . . . . 5  |-  (;; 2 0 0  +  0 )  = ;; 2 0 0
180177, 179eqtri 2303 . . . 4  |-  ( ( 2  x. ; 10 0 )  +  0 )  = ;; 2 0 0
18110, 17, 3, 167, 3, 3, 180, 172decmul2c 10172 . . 3  |-  ( 2  x. ;; 10 0 0 )  = ;;; 2 0 0 0
1828nncni 9756 . . . . . 6  |-  N  e.  CC
183182mul02i 9001 . . . . 5  |-  ( 0  x.  N )  =  0
184183oveq1i 5868 . . . 4  |-  ( ( 0  x.  N )  +  1 )  =  ( 0  +  1 )
18598, 117eqtr4i 2306 . . . 4  |-  ( 1  x.  1 )  =  ( 0  +  1 )
186184, 185eqtr4i 2306 . . 3  |-  ( ( 0  x.  N )  +  1 )  =  ( 1  x.  1 )
1878, 9, 18, 14, 15, 15, 166, 181, 186mod2xi 13084 . 2  |-  ( ( 2 ^;;; 2 0 0 0 )  mod  N )  =  ( 1  mod 
N )
188 eqid 2283 . . . 4  |- ;;; 2 0 0 0  = ;;; 2 0 0 0
18910, 10, 3, 38, 3, 3, 149, 172decmul2c 10172 . . . . . . . . 9  |-  ( 2  x. ; 2 0 )  = ; 4
0
190189oveq1i 5868 . . . . . . . 8  |-  ( ( 2  x. ; 2 0 )  +  0 )  =  (; 4
0  +  0 )
1914nn0cni 9977 . . . . . . . . 9  |- ; 4 0  e.  CC
192191addid1i 8999 . . . . . . . 8  |-  (; 4 0  +  0 )  = ; 4 0
193190, 192eqtri 2303 . . . . . . 7  |-  ( ( 2  x. ; 2 0 )  +  0 )  = ; 4 0
19410, 11, 3, 36, 3, 3, 193, 172decmul2c 10172 . . . . . 6  |-  ( 2  x. ;; 2 0 0 )  = ;; 4 0 0
195194oveq1i 5868 . . . . 5  |-  ( ( 2  x. ;; 2 0 0 )  +  0 )  =  (;; 4 0 0  +  0 )
1965nn0cni 9977 . . . . . 6  |- ;; 4 0 0  e.  CC
197196addid1i 8999 . . . . 5  |-  (;; 4 0 0  +  0 )  = ;; 4 0 0
198195, 197eqtri 2303 . . . 4  |-  ( ( 2  x. ;; 2 0 0 )  +  0 )  = ;; 4 0 0
19910, 12, 3, 188, 3, 3, 198, 172decmul2c 10172 . . 3  |-  ( 2  x. ;;; 2 0 0 0 )  = ;;; 4 0 0 0
200 eqid 2283 . . . . . . 7  |- ;;; 4 0 0 0  = ;;; 4 0 0 0
2015, 3, 117, 200decsuc 10147 . . . . . 6  |-  (;;; 4 0 0 0  +  1 )  = ;;; 4 0 0 1
2021, 201eqtr4i 2306 . . . . 5  |-  N  =  (;;; 4 0 0 0  +  1 )
203202oveq1i 5868 . . . 4  |-  ( N  -  1 )  =  ( (;;; 4 0 0 0  +  1 )  - 
1 )
2045, 3deccl 10138 . . . . . 6  |- ;;; 4 0 0 0  e.  NN0
205204nn0cni 9977 . . . . 5  |- ;;; 4 0 0 0  e.  CC
206 pncan 9057 . . . . 5  |-  ( (;;; 4 0 0 0  e.  CC  /\  1  e.  CC )  ->  (
(;;; 4 0 0 0  +  1 )  - 
1 )  = ;;; 4 0 0 0 )
207205, 97, 206mp2an 653 . . . 4  |-  ( (;;; 4 0 0 0  +  1 )  -  1 )  = ;;; 4 0 0 0
208203, 207eqtri 2303 . . 3  |-  ( N  -  1 )  = ;;; 4 0 0 0
209199, 208eqtr4i 2306 . 2  |-  ( 2  x. ;;; 2 0 0 0 )  =  ( N  -  1 )
2108, 9, 13, 14, 15, 15, 187, 209, 186mod2xi 13084 1  |-  ( ( 2 ^ ( N  -  1 ) )  mod  N )  =  ( 1  mod  N
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   2c2 9795   3c3 9796   4c4 9797   5c5 9798   6c6 9799   7c7 9800   8c8 9801   9c9 9802   10c10 9803  ;cdc 10124    mod cmo 10973   ^cexp 11104
This theorem is referenced by:  4001prm  13143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105
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