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Theorem 4001lem3 13462
Description: Lemma for 4001prm 13464. 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 10240 . . . . . 6  |-  4  e.  NN0
3 0nn0 10236 . . . . . 6  |-  0  e.  NN0
42, 3deccl 10396 . . . . 5  |- ; 4 0  e.  NN0
54, 3deccl 10396 . . . 4  |- ;; 4 0 0  e.  NN0
6 1nn 10011 . . . 4  |-  1  e.  NN
75, 6decnncl 10395 . . 3  |- ;;; 4 0 0 1  e.  NN
81, 7eqeltri 2506 . 2  |-  N  e.  NN
9 2nn 10133 . 2  |-  2  e.  NN
10 2nn0 10238 . . . . 5  |-  2  e.  NN0
1110, 3deccl 10396 . . . 4  |- ; 2 0  e.  NN0
1211, 3deccl 10396 . . 3  |- ;; 2 0 0  e.  NN0
1312, 3deccl 10396 . 2  |- ;;; 2 0 0 0  e.  NN0
14 0z 10293 . 2  |-  0  e.  ZZ
15 1nn0 10237 . 2  |-  1  e.  NN0
16 10nn0 10246 . . . . 5  |-  10  e.  NN0
1716, 3deccl 10396 . . . 4  |- ; 10 0  e.  NN0
1817, 3deccl 10396 . . 3  |- ;; 10 0 0  e.  NN0
19 8nn0 10244 . . . . . 6  |-  8  e.  NN0
2019, 3deccl 10396 . . . . 5  |- ; 8 0  e.  NN0
2120, 3deccl 10396 . . . 4  |- ;; 8 0 0  e.  NN0
22 5nn0 10241 . . . . . . 7  |-  5  e.  NN0
2322, 10deccl 10396 . . . . . 6  |- ; 5 2  e.  NN0
2423, 15deccl 10396 . . . . 5  |- ;; 5 2 1  e.  NN0
2524nn0zi 10306 . . . 4  |- ;; 5 2 1  e.  ZZ
26 3nn0 10239 . . . . . . 7  |-  3  e.  NN0
2710, 26deccl 10396 . . . . . 6  |- ; 2 3  e.  NN0
2827, 15deccl 10396 . . . . 5  |- ;; 2 3 1  e.  NN0
2928, 15deccl 10396 . . . 4  |- ;;; 2 3 1 1  e.  NN0
30 9nn0 10245 . . . . . 6  |-  9  e.  NN0
3130, 3deccl 10396 . . . . 5  |- ; 9 0  e.  NN0
3231, 10deccl 10396 . . . 4  |- ;; 9 0 2  e.  NN0
3314001lem2 13461 . . . 4  |-  ( ( 2 ^;; 8 0 0 )  mod 
N )  =  (;;; 2 3 1 1  mod 
N )
3414001lem1 13460 . . . 4  |-  ( ( 2 ^;; 2 0 0 )  mod 
N )  =  (;; 9 0 2  mod 
N )
35 eqid 2436 . . . . 5  |- ;; 8 0 0  = ;; 8 0 0
36 eqid 2436 . . . . 5  |- ;; 2 0 0  = ;; 2 0 0
37 eqid 2436 . . . . . 6  |- ; 8 0  = ; 8 0
38 eqid 2436 . . . . . 6  |- ; 2 0  = ; 2 0
39 8p2e10 10125 . . . . . 6  |-  ( 8  +  2 )  =  10
40 00id 9241 . . . . . 6  |-  ( 0  +  0 )  =  0
4119, 3, 10, 3, 37, 38, 39, 40decadd 10423 . . . . 5  |-  (; 8 0  + ; 2 0 )  = ; 10 0
4220, 3, 11, 3, 35, 36, 41, 40decadd 10423 . . . 4  |-  (;; 8 0 0  + ;; 2 0 0 )  = ;; 10 0 0
4315dec0h 10398 . . . . . 6  |-  1  = ; 0 1
44 eqid 2436 . . . . . . 7  |- ;; 4 0 0  = ;; 4 0 0
4523nn0cni 10233 . . . . . . . 8  |- ; 5 2  e.  CC
4645addid2i 9254 . . . . . . 7  |-  ( 0  + ; 5 2 )  = ; 5
2
47 eqid 2436 . . . . . . . 8  |- ; 4 0  = ; 4 0
48 5nn 10136 . . . . . . . . . . 11  |-  5  e.  NN
4948nncni 10010 . . . . . . . . . 10  |-  5  e.  CC
5049addid1i 9253 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
5122dec0h 10398 . . . . . . . . 9  |-  5  = ; 0 5
5250, 51eqtri 2456 . . . . . . . 8  |-  ( 5  +  0 )  = ; 0
5
53 eqid 2436 . . . . . . . . 9  |- ;; 5 2 1  = ;; 5 2 1
543dec0h 10398 . . . . . . . . . 10  |-  0  = ; 0 0
5540, 54eqtri 2456 . . . . . . . . 9  |-  ( 0  +  0 )  = ; 0
0
56 eqid 2436 . . . . . . . . . 10  |- ; 5 2  = ; 5 2
57 5t4e20 10457 . . . . . . . . . . . 12  |-  ( 5  x.  4 )  = ; 2
0
5857oveq1i 6091 . . . . . . . . . . 11  |-  ( ( 5  x.  4 )  +  0 )  =  (; 2 0  +  0 )
5911nn0cni 10233 . . . . . . . . . . . 12  |- ; 2 0  e.  CC
6059addid1i 9253 . . . . . . . . . . 11  |-  (; 2 0  +  0 )  = ; 2 0
6158, 60eqtri 2456 . . . . . . . . . 10  |-  ( ( 5  x.  4 )  +  0 )  = ; 2
0
62 4cn 10074 . . . . . . . . . . . . 13  |-  4  e.  CC
63 2cn 10070 . . . . . . . . . . . . 13  |-  2  e.  CC
64 4t2e8 10130 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
6562, 63, 64mulcomli 9097 . . . . . . . . . . . 12  |-  ( 2  x.  4 )  =  8
6665oveq1i 6091 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  0 )  =  ( 8  +  0 )
67 8nn 10139 . . . . . . . . . . . . 13  |-  8  e.  NN
6867nncni 10010 . . . . . . . . . . . 12  |-  8  e.  CC
6968addid1i 9253 . . . . . . . . . . 11  |-  ( 8  +  0 )  =  8
7066, 69eqtri 2456 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  0 )  =  8
7122, 10, 3, 3, 56, 55, 2, 61, 70decma 10420 . . . . . . . . 9  |-  ( (; 5
2  x.  4 )  +  ( 0  +  0 ) )  = ;; 2 0 8
7262mulid2i 9093 . . . . . . . . . . 11  |-  ( 1  x.  4 )  =  4
7372oveq1i 6091 . . . . . . . . . 10  |-  ( ( 1  x.  4 )  +  0 )  =  ( 4  +  0 )
7462addid1i 9253 . . . . . . . . . 10  |-  ( 4  +  0 )  =  4
752dec0h 10398 . . . . . . . . . 10  |-  4  = ; 0 4
7673, 74, 753eqtri 2460 . . . . . . . . 9  |-  ( ( 1  x.  4 )  +  0 )  = ; 0
4
7723, 15, 3, 3, 53, 55, 2, 2, 3, 71, 76decmac 10421 . . . . . . . 8  |-  ( (;; 5 2 1  x.  4 )  +  ( 0  +  0 ) )  = ;;; 2 0 8 4
7824nn0cni 10233 . . . . . . . . . . 11  |- ;; 5 2 1  e.  CC
7978mul01i 9256 . . . . . . . . . 10  |-  (;; 5 2 1  x.  0 )  =  0
8079oveq1i 6091 . . . . . . . . 9  |-  ( (;; 5 2 1  x.  0 )  +  5 )  =  ( 0  +  5 )
8149addid2i 9254 . . . . . . . . 9  |-  ( 0  +  5 )  =  5
8280, 81, 513eqtri 2460 . . . . . . . 8  |-  ( (;; 5 2 1  x.  0 )  +  5 )  = ; 0 5
832, 3, 3, 22, 47, 52, 24, 22, 3, 77, 82decma2c 10422 . . . . . . 7  |-  ( (;; 5 2 1  x. ; 4
0 )  +  ( 5  +  0 ) )  = ;;;; 2 0 8 4 5
8479oveq1i 6091 . . . . . . . 8  |-  ( (;; 5 2 1  x.  0 )  +  2 )  =  ( 0  +  2 )
8563addid2i 9254 . . . . . . . 8  |-  ( 0  +  2 )  =  2
8610dec0h 10398 . . . . . . . 8  |-  2  = ; 0 2
8784, 85, 863eqtri 2460 . . . . . . 7  |-  ( (;; 5 2 1  x.  0 )  +  2 )  = ; 0 2
884, 3, 22, 10, 44, 46, 24, 10, 3, 83, 87decma2c 10422 . . . . . 6  |-  ( (;; 5 2 1  x. ;; 4 0 0 )  +  ( 0  + ; 5
2 ) )  = ;;;;; 2 0 8 4 5 2
8949mulid1i 9092 . . . . . . . . . 10  |-  ( 5  x.  1 )  =  5
9089, 40oveq12i 6093 . . . . . . . . 9  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  ( 5  +  0 )
9190, 50eqtri 2456 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  5
9263mulid1i 9092 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
9392oveq1i 6091 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  0 )  =  ( 2  +  0 )
9463addid1i 9253 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
9593, 94, 863eqtri 2460 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  0 )  = ; 0
2
9622, 10, 3, 3, 56, 55, 15, 10, 3, 91, 95decmac 10421 . . . . . . 7  |-  ( (; 5
2  x.  1 )  +  ( 0  +  0 ) )  = ; 5
2
97 ax-1cn 9048 . . . . . . . . . 10  |-  1  e.  CC
9897mulid2i 9093 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
9998oveq1i 6091 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  1 )  =  ( 1  +  1 )
100 1p1e2 10094 . . . . . . . 8  |-  ( 1  +  1 )  =  2
10199, 100, 863eqtri 2460 . . . . . . 7  |-  ( ( 1  x.  1 )  +  1 )  = ; 0
2
10223, 15, 3, 15, 53, 43, 15, 10, 3, 96, 101decmac 10421 . . . . . 6  |-  ( (;; 5 2 1  x.  1 )  +  1 )  = ;; 5 2 2
1035, 15, 3, 15, 1, 43, 24, 10, 23, 88, 102decma2c 10422 . . . . 5  |-  ( (;; 5 2 1  x.  N )  +  1 )  = ;;;;;; 2 0 8 4 5 2 2
104 eqid 2436 . . . . . 6  |- ;; 9 0 2  = ;; 9 0 2
105 6nn0 10242 . . . . . . . 8  |-  6  e.  NN0
1062, 105deccl 10396 . . . . . . 7  |- ; 4 6  e.  NN0
107106, 10deccl 10396 . . . . . 6  |- ;; 4 6 2  e.  NN0
108 eqid 2436 . . . . . . 7  |- ; 9 0  = ; 9 0
109 eqid 2436 . . . . . . 7  |- ;; 4 6 2  = ;; 4 6 2
110 eqid 2436 . . . . . . . 8  |- ;;; 2 3 1 1  = ;;; 2 3 1 1
111106nn0cni 10233 . . . . . . . . 9  |- ; 4 6  e.  CC
112111addid1i 9253 . . . . . . . 8  |-  (; 4 6  +  0 )  = ; 4 6
113 eqid 2436 . . . . . . . . 9  |- ;; 2 3 1  = ;; 2 3 1
114 4p1e5 10105 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
115114, 51eqtri 2456 . . . . . . . . 9  |-  ( 4  +  1 )  = ; 0
5
116 eqid 2436 . . . . . . . . . 10  |- ; 2 3  = ; 2 3
11797addid2i 9254 . . . . . . . . . . 11  |-  ( 0  +  1 )  =  1
118117, 43eqtri 2456 . . . . . . . . . 10  |-  ( 0  +  1 )  = ; 0
1
11985oveq2i 6092 . . . . . . . . . . 11  |-  ( ( 2  x.  9 )  +  ( 0  +  2 ) )  =  ( ( 2  x.  9 )  +  2 )
120 9nn 10140 . . . . . . . . . . . . . 14  |-  9  e.  NN
121120nncni 10010 . . . . . . . . . . . . 13  |-  9  e.  CC
122 9t2e18 10477 . . . . . . . . . . . . 13  |-  ( 9  x.  2 )  = ; 1
8
123121, 63, 122mulcomli 9097 . . . . . . . . . . . 12  |-  ( 2  x.  9 )  = ; 1
8
12415, 19, 10, 123, 100, 39decaddci2 10428 . . . . . . . . . . 11  |-  ( ( 2  x.  9 )  +  2 )  = ; 2
0
125119, 124eqtri 2456 . . . . . . . . . 10  |-  ( ( 2  x.  9 )  +  ( 0  +  2 ) )  = ; 2
0
126 7nn0 10243 . . . . . . . . . . 11  |-  7  e.  NN0
127 7p1e8 10108 . . . . . . . . . . 11  |-  ( 7  +  1 )  =  8
128 3cn 10072 . . . . . . . . . . . 12  |-  3  e.  CC
129 9t3e27 10478 . . . . . . . . . . . 12  |-  ( 9  x.  3 )  = ; 2
7
130121, 128, 129mulcomli 9097 . . . . . . . . . . 11  |-  ( 3  x.  9 )  = ; 2
7
13110, 126, 127, 130decsuc 10405 . . . . . . . . . 10  |-  ( ( 3  x.  9 )  +  1 )  = ; 2
8
13210, 26, 3, 15, 116, 118, 30, 19, 10, 125, 131decmac 10421 . . . . . . . . 9  |-  ( (; 2
3  x.  9 )  +  ( 0  +  1 ) )  = ;; 2 0 8
133121mulid2i 9093 . . . . . . . . . . 11  |-  ( 1  x.  9 )  =  9
134133oveq1i 6091 . . . . . . . . . 10  |-  ( ( 1  x.  9 )  +  5 )  =  ( 9  +  5 )
135 9p5e14 10447 . . . . . . . . . 10  |-  ( 9  +  5 )  = ; 1
4
136134, 135eqtri 2456 . . . . . . . . 9  |-  ( ( 1  x.  9 )  +  5 )  = ; 1
4
13727, 15, 3, 22, 113, 115, 30, 2, 15, 132, 136decmac 10421 . . . . . . . 8  |-  ( (;; 2 3 1  x.  9 )  +  ( 4  +  1 ) )  = ;;; 2 0 8 4
138133oveq1i 6091 . . . . . . . . 9  |-  ( ( 1  x.  9 )  +  6 )  =  ( 9  +  6 )
139 9p6e15 10448 . . . . . . . . 9  |-  ( 9  +  6 )  = ; 1
5
140138, 139eqtri 2456 . . . . . . . 8  |-  ( ( 1  x.  9 )  +  6 )  = ; 1
5
14128, 15, 2, 105, 110, 112, 30, 22, 15, 137, 140decmac 10421 . . . . . . 7  |-  ( (;;; 2 3 1 1  x.  9 )  +  (; 4
6  +  0 ) )  = ;;;; 2 0 8 4 5
14229nn0cni 10233 . . . . . . . . . 10  |- ;;; 2 3 1 1  e.  CC
143142mul01i 9256 . . . . . . . . 9  |-  (;;; 2 3 1 1  x.  0 )  =  0
144143oveq1i 6091 . . . . . . . 8  |-  ( (;;; 2 3 1 1  x.  0 )  +  2 )  =  ( 0  +  2 )
145144, 85, 863eqtri 2460 . . . . . . 7  |-  ( (;;; 2 3 1 1  x.  0 )  +  2 )  = ; 0 2
14630, 3, 106, 10, 108, 109, 29, 10, 3, 141, 145decma2c 10422 . . . . . 6  |-  ( (;;; 2 3 1 1  x. ; 9
0 )  + ;; 4 6 2 )  = ;;;;; 2 0 8 4 5 2
147 2t2e4 10127 . . . . . . . . . . . . . . 15  |-  ( 2  x.  2 )  =  4
148147oveq1i 6091 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  0 )  =  ( 4  +  0 )
149148, 74eqtri 2456 . . . . . . . . . . . . 13  |-  ( ( 2  x.  2 )  +  0 )  =  4
150 3t2e6 10128 . . . . . . . . . . . . . 14  |-  ( 3  x.  2 )  =  6
151105dec0h 10398 . . . . . . . . . . . . . 14  |-  6  = ; 0 6
152150, 151eqtri 2456 . . . . . . . . . . . . 13  |-  ( 3  x.  2 )  = ; 0
6
15310, 10, 26, 116, 105, 3, 149, 152decmul1c 10429 . . . . . . . . . . . 12  |-  (; 2 3  x.  2 )  = ; 4 6
154153oveq1i 6091 . . . . . . . . . . 11  |-  ( (; 2
3  x.  2 )  +  0 )  =  (; 4 6  +  0 )
155154, 112eqtri 2456 . . . . . . . . . 10  |-  ( (; 2
3  x.  2 )  +  0 )  = ; 4
6
15663mulid2i 9093 . . . . . . . . . . 11  |-  ( 1  x.  2 )  =  2
157156, 86eqtri 2456 . . . . . . . . . 10  |-  ( 1  x.  2 )  = ; 0
2
15810, 27, 15, 113, 10, 3, 155, 157decmul1c 10429 . . . . . . . . 9  |-  (;; 2 3 1  x.  2 )  = ;; 4 6 2
159158oveq1i 6091 . . . . . . . 8  |-  ( (;; 2 3 1  x.  2 )  +  0 )  =  (;; 4 6 2  +  0 )
160107nn0cni 10233 . . . . . . . . 9  |- ;; 4 6 2  e.  CC
161160addid1i 9253 . . . . . . . 8  |-  (;; 4 6 2  +  0 )  = ;; 4 6 2
162159, 161eqtri 2456 . . . . . . 7  |-  ( (;; 2 3 1  x.  2 )  +  0 )  = ;; 4 6 2
16310, 28, 15, 110, 10, 3, 162, 157decmul1c 10429 . . . . . 6  |-  (;;; 2 3 1 1  x.  2 )  = ;;; 4 6 2 2
16429, 31, 10, 104, 10, 107, 146, 163decmul2c 10430 . . . . 5  |-  (;;; 2 3 1 1  x. ;; 9 0 2 )  = ;;;;;; 2 0 8 4 5 2 2
165103, 164eqtr4i 2459 . . . 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 13404 . . 3  |-  ( ( 2 ^;; 10 0 0 )  mod 
N )  =  ( 1  mod  N )
167 eqid 2436 . . . 4  |- ;; 10 0 0  = ;; 10 0 0
168 eqid 2436 . . . . . . 7  |- ; 10 0  = ; 10 0
169 dec10 10412 . . . . . . . . . 10  |-  10  = ; 1 0
17093, 94eqtri 2456 . . . . . . . . . 10  |-  ( ( 2  x.  1 )  +  0 )  =  2
17163mul01i 9256 . . . . . . . . . . 11  |-  ( 2  x.  0 )  =  0
172171, 54eqtri 2456 . . . . . . . . . 10  |-  ( 2  x.  0 )  = ; 0
0
17310, 15, 3, 169, 3, 3, 170, 172decmul2c 10430 . . . . . . . . 9  |-  ( 2  x.  10 )  = ; 2
0
174173oveq1i 6091 . . . . . . . 8  |-  ( ( 2  x.  10 )  +  0 )  =  (; 2 0  +  0 )
175174, 60eqtri 2456 . . . . . . 7  |-  ( ( 2  x.  10 )  +  0 )  = ; 2
0
17610, 16, 3, 168, 3, 3, 175, 172decmul2c 10430 . . . . . 6  |-  ( 2  x. ; 10 0 )  = ;; 2 0 0
177176oveq1i 6091 . . . . 5  |-  ( ( 2  x. ; 10 0 )  +  0 )  =  (;; 2 0 0  +  0 )
17812nn0cni 10233 . . . . . 6  |- ;; 2 0 0  e.  CC
179178addid1i 9253 . . . . 5  |-  (;; 2 0 0  +  0 )  = ;; 2 0 0
180177, 179eqtri 2456 . . . 4  |-  ( ( 2  x. ; 10 0 )  +  0 )  = ;; 2 0 0
18110, 17, 3, 167, 3, 3, 180, 172decmul2c 10430 . . 3  |-  ( 2  x. ;; 10 0 0 )  = ;;; 2 0 0 0
1828nncni 10010 . . . . . 6  |-  N  e.  CC
183182mul02i 9255 . . . . 5  |-  ( 0  x.  N )  =  0
184183oveq1i 6091 . . . 4  |-  ( ( 0  x.  N )  +  1 )  =  ( 0  +  1 )
18598, 117eqtr4i 2459 . . . 4  |-  ( 1  x.  1 )  =  ( 0  +  1 )
186184, 185eqtr4i 2459 . . 3  |-  ( ( 0  x.  N )  +  1 )  =  ( 1  x.  1 )
1878, 9, 18, 14, 15, 15, 166, 181, 186mod2xi 13405 . 2  |-  ( ( 2 ^;;; 2 0 0 0 )  mod  N )  =  ( 1  mod 
N )
188 eqid 2436 . . . 4  |- ;;; 2 0 0 0  = ;;; 2 0 0 0
18910, 10, 3, 38, 3, 3, 149, 172decmul2c 10430 . . . . . . . . 9  |-  ( 2  x. ; 2 0 )  = ; 4
0
190189oveq1i 6091 . . . . . . . 8  |-  ( ( 2  x. ; 2 0 )  +  0 )  =  (; 4
0  +  0 )
1914nn0cni 10233 . . . . . . . . 9  |- ; 4 0  e.  CC
192191addid1i 9253 . . . . . . . 8  |-  (; 4 0  +  0 )  = ; 4 0
193190, 192eqtri 2456 . . . . . . 7  |-  ( ( 2  x. ; 2 0 )  +  0 )  = ; 4 0
19410, 11, 3, 36, 3, 3, 193, 172decmul2c 10430 . . . . . 6  |-  ( 2  x. ;; 2 0 0 )  = ;; 4 0 0
195194oveq1i 6091 . . . . 5  |-  ( ( 2  x. ;; 2 0 0 )  +  0 )  =  (;; 4 0 0  +  0 )
1965nn0cni 10233 . . . . . 6  |- ;; 4 0 0  e.  CC
197196addid1i 9253 . . . . 5  |-  (;; 4 0 0  +  0 )  = ;; 4 0 0
198195, 197eqtri 2456 . . . 4  |-  ( ( 2  x. ;; 2 0 0 )  +  0 )  = ;; 4 0 0
19910, 12, 3, 188, 3, 3, 198, 172decmul2c 10430 . . 3  |-  ( 2  x. ;;; 2 0 0 0 )  = ;;; 4 0 0 0
200 eqid 2436 . . . . . . 7  |- ;;; 4 0 0 0  = ;;; 4 0 0 0
2015, 3, 117, 200decsuc 10405 . . . . . 6  |-  (;;; 4 0 0 0  +  1 )  = ;;; 4 0 0 1
2021, 201eqtr4i 2459 . . . . 5  |-  N  =  (;;; 4 0 0 0  +  1 )
203202oveq1i 6091 . . . 4  |-  ( N  -  1 )  =  ( (;;; 4 0 0 0  +  1 )  - 
1 )
2045, 3deccl 10396 . . . . . 6  |- ;;; 4 0 0 0  e.  NN0
205204nn0cni 10233 . . . . 5  |- ;;; 4 0 0 0  e.  CC
206 pncan 9311 . . . . 5  |-  ( (;;; 4 0 0 0  e.  CC  /\  1  e.  CC )  ->  (
(;;; 4 0 0 0  +  1 )  - 
1 )  = ;;; 4 0 0 0 )
207205, 97, 206mp2an 654 . . . 4  |-  ( (;;; 4 0 0 0  +  1 )  -  1 )  = ;;; 4 0 0 0
208203, 207eqtri 2456 . . 3  |-  ( N  -  1 )  = ;;; 4 0 0 0
209199, 208eqtr4i 2459 . 2  |-  ( 2  x. ;;; 2 0 0 0 )  =  ( N  -  1 )
2108, 9, 13, 14, 15, 15, 187, 209, 186mod2xi 13405 1  |-  ( ( 2 ^ ( N  -  1 ) )  mod  N )  =  ( 1  mod  N
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   NNcn 10000   2c2 10049   3c3 10050   4c4 10051   5c5 10052   6c6 10053   7c7 10054   8c8 10055   9c9 10056   10c10 10057  ;cdc 10382    mod cmo 11250   ^cexp 11382
This theorem is referenced by:  4001prm  13464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-rp 10613  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383
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