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Theorem 4001lem3 14474
Description: Lemma for 4001prm 14476. 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 10805 . . . . . 6  |-  4  e.  NN0
3 0nn0 10801 . . . . . 6  |-  0  e.  NN0
42, 3deccl 10981 . . . . 5  |- ; 4 0  e.  NN0
54, 3deccl 10981 . . . 4  |- ;; 4 0 0  e.  NN0
6 1nn 10538 . . . 4  |-  1  e.  NN
75, 6decnncl 10980 . . 3  |- ;;; 4 0 0 1  e.  NN
81, 7eqeltri 2546 . 2  |-  N  e.  NN
9 2nn 10684 . 2  |-  2  e.  NN
10 2nn0 10803 . . . . 5  |-  2  e.  NN0
1110, 3deccl 10981 . . . 4  |- ; 2 0  e.  NN0
1211, 3deccl 10981 . . 3  |- ;; 2 0 0  e.  NN0
1312, 3deccl 10981 . 2  |- ;;; 2 0 0 0  e.  NN0
14 0z 10866 . 2  |-  0  e.  ZZ
15 1nn0 10802 . 2  |-  1  e.  NN0
16 10nn0 10811 . . . . 5  |-  10  e.  NN0
1716, 3deccl 10981 . . . 4  |- ; 10 0  e.  NN0
1817, 3deccl 10981 . . 3  |- ;; 10 0 0  e.  NN0
19 8nn0 10809 . . . . . 6  |-  8  e.  NN0
2019, 3deccl 10981 . . . . 5  |- ; 8 0  e.  NN0
2120, 3deccl 10981 . . . 4  |- ;; 8 0 0  e.  NN0
22 5nn0 10806 . . . . . . 7  |-  5  e.  NN0
2322, 10deccl 10981 . . . . . 6  |- ; 5 2  e.  NN0
2423, 15deccl 10981 . . . . 5  |- ;; 5 2 1  e.  NN0
2524nn0zi 10880 . . . 4  |- ;; 5 2 1  e.  ZZ
26 3nn0 10804 . . . . . . 7  |-  3  e.  NN0
2710, 26deccl 10981 . . . . . 6  |- ; 2 3  e.  NN0
2827, 15deccl 10981 . . . . 5  |- ;; 2 3 1  e.  NN0
2928, 15deccl 10981 . . . 4  |- ;;; 2 3 1 1  e.  NN0
30 9nn0 10810 . . . . . 6  |-  9  e.  NN0
3130, 3deccl 10981 . . . . 5  |- ; 9 0  e.  NN0
3231, 10deccl 10981 . . . 4  |- ;; 9 0 2  e.  NN0
3314001lem2 14473 . . . 4  |-  ( ( 2 ^;; 8 0 0 )  mod 
N )  =  (;;; 2 3 1 1  mod 
N )
3414001lem1 14472 . . . 4  |-  ( ( 2 ^;; 2 0 0 )  mod 
N )  =  (;; 9 0 2  mod 
N )
35 eqid 2462 . . . . 5  |- ;; 8 0 0  = ;; 8 0 0
36 eqid 2462 . . . . 5  |- ;; 2 0 0  = ;; 2 0 0
37 eqid 2462 . . . . . 6  |- ; 8 0  = ; 8 0
38 eqid 2462 . . . . . 6  |- ; 2 0  = ; 2 0
39 8p2e10 10673 . . . . . 6  |-  ( 8  +  2 )  =  10
40 00id 9745 . . . . . 6  |-  ( 0  +  0 )  =  0
4119, 3, 10, 3, 37, 38, 39, 40decadd 11008 . . . . 5  |-  (; 8 0  + ; 2 0 )  = ; 10 0
4220, 3, 11, 3, 35, 36, 41, 40decadd 11008 . . . 4  |-  (;; 8 0 0  + ;; 2 0 0 )  = ;; 10 0 0
4315dec0h 10983 . . . . . 6  |-  1  = ; 0 1
44 eqid 2462 . . . . . . 7  |- ;; 4 0 0  = ;; 4 0 0
4523nn0cni 10798 . . . . . . . 8  |- ; 5 2  e.  CC
4645addid2i 9758 . . . . . . 7  |-  ( 0  + ; 5 2 )  = ; 5
2
47 eqid 2462 . . . . . . . 8  |- ; 4 0  = ; 4 0
48 5cn 10606 . . . . . . . . . 10  |-  5  e.  CC
4948addid1i 9757 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
5022dec0h 10983 . . . . . . . . 9  |-  5  = ; 0 5
5149, 50eqtri 2491 . . . . . . . 8  |-  ( 5  +  0 )  = ; 0
5
52 eqid 2462 . . . . . . . . 9  |- ;; 5 2 1  = ;; 5 2 1
533dec0h 10983 . . . . . . . . . 10  |-  0  = ; 0 0
5440, 53eqtri 2491 . . . . . . . . 9  |-  ( 0  +  0 )  = ; 0
0
55 eqid 2462 . . . . . . . . . 10  |- ; 5 2  = ; 5 2
56 5t4e20 11042 . . . . . . . . . . . 12  |-  ( 5  x.  4 )  = ; 2
0
5756oveq1i 6287 . . . . . . . . . . 11  |-  ( ( 5  x.  4 )  +  0 )  =  (; 2 0  +  0 )
5811nn0cni 10798 . . . . . . . . . . . 12  |- ; 2 0  e.  CC
5958addid1i 9757 . . . . . . . . . . 11  |-  (; 2 0  +  0 )  = ; 2 0
6057, 59eqtri 2491 . . . . . . . . . 10  |-  ( ( 5  x.  4 )  +  0 )  = ; 2
0
61 4cn 10604 . . . . . . . . . . . . 13  |-  4  e.  CC
62 2cn 10597 . . . . . . . . . . . . 13  |-  2  e.  CC
63 4t2e8 10680 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
6461, 62, 63mulcomli 9594 . . . . . . . . . . . 12  |-  ( 2  x.  4 )  =  8
6564oveq1i 6287 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  0 )  =  ( 8  +  0 )
66 8cn 10612 . . . . . . . . . . . 12  |-  8  e.  CC
6766addid1i 9757 . . . . . . . . . . 11  |-  ( 8  +  0 )  =  8
6865, 67eqtri 2491 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  0 )  =  8
6922, 10, 3, 3, 55, 54, 2, 60, 68decma 11005 . . . . . . . . 9  |-  ( (; 5
2  x.  4 )  +  ( 0  +  0 ) )  = ;; 2 0 8
7061mulid2i 9590 . . . . . . . . . . 11  |-  ( 1  x.  4 )  =  4
7170oveq1i 6287 . . . . . . . . . 10  |-  ( ( 1  x.  4 )  +  0 )  =  ( 4  +  0 )
7261addid1i 9757 . . . . . . . . . 10  |-  ( 4  +  0 )  =  4
732dec0h 10983 . . . . . . . . . 10  |-  4  = ; 0 4
7471, 72, 733eqtri 2495 . . . . . . . . 9  |-  ( ( 1  x.  4 )  +  0 )  = ; 0
4
7523, 15, 3, 3, 52, 54, 2, 2, 3, 69, 74decmac 11006 . . . . . . . 8  |-  ( (;; 5 2 1  x.  4 )  +  ( 0  +  0 ) )  = ;;; 2 0 8 4
7624nn0cni 10798 . . . . . . . . . . 11  |- ;; 5 2 1  e.  CC
7776mul01i 9760 . . . . . . . . . 10  |-  (;; 5 2 1  x.  0 )  =  0
7877oveq1i 6287 . . . . . . . . 9  |-  ( (;; 5 2 1  x.  0 )  +  5 )  =  ( 0  +  5 )
7948addid2i 9758 . . . . . . . . 9  |-  ( 0  +  5 )  =  5
8078, 79, 503eqtri 2495 . . . . . . . 8  |-  ( (;; 5 2 1  x.  0 )  +  5 )  = ; 0 5
812, 3, 3, 22, 47, 51, 24, 22, 3, 75, 80decma2c 11007 . . . . . . 7  |-  ( (;; 5 2 1  x. ; 4
0 )  +  ( 5  +  0 ) )  = ;;;; 2 0 8 4 5
8277oveq1i 6287 . . . . . . . 8  |-  ( (;; 5 2 1  x.  0 )  +  2 )  =  ( 0  +  2 )
8362addid2i 9758 . . . . . . . 8  |-  ( 0  +  2 )  =  2
8410dec0h 10983 . . . . . . . 8  |-  2  = ; 0 2
8582, 83, 843eqtri 2495 . . . . . . 7  |-  ( (;; 5 2 1  x.  0 )  +  2 )  = ; 0 2
864, 3, 22, 10, 44, 46, 24, 10, 3, 81, 85decma2c 11007 . . . . . 6  |-  ( (;; 5 2 1  x. ;; 4 0 0 )  +  ( 0  + ; 5
2 ) )  = ;;;;; 2 0 8 4 5 2
8748mulid1i 9589 . . . . . . . . . 10  |-  ( 5  x.  1 )  =  5
8887, 40oveq12i 6289 . . . . . . . . 9  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  ( 5  +  0 )
8988, 49eqtri 2491 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  5
9062mulid1i 9589 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
9190oveq1i 6287 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  0 )  =  ( 2  +  0 )
9262addid1i 9757 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
9391, 92, 843eqtri 2495 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  0 )  = ; 0
2
9422, 10, 3, 3, 55, 54, 15, 10, 3, 89, 93decmac 11006 . . . . . . 7  |-  ( (; 5
2  x.  1 )  +  ( 0  +  0 ) )  = ; 5
2
95 ax-1cn 9541 . . . . . . . . . 10  |-  1  e.  CC
9695mulid2i 9590 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
9796oveq1i 6287 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  1 )  =  ( 1  +  1 )
98 1p1e2 10640 . . . . . . . 8  |-  ( 1  +  1 )  =  2
9997, 98, 843eqtri 2495 . . . . . . 7  |-  ( ( 1  x.  1 )  +  1 )  = ; 0
2
10023, 15, 3, 15, 52, 43, 15, 10, 3, 94, 99decmac 11006 . . . . . 6  |-  ( (;; 5 2 1  x.  1 )  +  1 )  = ;; 5 2 2
1015, 15, 3, 15, 1, 43, 24, 10, 23, 86, 100decma2c 11007 . . . . 5  |-  ( (;; 5 2 1  x.  N )  +  1 )  = ;;;;;; 2 0 8 4 5 2 2
102 eqid 2462 . . . . . 6  |- ;; 9 0 2  = ;; 9 0 2
103 6nn0 10807 . . . . . . . 8  |-  6  e.  NN0
1042, 103deccl 10981 . . . . . . 7  |- ; 4 6  e.  NN0
105104, 10deccl 10981 . . . . . 6  |- ;; 4 6 2  e.  NN0
106 eqid 2462 . . . . . . 7  |- ; 9 0  = ; 9 0
107 eqid 2462 . . . . . . 7  |- ;; 4 6 2  = ;; 4 6 2
108 eqid 2462 . . . . . . . 8  |- ;;; 2 3 1 1  = ;;; 2 3 1 1
109104nn0cni 10798 . . . . . . . . 9  |- ; 4 6  e.  CC
110109addid1i 9757 . . . . . . . 8  |-  (; 4 6  +  0 )  = ; 4 6
111 eqid 2462 . . . . . . . . 9  |- ;; 2 3 1  = ;; 2 3 1
112 4p1e5 10653 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
113112, 50eqtri 2491 . . . . . . . . 9  |-  ( 4  +  1 )  = ; 0
5
114 eqid 2462 . . . . . . . . . 10  |- ; 2 3  = ; 2 3
11595addid2i 9758 . . . . . . . . . . 11  |-  ( 0  +  1 )  =  1
116115, 43eqtri 2491 . . . . . . . . . 10  |-  ( 0  +  1 )  = ; 0
1
11783oveq2i 6288 . . . . . . . . . . 11  |-  ( ( 2  x.  9 )  +  ( 0  +  2 ) )  =  ( ( 2  x.  9 )  +  2 )
118 9cn 10614 . . . . . . . . . . . . 13  |-  9  e.  CC
119 9t2e18 11062 . . . . . . . . . . . . 13  |-  ( 9  x.  2 )  = ; 1
8
120118, 62, 119mulcomli 9594 . . . . . . . . . . . 12  |-  ( 2  x.  9 )  = ; 1
8
12115, 19, 10, 120, 98, 39decaddci2 11013 . . . . . . . . . . 11  |-  ( ( 2  x.  9 )  +  2 )  = ; 2
0
122117, 121eqtri 2491 . . . . . . . . . 10  |-  ( ( 2  x.  9 )  +  ( 0  +  2 ) )  = ; 2
0
123 7nn0 10808 . . . . . . . . . . 11  |-  7  e.  NN0
124 7p1e8 10656 . . . . . . . . . . 11  |-  ( 7  +  1 )  =  8
125 3cn 10601 . . . . . . . . . . . 12  |-  3  e.  CC
126 9t3e27 11063 . . . . . . . . . . . 12  |-  ( 9  x.  3 )  = ; 2
7
127118, 125, 126mulcomli 9594 . . . . . . . . . . 11  |-  ( 3  x.  9 )  = ; 2
7
12810, 123, 124, 127decsuc 10990 . . . . . . . . . 10  |-  ( ( 3  x.  9 )  +  1 )  = ; 2
8
12910, 26, 3, 15, 114, 116, 30, 19, 10, 122, 128decmac 11006 . . . . . . . . 9  |-  ( (; 2
3  x.  9 )  +  ( 0  +  1 ) )  = ;; 2 0 8
130118mulid2i 9590 . . . . . . . . . . 11  |-  ( 1  x.  9 )  =  9
131130oveq1i 6287 . . . . . . . . . 10  |-  ( ( 1  x.  9 )  +  5 )  =  ( 9  +  5 )
132 9p5e14 11032 . . . . . . . . . 10  |-  ( 9  +  5 )  = ; 1
4
133131, 132eqtri 2491 . . . . . . . . 9  |-  ( ( 1  x.  9 )  +  5 )  = ; 1
4
13427, 15, 3, 22, 111, 113, 30, 2, 15, 129, 133decmac 11006 . . . . . . . 8  |-  ( (;; 2 3 1  x.  9 )  +  ( 4  +  1 ) )  = ;;; 2 0 8 4
135130oveq1i 6287 . . . . . . . . 9  |-  ( ( 1  x.  9 )  +  6 )  =  ( 9  +  6 )
136 9p6e15 11033 . . . . . . . . 9  |-  ( 9  +  6 )  = ; 1
5
137135, 136eqtri 2491 . . . . . . . 8  |-  ( ( 1  x.  9 )  +  6 )  = ; 1
5
13828, 15, 2, 103, 108, 110, 30, 22, 15, 134, 137decmac 11006 . . . . . . 7  |-  ( (;;; 2 3 1 1  x.  9 )  +  (; 4
6  +  0 ) )  = ;;;; 2 0 8 4 5
13929nn0cni 10798 . . . . . . . . . 10  |- ;;; 2 3 1 1  e.  CC
140139mul01i 9760 . . . . . . . . 9  |-  (;;; 2 3 1 1  x.  0 )  =  0
141140oveq1i 6287 . . . . . . . 8  |-  ( (;;; 2 3 1 1  x.  0 )  +  2 )  =  ( 0  +  2 )
142141, 83, 843eqtri 2495 . . . . . . 7  |-  ( (;;; 2 3 1 1  x.  0 )  +  2 )  = ; 0 2
14330, 3, 104, 10, 106, 107, 29, 10, 3, 138, 142decma2c 11007 . . . . . 6  |-  ( (;;; 2 3 1 1  x. ; 9
0 )  + ;; 4 6 2 )  = ;;;;; 2 0 8 4 5 2
144 2t2e4 10676 . . . . . . . . . . . . . . 15  |-  ( 2  x.  2 )  =  4
145144oveq1i 6287 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  0 )  =  ( 4  +  0 )
146145, 72eqtri 2491 . . . . . . . . . . . . 13  |-  ( ( 2  x.  2 )  +  0 )  =  4
147 3t2e6 10678 . . . . . . . . . . . . . 14  |-  ( 3  x.  2 )  =  6
148103dec0h 10983 . . . . . . . . . . . . . 14  |-  6  = ; 0 6
149147, 148eqtri 2491 . . . . . . . . . . . . 13  |-  ( 3  x.  2 )  = ; 0
6
15010, 10, 26, 114, 103, 3, 146, 149decmul1c 11014 . . . . . . . . . . . 12  |-  (; 2 3  x.  2 )  = ; 4 6
151150oveq1i 6287 . . . . . . . . . . 11  |-  ( (; 2
3  x.  2 )  +  0 )  =  (; 4 6  +  0 )
152151, 110eqtri 2491 . . . . . . . . . 10  |-  ( (; 2
3  x.  2 )  +  0 )  = ; 4
6
15362mulid2i 9590 . . . . . . . . . . 11  |-  ( 1  x.  2 )  =  2
154153, 84eqtri 2491 . . . . . . . . . 10  |-  ( 1  x.  2 )  = ; 0
2
15510, 27, 15, 111, 10, 3, 152, 154decmul1c 11014 . . . . . . . . 9  |-  (;; 2 3 1  x.  2 )  = ;; 4 6 2
156155oveq1i 6287 . . . . . . . 8  |-  ( (;; 2 3 1  x.  2 )  +  0 )  =  (;; 4 6 2  +  0 )
157105nn0cni 10798 . . . . . . . . 9  |- ;; 4 6 2  e.  CC
158157addid1i 9757 . . . . . . . 8  |-  (;; 4 6 2  +  0 )  = ;; 4 6 2
159156, 158eqtri 2491 . . . . . . 7  |-  ( (;; 2 3 1  x.  2 )  +  0 )  = ;; 4 6 2
16010, 28, 15, 108, 10, 3, 159, 154decmul1c 11014 . . . . . 6  |-  (;;; 2 3 1 1  x.  2 )  = ;;; 4 6 2 2
16129, 31, 10, 102, 10, 105, 143, 160decmul2c 11015 . . . . 5  |-  (;;; 2 3 1 1  x. ;; 9 0 2 )  = ;;;;;; 2 0 8 4 5 2 2
162101, 161eqtr4i 2494 . . . 4  |-  ( (;; 5 2 1  x.  N )  +  1 )  =  (;;; 2 3 1 1  x. ;; 9 0 2 )
1638, 9, 21, 25, 29, 15, 12, 32, 33, 34, 42, 162modxai 14404 . . 3  |-  ( ( 2 ^;; 10 0 0 )  mod 
N )  =  ( 1  mod  N )
164 eqid 2462 . . . 4  |- ;; 10 0 0  = ;; 10 0 0
165 eqid 2462 . . . . . . 7  |- ; 10 0  = ; 10 0
166 dec10 10997 . . . . . . . . . 10  |-  10  = ; 1 0
16791, 92eqtri 2491 . . . . . . . . . 10  |-  ( ( 2  x.  1 )  +  0 )  =  2
16862mul01i 9760 . . . . . . . . . . 11  |-  ( 2  x.  0 )  =  0
169168, 53eqtri 2491 . . . . . . . . . 10  |-  ( 2  x.  0 )  = ; 0
0
17010, 15, 3, 166, 3, 3, 167, 169decmul2c 11015 . . . . . . . . 9  |-  ( 2  x.  10 )  = ; 2
0
171170oveq1i 6287 . . . . . . . 8  |-  ( ( 2  x.  10 )  +  0 )  =  (; 2 0  +  0 )
172171, 59eqtri 2491 . . . . . . 7  |-  ( ( 2  x.  10 )  +  0 )  = ; 2
0
17310, 16, 3, 165, 3, 3, 172, 169decmul2c 11015 . . . . . 6  |-  ( 2  x. ; 10 0 )  = ;; 2 0 0
174173oveq1i 6287 . . . . 5  |-  ( ( 2  x. ; 10 0 )  +  0 )  =  (;; 2 0 0  +  0 )
17512nn0cni 10798 . . . . . 6  |- ;; 2 0 0  e.  CC
176175addid1i 9757 . . . . 5  |-  (;; 2 0 0  +  0 )  = ;; 2 0 0
177174, 176eqtri 2491 . . . 4  |-  ( ( 2  x. ; 10 0 )  +  0 )  = ;; 2 0 0
17810, 17, 3, 164, 3, 3, 177, 169decmul2c 11015 . . 3  |-  ( 2  x. ;; 10 0 0 )  = ;;; 2 0 0 0
1798nncni 10537 . . . . . 6  |-  N  e.  CC
180179mul02i 9759 . . . . 5  |-  ( 0  x.  N )  =  0
181180oveq1i 6287 . . . 4  |-  ( ( 0  x.  N )  +  1 )  =  ( 0  +  1 )
18296, 115eqtr4i 2494 . . . 4  |-  ( 1  x.  1 )  =  ( 0  +  1 )
183181, 182eqtr4i 2494 . . 3  |-  ( ( 0  x.  N )  +  1 )  =  ( 1  x.  1 )
1848, 9, 18, 14, 15, 15, 163, 178, 183mod2xi 14405 . 2  |-  ( ( 2 ^;;; 2 0 0 0 )  mod  N )  =  ( 1  mod 
N )
185 eqid 2462 . . . 4  |- ;;; 2 0 0 0  = ;;; 2 0 0 0
18610, 10, 3, 38, 3, 3, 146, 169decmul2c 11015 . . . . . . . . 9  |-  ( 2  x. ; 2 0 )  = ; 4
0
187186oveq1i 6287 . . . . . . . 8  |-  ( ( 2  x. ; 2 0 )  +  0 )  =  (; 4
0  +  0 )
1884nn0cni 10798 . . . . . . . . 9  |- ; 4 0  e.  CC
189188addid1i 9757 . . . . . . . 8  |-  (; 4 0  +  0 )  = ; 4 0
190187, 189eqtri 2491 . . . . . . 7  |-  ( ( 2  x. ; 2 0 )  +  0 )  = ; 4 0
19110, 11, 3, 36, 3, 3, 190, 169decmul2c 11015 . . . . . 6  |-  ( 2  x. ;; 2 0 0 )  = ;; 4 0 0
192191oveq1i 6287 . . . . 5  |-  ( ( 2  x. ;; 2 0 0 )  +  0 )  =  (;; 4 0 0  +  0 )
1935nn0cni 10798 . . . . . 6  |- ;; 4 0 0  e.  CC
194193addid1i 9757 . . . . 5  |-  (;; 4 0 0  +  0 )  = ;; 4 0 0
195192, 194eqtri 2491 . . . 4  |-  ( ( 2  x. ;; 2 0 0 )  +  0 )  = ;; 4 0 0
19610, 12, 3, 185, 3, 3, 195, 169decmul2c 11015 . . 3  |-  ( 2  x. ;;; 2 0 0 0 )  = ;;; 4 0 0 0
197 eqid 2462 . . . . . . 7  |- ;;; 4 0 0 0  = ;;; 4 0 0 0
1985, 3, 115, 197decsuc 10990 . . . . . 6  |-  (;;; 4 0 0 0  +  1 )  = ;;; 4 0 0 1
1991, 198eqtr4i 2494 . . . . 5  |-  N  =  (;;; 4 0 0 0  +  1 )
200199oveq1i 6287 . . . 4  |-  ( N  -  1 )  =  ( (;;; 4 0 0 0  +  1 )  - 
1 )
2015, 3deccl 10981 . . . . . 6  |- ;;; 4 0 0 0  e.  NN0
202201nn0cni 10798 . . . . 5  |- ;;; 4 0 0 0  e.  CC
203 pncan 9817 . . . . 5  |-  ( (;;; 4 0 0 0  e.  CC  /\  1  e.  CC )  ->  (
(;;; 4 0 0 0  +  1 )  - 
1 )  = ;;; 4 0 0 0 )
204202, 95, 203mp2an 672 . . . 4  |-  ( (;;; 4 0 0 0  +  1 )  -  1 )  = ;;; 4 0 0 0
205200, 204eqtri 2491 . . 3  |-  ( N  -  1 )  = ;;; 4 0 0 0
206196, 205eqtr4i 2494 . 2  |-  ( 2  x. ;;; 2 0 0 0 )  =  ( N  -  1 )
2078, 9, 13, 14, 15, 15, 184, 206, 183mod2xi 14405 1  |-  ( ( 2 ^ ( N  -  1 ) )  mod  N )  =  ( 1  mod  N
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762  (class class class)co 6277   CCcc 9481   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488    - cmin 9796   NNcn 10527   2c2 10576   3c3 10577   4c4 10578   5c5 10579   6c6 10580   7c7 10581   8c8 10582   9c9 10583   10c10 10584  ;cdc 10967    mod cmo 11954   ^cexp 12124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-rp 11212  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125
This theorem is referenced by:  4001prm  14476
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