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Theorem 4001lem3 14163
Description: Lemma for 4001prm 14165. 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 10594 . . . . . 6  |-  4  e.  NN0
3 0nn0 10590 . . . . . 6  |-  0  e.  NN0
42, 3deccl 10765 . . . . 5  |- ; 4 0  e.  NN0
54, 3deccl 10765 . . . 4  |- ;; 4 0 0  e.  NN0
6 1nn 10329 . . . 4  |-  1  e.  NN
75, 6decnncl 10764 . . 3  |- ;;; 4 0 0 1  e.  NN
81, 7eqeltri 2511 . 2  |-  N  e.  NN
9 2nn 10475 . 2  |-  2  e.  NN
10 2nn0 10592 . . . . 5  |-  2  e.  NN0
1110, 3deccl 10765 . . . 4  |- ; 2 0  e.  NN0
1211, 3deccl 10765 . . 3  |- ;; 2 0 0  e.  NN0
1312, 3deccl 10765 . 2  |- ;;; 2 0 0 0  e.  NN0
14 0z 10653 . 2  |-  0  e.  ZZ
15 1nn0 10591 . 2  |-  1  e.  NN0
16 10nn0 10600 . . . . 5  |-  10  e.  NN0
1716, 3deccl 10765 . . . 4  |- ; 10 0  e.  NN0
1817, 3deccl 10765 . . 3  |- ;; 10 0 0  e.  NN0
19 8nn0 10598 . . . . . 6  |-  8  e.  NN0
2019, 3deccl 10765 . . . . 5  |- ; 8 0  e.  NN0
2120, 3deccl 10765 . . . 4  |- ;; 8 0 0  e.  NN0
22 5nn0 10595 . . . . . . 7  |-  5  e.  NN0
2322, 10deccl 10765 . . . . . 6  |- ; 5 2  e.  NN0
2423, 15deccl 10765 . . . . 5  |- ;; 5 2 1  e.  NN0
2524nn0zi 10667 . . . 4  |- ;; 5 2 1  e.  ZZ
26 3nn0 10593 . . . . . . 7  |-  3  e.  NN0
2710, 26deccl 10765 . . . . . 6  |- ; 2 3  e.  NN0
2827, 15deccl 10765 . . . . 5  |- ;; 2 3 1  e.  NN0
2928, 15deccl 10765 . . . 4  |- ;;; 2 3 1 1  e.  NN0
30 9nn0 10599 . . . . . 6  |-  9  e.  NN0
3130, 3deccl 10765 . . . . 5  |- ; 9 0  e.  NN0
3231, 10deccl 10765 . . . 4  |- ;; 9 0 2  e.  NN0
3314001lem2 14162 . . . 4  |-  ( ( 2 ^;; 8 0 0 )  mod 
N )  =  (;;; 2 3 1 1  mod 
N )
3414001lem1 14161 . . . 4  |-  ( ( 2 ^;; 2 0 0 )  mod 
N )  =  (;; 9 0 2  mod 
N )
35 eqid 2441 . . . . 5  |- ;; 8 0 0  = ;; 8 0 0
36 eqid 2441 . . . . 5  |- ;; 2 0 0  = ;; 2 0 0
37 eqid 2441 . . . . . 6  |- ; 8 0  = ; 8 0
38 eqid 2441 . . . . . 6  |- ; 2 0  = ; 2 0
39 8p2e10 10464 . . . . . 6  |-  ( 8  +  2 )  =  10
40 00id 9540 . . . . . 6  |-  ( 0  +  0 )  =  0
4119, 3, 10, 3, 37, 38, 39, 40decadd 10792 . . . . 5  |-  (; 8 0  + ; 2 0 )  = ; 10 0
4220, 3, 11, 3, 35, 36, 41, 40decadd 10792 . . . 4  |-  (;; 8 0 0  + ;; 2 0 0 )  = ;; 10 0 0
4315dec0h 10767 . . . . . 6  |-  1  = ; 0 1
44 eqid 2441 . . . . . . 7  |- ;; 4 0 0  = ;; 4 0 0
4523nn0cni 10587 . . . . . . . 8  |- ; 5 2  e.  CC
4645addid2i 9553 . . . . . . 7  |-  ( 0  + ; 5 2 )  = ; 5
2
47 eqid 2441 . . . . . . . 8  |- ; 4 0  = ; 4 0
48 5cn 10397 . . . . . . . . . 10  |-  5  e.  CC
4948addid1i 9552 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
5022dec0h 10767 . . . . . . . . 9  |-  5  = ; 0 5
5149, 50eqtri 2461 . . . . . . . 8  |-  ( 5  +  0 )  = ; 0
5
52 eqid 2441 . . . . . . . . 9  |- ;; 5 2 1  = ;; 5 2 1
533dec0h 10767 . . . . . . . . . 10  |-  0  = ; 0 0
5440, 53eqtri 2461 . . . . . . . . 9  |-  ( 0  +  0 )  = ; 0
0
55 eqid 2441 . . . . . . . . . 10  |- ; 5 2  = ; 5 2
56 5t4e20 10826 . . . . . . . . . . . 12  |-  ( 5  x.  4 )  = ; 2
0
5756oveq1i 6100 . . . . . . . . . . 11  |-  ( ( 5  x.  4 )  +  0 )  =  (; 2 0  +  0 )
5811nn0cni 10587 . . . . . . . . . . . 12  |- ; 2 0  e.  CC
5958addid1i 9552 . . . . . . . . . . 11  |-  (; 2 0  +  0 )  = ; 2 0
6057, 59eqtri 2461 . . . . . . . . . 10  |-  ( ( 5  x.  4 )  +  0 )  = ; 2
0
61 4cn 10395 . . . . . . . . . . . . 13  |-  4  e.  CC
62 2cn 10388 . . . . . . . . . . . . 13  |-  2  e.  CC
63 4t2e8 10471 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
6461, 62, 63mulcomli 9389 . . . . . . . . . . . 12  |-  ( 2  x.  4 )  =  8
6564oveq1i 6100 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  0 )  =  ( 8  +  0 )
66 8cn 10403 . . . . . . . . . . . 12  |-  8  e.  CC
6766addid1i 9552 . . . . . . . . . . 11  |-  ( 8  +  0 )  =  8
6865, 67eqtri 2461 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  0 )  =  8
6922, 10, 3, 3, 55, 54, 2, 60, 68decma 10789 . . . . . . . . 9  |-  ( (; 5
2  x.  4 )  +  ( 0  +  0 ) )  = ;; 2 0 8
7061mulid2i 9385 . . . . . . . . . . 11  |-  ( 1  x.  4 )  =  4
7170oveq1i 6100 . . . . . . . . . 10  |-  ( ( 1  x.  4 )  +  0 )  =  ( 4  +  0 )
7261addid1i 9552 . . . . . . . . . 10  |-  ( 4  +  0 )  =  4
732dec0h 10767 . . . . . . . . . 10  |-  4  = ; 0 4
7471, 72, 733eqtri 2465 . . . . . . . . 9  |-  ( ( 1  x.  4 )  +  0 )  = ; 0
4
7523, 15, 3, 3, 52, 54, 2, 2, 3, 69, 74decmac 10790 . . . . . . . 8  |-  ( (;; 5 2 1  x.  4 )  +  ( 0  +  0 ) )  = ;;; 2 0 8 4
7624nn0cni 10587 . . . . . . . . . . 11  |- ;; 5 2 1  e.  CC
7776mul01i 9555 . . . . . . . . . 10  |-  (;; 5 2 1  x.  0 )  =  0
7877oveq1i 6100 . . . . . . . . 9  |-  ( (;; 5 2 1  x.  0 )  +  5 )  =  ( 0  +  5 )
7948addid2i 9553 . . . . . . . . 9  |-  ( 0  +  5 )  =  5
8078, 79, 503eqtri 2465 . . . . . . . 8  |-  ( (;; 5 2 1  x.  0 )  +  5 )  = ; 0 5
812, 3, 3, 22, 47, 51, 24, 22, 3, 75, 80decma2c 10791 . . . . . . 7  |-  ( (;; 5 2 1  x. ; 4
0 )  +  ( 5  +  0 ) )  = ;;;; 2 0 8 4 5
8277oveq1i 6100 . . . . . . . 8  |-  ( (;; 5 2 1  x.  0 )  +  2 )  =  ( 0  +  2 )
8362addid2i 9553 . . . . . . . 8  |-  ( 0  +  2 )  =  2
8410dec0h 10767 . . . . . . . 8  |-  2  = ; 0 2
8582, 83, 843eqtri 2465 . . . . . . 7  |-  ( (;; 5 2 1  x.  0 )  +  2 )  = ; 0 2
864, 3, 22, 10, 44, 46, 24, 10, 3, 81, 85decma2c 10791 . . . . . 6  |-  ( (;; 5 2 1  x. ;; 4 0 0 )  +  ( 0  + ; 5
2 ) )  = ;;;;; 2 0 8 4 5 2
8748mulid1i 9384 . . . . . . . . . 10  |-  ( 5  x.  1 )  =  5
8887, 40oveq12i 6102 . . . . . . . . 9  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  ( 5  +  0 )
8988, 49eqtri 2461 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  5
9062mulid1i 9384 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
9190oveq1i 6100 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  0 )  =  ( 2  +  0 )
9262addid1i 9552 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
9391, 92, 843eqtri 2465 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  0 )  = ; 0
2
9422, 10, 3, 3, 55, 54, 15, 10, 3, 89, 93decmac 10790 . . . . . . 7  |-  ( (; 5
2  x.  1 )  +  ( 0  +  0 ) )  = ; 5
2
95 ax-1cn 9336 . . . . . . . . . 10  |-  1  e.  CC
9695mulid2i 9385 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
9796oveq1i 6100 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  1 )  =  ( 1  +  1 )
98 1p1e2 10431 . . . . . . . 8  |-  ( 1  +  1 )  =  2
9997, 98, 843eqtri 2465 . . . . . . 7  |-  ( ( 1  x.  1 )  +  1 )  = ; 0
2
10023, 15, 3, 15, 52, 43, 15, 10, 3, 94, 99decmac 10790 . . . . . 6  |-  ( (;; 5 2 1  x.  1 )  +  1 )  = ;; 5 2 2
1015, 15, 3, 15, 1, 43, 24, 10, 23, 86, 100decma2c 10791 . . . . 5  |-  ( (;; 5 2 1  x.  N )  +  1 )  = ;;;;;; 2 0 8 4 5 2 2
102 eqid 2441 . . . . . 6  |- ;; 9 0 2  = ;; 9 0 2
103 6nn0 10596 . . . . . . . 8  |-  6  e.  NN0
1042, 103deccl 10765 . . . . . . 7  |- ; 4 6  e.  NN0
105104, 10deccl 10765 . . . . . 6  |- ;; 4 6 2  e.  NN0
106 eqid 2441 . . . . . . 7  |- ; 9 0  = ; 9 0
107 eqid 2441 . . . . . . 7  |- ;; 4 6 2  = ;; 4 6 2
108 eqid 2441 . . . . . . . 8  |- ;;; 2 3 1 1  = ;;; 2 3 1 1
109104nn0cni 10587 . . . . . . . . 9  |- ; 4 6  e.  CC
110109addid1i 9552 . . . . . . . 8  |-  (; 4 6  +  0 )  = ; 4 6
111 eqid 2441 . . . . . . . . 9  |- ;; 2 3 1  = ;; 2 3 1
112 4p1e5 10444 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
113112, 50eqtri 2461 . . . . . . . . 9  |-  ( 4  +  1 )  = ; 0
5
114 eqid 2441 . . . . . . . . . 10  |- ; 2 3  = ; 2 3
11595addid2i 9553 . . . . . . . . . . 11  |-  ( 0  +  1 )  =  1
116115, 43eqtri 2461 . . . . . . . . . 10  |-  ( 0  +  1 )  = ; 0
1
11783oveq2i 6101 . . . . . . . . . . 11  |-  ( ( 2  x.  9 )  +  ( 0  +  2 ) )  =  ( ( 2  x.  9 )  +  2 )
118 9cn 10405 . . . . . . . . . . . . 13  |-  9  e.  CC
119 9t2e18 10846 . . . . . . . . . . . . 13  |-  ( 9  x.  2 )  = ; 1
8
120118, 62, 119mulcomli 9389 . . . . . . . . . . . 12  |-  ( 2  x.  9 )  = ; 1
8
12115, 19, 10, 120, 98, 39decaddci2 10797 . . . . . . . . . . 11  |-  ( ( 2  x.  9 )  +  2 )  = ; 2
0
122117, 121eqtri 2461 . . . . . . . . . 10  |-  ( ( 2  x.  9 )  +  ( 0  +  2 ) )  = ; 2
0
123 7nn0 10597 . . . . . . . . . . 11  |-  7  e.  NN0
124 7p1e8 10447 . . . . . . . . . . 11  |-  ( 7  +  1 )  =  8
125 3cn 10392 . . . . . . . . . . . 12  |-  3  e.  CC
126 9t3e27 10847 . . . . . . . . . . . 12  |-  ( 9  x.  3 )  = ; 2
7
127118, 125, 126mulcomli 9389 . . . . . . . . . . 11  |-  ( 3  x.  9 )  = ; 2
7
12810, 123, 124, 127decsuc 10774 . . . . . . . . . 10  |-  ( ( 3  x.  9 )  +  1 )  = ; 2
8
12910, 26, 3, 15, 114, 116, 30, 19, 10, 122, 128decmac 10790 . . . . . . . . 9  |-  ( (; 2
3  x.  9 )  +  ( 0  +  1 ) )  = ;; 2 0 8
130118mulid2i 9385 . . . . . . . . . . 11  |-  ( 1  x.  9 )  =  9
131130oveq1i 6100 . . . . . . . . . 10  |-  ( ( 1  x.  9 )  +  5 )  =  ( 9  +  5 )
132 9p5e14 10816 . . . . . . . . . 10  |-  ( 9  +  5 )  = ; 1
4
133131, 132eqtri 2461 . . . . . . . . 9  |-  ( ( 1  x.  9 )  +  5 )  = ; 1
4
13427, 15, 3, 22, 111, 113, 30, 2, 15, 129, 133decmac 10790 . . . . . . . 8  |-  ( (;; 2 3 1  x.  9 )  +  ( 4  +  1 ) )  = ;;; 2 0 8 4
135130oveq1i 6100 . . . . . . . . 9  |-  ( ( 1  x.  9 )  +  6 )  =  ( 9  +  6 )
136 9p6e15 10817 . . . . . . . . 9  |-  ( 9  +  6 )  = ; 1
5
137135, 136eqtri 2461 . . . . . . . 8  |-  ( ( 1  x.  9 )  +  6 )  = ; 1
5
13828, 15, 2, 103, 108, 110, 30, 22, 15, 134, 137decmac 10790 . . . . . . 7  |-  ( (;;; 2 3 1 1  x.  9 )  +  (; 4
6  +  0 ) )  = ;;;; 2 0 8 4 5
13929nn0cni 10587 . . . . . . . . . 10  |- ;;; 2 3 1 1  e.  CC
140139mul01i 9555 . . . . . . . . 9  |-  (;;; 2 3 1 1  x.  0 )  =  0
141140oveq1i 6100 . . . . . . . 8  |-  ( (;;; 2 3 1 1  x.  0 )  +  2 )  =  ( 0  +  2 )
142141, 83, 843eqtri 2465 . . . . . . 7  |-  ( (;;; 2 3 1 1  x.  0 )  +  2 )  = ; 0 2
14330, 3, 104, 10, 106, 107, 29, 10, 3, 138, 142decma2c 10791 . . . . . 6  |-  ( (;;; 2 3 1 1  x. ; 9
0 )  + ;; 4 6 2 )  = ;;;;; 2 0 8 4 5 2
144 2t2e4 10467 . . . . . . . . . . . . . . 15  |-  ( 2  x.  2 )  =  4
145144oveq1i 6100 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  0 )  =  ( 4  +  0 )
146145, 72eqtri 2461 . . . . . . . . . . . . 13  |-  ( ( 2  x.  2 )  +  0 )  =  4
147 3t2e6 10469 . . . . . . . . . . . . . 14  |-  ( 3  x.  2 )  =  6
148103dec0h 10767 . . . . . . . . . . . . . 14  |-  6  = ; 0 6
149147, 148eqtri 2461 . . . . . . . . . . . . 13  |-  ( 3  x.  2 )  = ; 0
6
15010, 10, 26, 114, 103, 3, 146, 149decmul1c 10798 . . . . . . . . . . . 12  |-  (; 2 3  x.  2 )  = ; 4 6
151150oveq1i 6100 . . . . . . . . . . 11  |-  ( (; 2
3  x.  2 )  +  0 )  =  (; 4 6  +  0 )
152151, 110eqtri 2461 . . . . . . . . . 10  |-  ( (; 2
3  x.  2 )  +  0 )  = ; 4
6
15362mulid2i 9385 . . . . . . . . . . 11  |-  ( 1  x.  2 )  =  2
154153, 84eqtri 2461 . . . . . . . . . 10  |-  ( 1  x.  2 )  = ; 0
2
15510, 27, 15, 111, 10, 3, 152, 154decmul1c 10798 . . . . . . . . 9  |-  (;; 2 3 1  x.  2 )  = ;; 4 6 2
156155oveq1i 6100 . . . . . . . 8  |-  ( (;; 2 3 1  x.  2 )  +  0 )  =  (;; 4 6 2  +  0 )
157105nn0cni 10587 . . . . . . . . 9  |- ;; 4 6 2  e.  CC
158157addid1i 9552 . . . . . . . 8  |-  (;; 4 6 2  +  0 )  = ;; 4 6 2
159156, 158eqtri 2461 . . . . . . 7  |-  ( (;; 2 3 1  x.  2 )  +  0 )  = ;; 4 6 2
16010, 28, 15, 108, 10, 3, 159, 154decmul1c 10798 . . . . . 6  |-  (;;; 2 3 1 1  x.  2 )  = ;;; 4 6 2 2
16129, 31, 10, 102, 10, 105, 143, 160decmul2c 10799 . . . . 5  |-  (;;; 2 3 1 1  x. ;; 9 0 2 )  = ;;;;;; 2 0 8 4 5 2 2
162101, 161eqtr4i 2464 . . . 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 14093 . . 3  |-  ( ( 2 ^;; 10 0 0 )  mod 
N )  =  ( 1  mod  N )
164 eqid 2441 . . . 4  |- ;; 10 0 0  = ;; 10 0 0
165 eqid 2441 . . . . . . 7  |- ; 10 0  = ; 10 0
166 dec10 10781 . . . . . . . . . 10  |-  10  = ; 1 0
16791, 92eqtri 2461 . . . . . . . . . 10  |-  ( ( 2  x.  1 )  +  0 )  =  2
16862mul01i 9555 . . . . . . . . . . 11  |-  ( 2  x.  0 )  =  0
169168, 53eqtri 2461 . . . . . . . . . 10  |-  ( 2  x.  0 )  = ; 0
0
17010, 15, 3, 166, 3, 3, 167, 169decmul2c 10799 . . . . . . . . 9  |-  ( 2  x.  10 )  = ; 2
0
171170oveq1i 6100 . . . . . . . 8  |-  ( ( 2  x.  10 )  +  0 )  =  (; 2 0  +  0 )
172171, 59eqtri 2461 . . . . . . 7  |-  ( ( 2  x.  10 )  +  0 )  = ; 2
0
17310, 16, 3, 165, 3, 3, 172, 169decmul2c 10799 . . . . . 6  |-  ( 2  x. ; 10 0 )  = ;; 2 0 0
174173oveq1i 6100 . . . . 5  |-  ( ( 2  x. ; 10 0 )  +  0 )  =  (;; 2 0 0  +  0 )
17512nn0cni 10587 . . . . . 6  |- ;; 2 0 0  e.  CC
176175addid1i 9552 . . . . 5  |-  (;; 2 0 0  +  0 )  = ;; 2 0 0
177174, 176eqtri 2461 . . . 4  |-  ( ( 2  x. ; 10 0 )  +  0 )  = ;; 2 0 0
17810, 17, 3, 164, 3, 3, 177, 169decmul2c 10799 . . 3  |-  ( 2  x. ;; 10 0 0 )  = ;;; 2 0 0 0
1798nncni 10328 . . . . . 6  |-  N  e.  CC
180179mul02i 9554 . . . . 5  |-  ( 0  x.  N )  =  0
181180oveq1i 6100 . . . 4  |-  ( ( 0  x.  N )  +  1 )  =  ( 0  +  1 )
18296, 115eqtr4i 2464 . . . 4  |-  ( 1  x.  1 )  =  ( 0  +  1 )
183181, 182eqtr4i 2464 . . 3  |-  ( ( 0  x.  N )  +  1 )  =  ( 1  x.  1 )
1848, 9, 18, 14, 15, 15, 163, 178, 183mod2xi 14094 . 2  |-  ( ( 2 ^;;; 2 0 0 0 )  mod  N )  =  ( 1  mod 
N )
185 eqid 2441 . . . 4  |- ;;; 2 0 0 0  = ;;; 2 0 0 0
18610, 10, 3, 38, 3, 3, 146, 169decmul2c 10799 . . . . . . . . 9  |-  ( 2  x. ; 2 0 )  = ; 4
0
187186oveq1i 6100 . . . . . . . 8  |-  ( ( 2  x. ; 2 0 )  +  0 )  =  (; 4
0  +  0 )
1884nn0cni 10587 . . . . . . . . 9  |- ; 4 0  e.  CC
189188addid1i 9552 . . . . . . . 8  |-  (; 4 0  +  0 )  = ; 4 0
190187, 189eqtri 2461 . . . . . . 7  |-  ( ( 2  x. ; 2 0 )  +  0 )  = ; 4 0
19110, 11, 3, 36, 3, 3, 190, 169decmul2c 10799 . . . . . 6  |-  ( 2  x. ;; 2 0 0 )  = ;; 4 0 0
192191oveq1i 6100 . . . . 5  |-  ( ( 2  x. ;; 2 0 0 )  +  0 )  =  (;; 4 0 0  +  0 )
1935nn0cni 10587 . . . . . 6  |- ;; 4 0 0  e.  CC
194193addid1i 9552 . . . . 5  |-  (;; 4 0 0  +  0 )  = ;; 4 0 0
195192, 194eqtri 2461 . . . 4  |-  ( ( 2  x. ;; 2 0 0 )  +  0 )  = ;; 4 0 0
19610, 12, 3, 185, 3, 3, 195, 169decmul2c 10799 . . 3  |-  ( 2  x. ;;; 2 0 0 0 )  = ;;; 4 0 0 0
197 eqid 2441 . . . . . . 7  |- ;;; 4 0 0 0  = ;;; 4 0 0 0
1985, 3, 115, 197decsuc 10774 . . . . . 6  |-  (;;; 4 0 0 0  +  1 )  = ;;; 4 0 0 1
1991, 198eqtr4i 2464 . . . . 5  |-  N  =  (;;; 4 0 0 0  +  1 )
200199oveq1i 6100 . . . 4  |-  ( N  -  1 )  =  ( (;;; 4 0 0 0  +  1 )  - 
1 )
2015, 3deccl 10765 . . . . . 6  |- ;;; 4 0 0 0  e.  NN0
202201nn0cni 10587 . . . . 5  |- ;;; 4 0 0 0  e.  CC
203 pncan 9612 . . . . 5  |-  ( (;;; 4 0 0 0  e.  CC  /\  1  e.  CC )  ->  (
(;;; 4 0 0 0  +  1 )  - 
1 )  = ;;; 4 0 0 0 )
204202, 95, 203mp2an 667 . . . 4  |-  ( (;;; 4 0 0 0  +  1 )  -  1 )  = ;;; 4 0 0 0
205200, 204eqtri 2461 . . 3  |-  ( N  -  1 )  = ;;; 4 0 0 0
206196, 205eqtr4i 2464 . 2  |-  ( 2  x. ;;; 2 0 0 0 )  =  ( N  -  1 )
2078, 9, 13, 14, 15, 15, 184, 206, 183mod2xi 14094 1  |-  ( ( 2 ^ ( N  -  1 ) )  mod  N )  =  ( 1  mod  N
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364    e. wcel 1761  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    - cmin 9591   NNcn 10318   2c2 10367   3c3 10368   4c4 10369   5c5 10370   6c6 10371   7c7 10372   8c8 10373   9c9 10374   10c10 10375  ;cdc 10751    mod cmo 11704   ^cexp 11861
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-rp 10988  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862
This theorem is referenced by:  4001prm  14165
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