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Theorem 4001lem2 14473
Description: Lemma for 4001prm 14476. Calculate a power mod. In decimal, we calculate  2 ^ 4 0 0  =  ( 2 ^ 2 0 0 ) ^ 2  ==  9 0 2 ^ 2  =  2 0 3 N  + 
1 4 0 1 and  2 ^ 8 0 0  =  ( 2 ^ 4 0 0 ) ^ 2  ==  1 4 0 1 ^ 2  =  4 9 0 N  +  2 3 1 1  ==  2 3 1 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
4001lem2  |-  ( ( 2 ^;; 8 0 0 )  mod 
N )  =  (;;; 2 3 1 1  mod 
N )

Proof of Theorem 4001lem2
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 9nn0 10810 . . . . 5  |-  9  e.  NN0
112, 10deccl 10981 . . . 4  |- ; 4 9  e.  NN0
1211, 3deccl 10981 . . 3  |- ;; 4 9 0  e.  NN0
1312nn0zi 10880 . 2  |- ;; 4 9 0  e.  ZZ
14 1nn0 10802 . . . . 5  |-  1  e.  NN0
1514, 2deccl 10981 . . . 4  |- ; 1 4  e.  NN0
1615, 3deccl 10981 . . 3  |- ;; 1 4 0  e.  NN0
1716, 14deccl 10981 . 2  |- ;;; 1 4 0 1  e.  NN0
18 2nn0 10803 . . . . 5  |-  2  e.  NN0
19 3nn0 10804 . . . . 5  |-  3  e.  NN0
2018, 19deccl 10981 . . . 4  |- ; 2 3  e.  NN0
2120, 14deccl 10981 . . 3  |- ;; 2 3 1  e.  NN0
2221, 14deccl 10981 . 2  |- ;;; 2 3 1 1  e.  NN0
2318, 3deccl 10981 . . . 4  |- ; 2 0  e.  NN0
2423, 3deccl 10981 . . 3  |- ;; 2 0 0  e.  NN0
2523, 19deccl 10981 . . . 4  |- ;; 2 0 3  e.  NN0
2625nn0zi 10880 . . 3  |- ;; 2 0 3  e.  ZZ
2710, 3deccl 10981 . . . 4  |- ; 9 0  e.  NN0
2827, 18deccl 10981 . . 3  |- ;; 9 0 2  e.  NN0
2914001lem1 14472 . . 3  |-  ( ( 2 ^;; 2 0 0 )  mod 
N )  =  (;; 9 0 2  mod 
N )
30 eqid 2462 . . . 4  |- ;; 2 0 0  = ;; 2 0 0
31 eqid 2462 . . . . . . 7  |- ; 2 0  = ; 2 0
32 2t2e4 10676 . . . . . . . . 9  |-  ( 2  x.  2 )  =  4
3332oveq1i 6287 . . . . . . . 8  |-  ( ( 2  x.  2 )  +  0 )  =  ( 4  +  0 )
34 4cn 10604 . . . . . . . . 9  |-  4  e.  CC
3534addid1i 9757 . . . . . . . 8  |-  ( 4  +  0 )  =  4
3633, 35eqtri 2491 . . . . . . 7  |-  ( ( 2  x.  2 )  +  0 )  =  4
37 2cn 10597 . . . . . . . . 9  |-  2  e.  CC
3837mul01i 9760 . . . . . . . 8  |-  ( 2  x.  0 )  =  0
393dec0h 10983 . . . . . . . 8  |-  0  = ; 0 0
4038, 39eqtri 2491 . . . . . . 7  |-  ( 2  x.  0 )  = ; 0
0
4118, 18, 3, 31, 3, 3, 36, 40decmul2c 11015 . . . . . 6  |-  ( 2  x. ; 2 0 )  = ; 4
0
4241oveq1i 6287 . . . . 5  |-  ( ( 2  x. ; 2 0 )  +  0 )  =  (; 4
0  +  0 )
434nn0cni 10798 . . . . . 6  |- ; 4 0  e.  CC
4443addid1i 9757 . . . . 5  |-  (; 4 0  +  0 )  = ; 4 0
4542, 44eqtri 2491 . . . 4  |-  ( ( 2  x. ; 2 0 )  +  0 )  = ; 4 0
4618, 23, 3, 30, 3, 3, 45, 40decmul2c 11015 . . 3  |-  ( 2  x. ;; 2 0 0 )  = ;; 4 0 0
47 eqid 2462 . . . . 5  |- ;;; 1 4 0 1  = ;;; 1 4 0 1
48 6nn0 10807 . . . . . . 7  |-  6  e.  NN0
4914, 48deccl 10981 . . . . . 6  |- ; 1 6  e.  NN0
50 eqid 2462 . . . . . 6  |- ;; 4 0 0  = ;; 4 0 0
51 eqid 2462 . . . . . . 7  |- ;; 1 4 0  = ;; 1 4 0
52 eqid 2462 . . . . . . . 8  |- ; 1 4  = ; 1 4
53 4p2e6 10661 . . . . . . . 8  |-  ( 4  +  2 )  =  6
5414, 2, 18, 52, 53decaddi 11011 . . . . . . 7  |-  (; 1 4  +  2 )  = ; 1 6
55 00id 9745 . . . . . . 7  |-  ( 0  +  0 )  =  0
5615, 3, 18, 3, 51, 31, 54, 55decadd 11008 . . . . . 6  |-  (;; 1 4 0  + ; 2 0 )  = ;; 1 6 0
57 eqid 2462 . . . . . . 7  |- ; 4 0  = ; 4 0
5849nn0cni 10798 . . . . . . . 8  |- ; 1 6  e.  CC
5958addid1i 9757 . . . . . . 7  |-  (; 1 6  +  0 )  = ; 1 6
60 eqid 2462 . . . . . . . 8  |- ;; 2 0 3  = ;; 2 0 3
61 ax-1cn 9541 . . . . . . . . . 10  |-  1  e.  CC
6261addid1i 9757 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
6314dec0h 10983 . . . . . . . . 9  |-  1  = ; 0 1
6462, 63eqtri 2491 . . . . . . . 8  |-  ( 1  +  0 )  = ; 0
1
6561addid2i 9758 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
6665, 63eqtri 2491 . . . . . . . . 9  |-  ( 0  +  1 )  = ; 0
1
67 4t2e8 10680 . . . . . . . . . . . 12  |-  ( 4  x.  2 )  =  8
6834, 37, 67mulcomli 9594 . . . . . . . . . . 11  |-  ( 2  x.  4 )  =  8
6968, 55oveq12i 6289 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  ( 0  +  0 ) )  =  ( 8  +  0 )
70 8cn 10612 . . . . . . . . . . 11  |-  8  e.  CC
7170addid1i 9757 . . . . . . . . . 10  |-  ( 8  +  0 )  =  8
7269, 71eqtri 2491 . . . . . . . . 9  |-  ( ( 2  x.  4 )  +  ( 0  +  0 ) )  =  8
7334mul02i 9759 . . . . . . . . . . 11  |-  ( 0  x.  4 )  =  0
7473oveq1i 6287 . . . . . . . . . 10  |-  ( ( 0  x.  4 )  +  1 )  =  ( 0  +  1 )
7574, 65, 633eqtri 2495 . . . . . . . . 9  |-  ( ( 0  x.  4 )  +  1 )  = ; 0
1
7618, 3, 3, 14, 31, 66, 2, 14, 3, 72, 75decmac 11006 . . . . . . . 8  |-  ( (; 2
0  x.  4 )  +  ( 0  +  1 ) )  = ; 8
1
77 2p1e3 10650 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
78 3cn 10601 . . . . . . . . . 10  |-  3  e.  CC
79 4t3e12 11039 . . . . . . . . . 10  |-  ( 4  x.  3 )  = ; 1
2
8034, 78, 79mulcomli 9594 . . . . . . . . 9  |-  ( 3  x.  4 )  = ; 1
2
8114, 18, 77, 80decsuc 10990 . . . . . . . 8  |-  ( ( 3  x.  4 )  +  1 )  = ; 1
3
8223, 19, 3, 14, 60, 64, 2, 19, 14, 76, 81decmac 11006 . . . . . . 7  |-  ( (;; 2 0 3  x.  4 )  +  ( 1  +  0 ) )  = ;; 8 1 3
8325nn0cni 10798 . . . . . . . . . 10  |- ;; 2 0 3  e.  CC
8483mul01i 9760 . . . . . . . . 9  |-  (;; 2 0 3  x.  0 )  =  0
8584oveq1i 6287 . . . . . . . 8  |-  ( (;; 2 0 3  x.  0 )  +  6 )  =  ( 0  +  6 )
86 6cn 10608 . . . . . . . . 9  |-  6  e.  CC
8786addid2i 9758 . . . . . . . 8  |-  ( 0  +  6 )  =  6
8848dec0h 10983 . . . . . . . 8  |-  6  = ; 0 6
8985, 87, 883eqtri 2495 . . . . . . 7  |-  ( (;; 2 0 3  x.  0 )  +  6 )  = ; 0 6
902, 3, 14, 48, 57, 59, 25, 48, 3, 82, 89decma2c 11007 . . . . . 6  |-  ( (;; 2 0 3  x. ; 4
0 )  +  (; 1
6  +  0 ) )  = ;;; 8 1 3 6
9184oveq1i 6287 . . . . . . 7  |-  ( (;; 2 0 3  x.  0 )  +  0 )  =  ( 0  +  0 )
9291, 55, 393eqtri 2495 . . . . . 6  |-  ( (;; 2 0 3  x.  0 )  +  0 )  = ; 0 0
934, 3, 49, 3, 50, 56, 25, 3, 3, 90, 92decma2c 11007 . . . . 5  |-  ( (;; 2 0 3  x. ;; 4 0 0 )  +  (;; 1 4 0  + ; 2 0 ) )  = ;;;; 8 1 3 6 0
9455, 39eqtri 2491 . . . . . . 7  |-  ( 0  +  0 )  = ; 0
0
9537mulid1i 9589 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
9695, 55oveq12i 6289 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
9737addid1i 9757 . . . . . . . 8  |-  ( 2  +  0 )  =  2
9896, 97eqtri 2491 . . . . . . 7  |-  ( ( 2  x.  1 )  +  ( 0  +  0 ) )  =  2
9961mul02i 9759 . . . . . . . . 9  |-  ( 0  x.  1 )  =  0
10099oveq1i 6287 . . . . . . . 8  |-  ( ( 0  x.  1 )  +  0 )  =  ( 0  +  0 )
101100, 55, 393eqtri 2495 . . . . . . 7  |-  ( ( 0  x.  1 )  +  0 )  = ; 0
0
10218, 3, 3, 3, 31, 94, 14, 3, 3, 98, 101decmac 11006 . . . . . 6  |-  ( (; 2
0  x.  1 )  +  ( 0  +  0 ) )  = ; 2
0
10378mulid1i 9589 . . . . . . . 8  |-  ( 3  x.  1 )  =  3
104103oveq1i 6287 . . . . . . 7  |-  ( ( 3  x.  1 )  +  1 )  =  ( 3  +  1 )
105 3p1e4 10652 . . . . . . 7  |-  ( 3  +  1 )  =  4
1062dec0h 10983 . . . . . . 7  |-  4  = ; 0 4
107104, 105, 1063eqtri 2495 . . . . . 6  |-  ( ( 3  x.  1 )  +  1 )  = ; 0
4
10823, 19, 3, 14, 60, 63, 14, 2, 3, 102, 107decmac 11006 . . . . 5  |-  ( (;; 2 0 3  x.  1 )  +  1 )  = ;; 2 0 4
1095, 14, 16, 14, 1, 47, 25, 2, 23, 93, 108decma2c 11007 . . . 4  |-  ( (;; 2 0 3  x.  N )  + ;;; 1 4 0 1 )  = ;;;;; 8 1 3 6 0 4
110 eqid 2462 . . . . 5  |- ;; 9 0 2  = ;; 9 0 2
111 8nn0 10809 . . . . . . 7  |-  8  e.  NN0
11214, 111deccl 10981 . . . . . 6  |- ; 1 8  e.  NN0
113112, 3deccl 10981 . . . . 5  |- ;; 1 8 0  e.  NN0
114 eqid 2462 . . . . . 6  |- ; 9 0  = ; 9 0
115 eqid 2462 . . . . . 6  |- ;; 1 8 0  = ;; 1 8 0
116112nn0cni 10798 . . . . . . . 8  |- ; 1 8  e.  CC
117116addid1i 9757 . . . . . . 7  |-  (; 1 8  +  0 )  = ; 1 8
118 1p2e3 10651 . . . . . . . . 9  |-  ( 1  +  2 )  =  3
11919dec0h 10983 . . . . . . . . 9  |-  3  = ; 0 3
120118, 119eqtri 2491 . . . . . . . 8  |-  ( 1  +  2 )  = ; 0
3
121 9t9e81 11069 . . . . . . . . . 10  |-  ( 9  x.  9 )  = ; 8
1
122121oveq1i 6287 . . . . . . . . 9  |-  ( ( 9  x.  9 )  +  0 )  =  (; 8 1  +  0 )
123111, 14deccl 10981 . . . . . . . . . . 11  |- ; 8 1  e.  NN0
124123nn0cni 10798 . . . . . . . . . 10  |- ; 8 1  e.  CC
125124addid1i 9757 . . . . . . . . 9  |-  (; 8 1  +  0 )  = ; 8 1
126122, 125eqtri 2491 . . . . . . . 8  |-  ( ( 9  x.  9 )  +  0 )  = ; 8
1
127 9cn 10614 . . . . . . . . . . 11  |-  9  e.  CC
128127mul02i 9759 . . . . . . . . . 10  |-  ( 0  x.  9 )  =  0
129128oveq1i 6287 . . . . . . . . 9  |-  ( ( 0  x.  9 )  +  3 )  =  ( 0  +  3 )
13078addid2i 9758 . . . . . . . . 9  |-  ( 0  +  3 )  =  3
131129, 130eqtri 2491 . . . . . . . 8  |-  ( ( 0  x.  9 )  +  3 )  =  3
13210, 3, 3, 19, 114, 120, 10, 126, 131decma 11005 . . . . . . 7  |-  ( (; 9
0  x.  9 )  +  ( 1  +  2 ) )  = ;; 8 1 3
133 9t2e18 11062 . . . . . . . . 9  |-  ( 9  x.  2 )  = ; 1
8
134127, 37, 133mulcomli 9594 . . . . . . . 8  |-  ( 2  x.  9 )  = ; 1
8
135 1p1e2 10640 . . . . . . . 8  |-  ( 1  +  1 )  =  2
136 8p8e16 11028 . . . . . . . 8  |-  ( 8  +  8 )  = ; 1
6
13714, 111, 111, 134, 135, 48, 136decaddci 11012 . . . . . . 7  |-  ( ( 2  x.  9 )  +  8 )  = ; 2
6
13827, 18, 14, 111, 110, 117, 10, 48, 18, 132, 137decmac 11006 . . . . . 6  |-  ( (;; 9 0 2  x.  9 )  +  (; 1
8  +  0 ) )  = ;;; 8 1 3 6
13928nn0cni 10798 . . . . . . . . 9  |- ;; 9 0 2  e.  CC
140139mul01i 9760 . . . . . . . 8  |-  (;; 9 0 2  x.  0 )  =  0
141140oveq1i 6287 . . . . . . 7  |-  ( (;; 9 0 2  x.  0 )  +  0 )  =  ( 0  +  0 )
142141, 55, 393eqtri 2495 . . . . . 6  |-  ( (;; 9 0 2  x.  0 )  +  0 )  = ; 0 0
14310, 3, 112, 3, 114, 115, 28, 3, 3, 138, 142decma2c 11007 . . . . 5  |-  ( (;; 9 0 2  x. ; 9
0 )  + ;; 1 8 0 )  = ;;;; 8 1 3 6 0
144133oveq1i 6287 . . . . . . . . . 10  |-  ( ( 9  x.  2 )  +  0 )  =  (; 1 8  +  0 )
145144, 117eqtri 2491 . . . . . . . . 9  |-  ( ( 9  x.  2 )  +  0 )  = ; 1
8
14637mul02i 9759 . . . . . . . . . 10  |-  ( 0  x.  2 )  =  0
147146, 39eqtri 2491 . . . . . . . . 9  |-  ( 0  x.  2 )  = ; 0
0
14818, 10, 3, 114, 3, 3, 145, 147decmul1c 11014 . . . . . . . 8  |-  (; 9 0  x.  2 )  = ;; 1 8 0
149148oveq1i 6287 . . . . . . 7  |-  ( (; 9
0  x.  2 )  +  0 )  =  (;; 1 8 0  +  0 )
150113nn0cni 10798 . . . . . . . 8  |- ;; 1 8 0  e.  CC
151150addid1i 9757 . . . . . . 7  |-  (;; 1 8 0  +  0 )  = ;; 1 8 0
152149, 151eqtri 2491 . . . . . 6  |-  ( (; 9
0  x.  2 )  +  0 )  = ;; 1 8 0
15332, 106eqtri 2491 . . . . . 6  |-  ( 2  x.  2 )  = ; 0
4
15418, 27, 18, 110, 2, 3, 152, 153decmul1c 11014 . . . . 5  |-  (;; 9 0 2  x.  2 )  = ;;; 1 8 0 4
15528, 27, 18, 110, 2, 113, 143, 154decmul2c 11015 . . . 4  |-  (;; 9 0 2  x. ;; 9 0 2 )  = ;;;;; 8 1 3 6 0 4
156109, 155eqtr4i 2494 . . 3  |-  ( (;; 2 0 3  x.  N )  + ;;; 1 4 0 1 )  =  (;; 9 0 2  x. ;; 9 0 2 )
1578, 9, 24, 26, 28, 17, 29, 46, 156mod2xi 14405 . 2  |-  ( ( 2 ^;; 4 0 0 )  mod 
N )  =  (;;; 1 4 0 1  mod 
N )
15868oveq1i 6287 . . . . . . 7  |-  ( ( 2  x.  4 )  +  0 )  =  ( 8  +  0 )
159158, 71eqtri 2491 . . . . . 6  |-  ( ( 2  x.  4 )  +  0 )  =  8
16018, 2, 3, 57, 3, 3, 159, 40decmul2c 11015 . . . . 5  |-  ( 2  x. ; 4 0 )  = ; 8
0
161160oveq1i 6287 . . . 4  |-  ( ( 2  x. ; 4 0 )  +  0 )  =  (; 8
0  +  0 )
162111, 3deccl 10981 . . . . . 6  |- ; 8 0  e.  NN0
163162nn0cni 10798 . . . . 5  |- ; 8 0  e.  CC
164163addid1i 9757 . . . 4  |-  (; 8 0  +  0 )  = ; 8 0
165161, 164eqtri 2491 . . 3  |-  ( ( 2  x. ; 4 0 )  +  0 )  = ; 8 0
16618, 4, 3, 50, 3, 3, 165, 40decmul2c 11015 . 2  |-  ( 2  x. ;; 4 0 0 )  = ;; 8 0 0
167 eqid 2462 . . . 4  |- ;;; 2 3 1 1  = ;;; 2 3 1 1
16818, 111deccl 10981 . . . . 5  |- ; 2 8  e.  NN0
169 eqid 2462 . . . . . 6  |- ;; 2 3 1  = ;; 2 3 1
170 eqid 2462 . . . . . 6  |- ; 4 9  = ; 4 9
171 7nn0 10808 . . . . . . 7  |-  7  e.  NN0
172 7p1e8 10656 . . . . . . 7  |-  ( 7  +  1 )  =  8
173 eqid 2462 . . . . . . . 8  |- ; 2 3  = ; 2 3
174 4p3e7 10662 . . . . . . . . 9  |-  ( 4  +  3 )  =  7
17534, 78, 174addcomli 9762 . . . . . . . 8  |-  ( 3  +  4 )  =  7
17618, 19, 2, 173, 175decaddi 11011 . . . . . . 7  |-  (; 2 3  +  4 )  = ; 2 7
17718, 171, 172, 176decsuc 10990 . . . . . 6  |-  ( (; 2
3  +  4 )  +  1 )  = ; 2
8
178 9p1e10 10658 . . . . . . 7  |-  ( 9  +  1 )  =  10
179127, 61, 178addcomli 9762 . . . . . 6  |-  ( 1  +  9 )  =  10
18020, 14, 2, 10, 169, 170, 177, 179decaddc2 11010 . . . . 5  |-  (;; 2 3 1  + ; 4 9 )  = ;; 2 8 0
181168nn0cni 10798 . . . . . . 7  |- ; 2 8  e.  CC
182181addid1i 9757 . . . . . 6  |-  (; 2 8  +  0 )  = ; 2 8
183 eqid 2462 . . . . . . 7  |- ;; 4 9 0  = ;; 4 9 0
18418dec0h 10983 . . . . . . . 8  |-  2  = ; 0 2
18597, 184eqtri 2491 . . . . . . 7  |-  ( 2  +  0 )  = ; 0
2
186130oveq2i 6288 . . . . . . . . 9  |-  ( ( 4  x.  4 )  +  ( 0  +  3 ) )  =  ( ( 4  x.  4 )  +  3 )
187 4t4e16 11040 . . . . . . . . . 10  |-  ( 4  x.  4 )  = ; 1
6
188 6p3e9 10669 . . . . . . . . . 10  |-  ( 6  +  3 )  =  9
18914, 48, 19, 187, 188decaddi 11011 . . . . . . . . 9  |-  ( ( 4  x.  4 )  +  3 )  = ; 1
9
190186, 189eqtri 2491 . . . . . . . 8  |-  ( ( 4  x.  4 )  +  ( 0  +  3 ) )  = ; 1
9
191 9t4e36 11064 . . . . . . . . . 10  |-  ( 9  x.  4 )  = ; 3
6
192191oveq1i 6287 . . . . . . . . 9  |-  ( ( 9  x.  4 )  +  0 )  =  (; 3 6  +  0 )
19319, 48deccl 10981 . . . . . . . . . . 11  |- ; 3 6  e.  NN0
194193nn0cni 10798 . . . . . . . . . 10  |- ; 3 6  e.  CC
195194addid1i 9757 . . . . . . . . 9  |-  (; 3 6  +  0 )  = ; 3 6
196192, 195eqtri 2491 . . . . . . . 8  |-  ( ( 9  x.  4 )  +  0 )  = ; 3
6
1972, 10, 3, 3, 170, 94, 2, 48, 19, 190, 196decmac 11006 . . . . . . 7  |-  ( (; 4
9  x.  4 )  +  ( 0  +  0 ) )  = ;; 1 9 6
19873oveq1i 6287 . . . . . . . 8  |-  ( ( 0  x.  4 )  +  2 )  =  ( 0  +  2 )
19937addid2i 9758 . . . . . . . 8  |-  ( 0  +  2 )  =  2
200198, 199, 1843eqtri 2495 . . . . . . 7  |-  ( ( 0  x.  4 )  +  2 )  = ; 0
2
20111, 3, 3, 18, 183, 185, 2, 18, 3, 197, 200decmac 11006 . . . . . 6  |-  ( (;; 4 9 0  x.  4 )  +  ( 2  +  0 ) )  = ;;; 1 9 6 2
20212nn0cni 10798 . . . . . . . . 9  |- ;; 4 9 0  e.  CC
203202mul01i 9760 . . . . . . . 8  |-  (;; 4 9 0  x.  0 )  =  0
204203oveq1i 6287 . . . . . . 7  |-  ( (;; 4 9 0  x.  0 )  +  8 )  =  ( 0  +  8 )
20570addid2i 9758 . . . . . . 7  |-  ( 0  +  8 )  =  8
206111dec0h 10983 . . . . . . 7  |-  8  = ; 0 8
207204, 205, 2063eqtri 2495 . . . . . 6  |-  ( (;; 4 9 0  x.  0 )  +  8 )  = ; 0 8
2082, 3, 18, 111, 57, 182, 12, 111, 3, 201, 207decma2c 11007 . . . . 5  |-  ( (;; 4 9 0  x. ; 4
0 )  +  (; 2
8  +  0 ) )  = ;;;; 1 9 6 2 8
209203oveq1i 6287 . . . . . 6  |-  ( (;; 4 9 0  x.  0 )  +  0 )  =  ( 0  +  0 )
210209, 55, 393eqtri 2495 . . . . 5  |-  ( (;; 4 9 0  x.  0 )  +  0 )  = ; 0 0
2114, 3, 168, 3, 50, 180, 12, 3, 3, 208, 210decma2c 11007 . . . 4  |-  ( (;; 4 9 0  x. ;; 4 0 0 )  +  (;; 2 3 1  + ; 4 9 ) )  = ;;;;; 1 9 6 2 8 0
21234mulid1i 9589 . . . . . . . 8  |-  ( 4  x.  1 )  =  4
213212, 55oveq12i 6289 . . . . . . 7  |-  ( ( 4  x.  1 )  +  ( 0  +  0 ) )  =  ( 4  +  0 )
214213, 35eqtri 2491 . . . . . 6  |-  ( ( 4  x.  1 )  +  ( 0  +  0 ) )  =  4
215127mulid1i 9589 . . . . . . . 8  |-  ( 9  x.  1 )  =  9
216215oveq1i 6287 . . . . . . 7  |-  ( ( 9  x.  1 )  +  0 )  =  ( 9  +  0 )
217127addid1i 9757 . . . . . . 7  |-  ( 9  +  0 )  =  9
21810dec0h 10983 . . . . . . 7  |-  9  = ; 0 9
219216, 217, 2183eqtri 2495 . . . . . 6  |-  ( ( 9  x.  1 )  +  0 )  = ; 0
9
2202, 10, 3, 3, 170, 94, 14, 10, 3, 214, 219decmac 11006 . . . . 5  |-  ( (; 4
9  x.  1 )  +  ( 0  +  0 ) )  = ; 4
9
22199oveq1i 6287 . . . . . 6  |-  ( ( 0  x.  1 )  +  1 )  =  ( 0  +  1 )
222221, 65, 633eqtri 2495 . . . . 5  |-  ( ( 0  x.  1 )  +  1 )  = ; 0
1
22311, 3, 3, 14, 183, 63, 14, 14, 3, 220, 222decmac 11006 . . . 4  |-  ( (;; 4 9 0  x.  1 )  +  1 )  = ;; 4 9 1
2245, 14, 21, 14, 1, 167, 12, 14, 11, 211, 223decma2c 11007 . . 3  |-  ( (;; 4 9 0  x.  N )  + ;;; 2 3 1 1 )  = ;;;;;; 1 9 6 2 8 0 1
22515nn0cni 10798 . . . . . . 7  |- ; 1 4  e.  CC
226225addid1i 9757 . . . . . 6  |-  (; 1 4  +  0 )  = ; 1 4
227 5nn0 10806 . . . . . . . 8  |-  5  e.  NN0
228227, 48deccl 10981 . . . . . . 7  |- ; 5 6  e.  NN0
229228, 3deccl 10981 . . . . . 6  |- ;; 5 6 0  e.  NN0
230 eqid 2462 . . . . . . . 8  |- ;; 5 6 0  = ;; 5 6 0
231228nn0cni 10798 . . . . . . . . 9  |- ; 5 6  e.  CC
232231addid2i 9758 . . . . . . . 8  |-  ( 0  + ; 5 6 )  = ; 5
6
2333, 14, 228, 3, 63, 230, 232, 62decadd 11008 . . . . . . 7  |-  ( 1  + ;; 5 6 0 )  = ;; 5 6 1
234231addid1i 9757 . . . . . . . 8  |-  (; 5 6  +  0 )  = ; 5 6
235 5cn 10606 . . . . . . . . . . 11  |-  5  e.  CC
236235addid1i 9757 . . . . . . . . . 10  |-  ( 5  +  0 )  =  5
237227dec0h 10983 . . . . . . . . . 10  |-  5  = ; 0 5
238236, 237eqtri 2491 . . . . . . . . 9  |-  ( 5  +  0 )  = ; 0
5
23961mulid1i 9589 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
240239, 55oveq12i 6289 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
241240, 62eqtri 2491 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
242212oveq1i 6287 . . . . . . . . . 10  |-  ( ( 4  x.  1 )  +  5 )  =  ( 4  +  5 )
243 5p4e9 10666 . . . . . . . . . . 11  |-  ( 5  +  4 )  =  9
244235, 34, 243addcomli 9762 . . . . . . . . . 10  |-  ( 4  +  5 )  =  9
245242, 244, 2183eqtri 2495 . . . . . . . . 9  |-  ( ( 4  x.  1 )  +  5 )  = ; 0
9
24614, 2, 3, 227, 52, 238, 14, 10, 3, 241, 245decmac 11006 . . . . . . . 8  |-  ( (; 1
4  x.  1 )  +  ( 5  +  0 ) )  = ; 1
9
24799oveq1i 6287 . . . . . . . . 9  |-  ( ( 0  x.  1 )  +  6 )  =  ( 0  +  6 )
248247, 87, 883eqtri 2495 . . . . . . . 8  |-  ( ( 0  x.  1 )  +  6 )  = ; 0
6
24915, 3, 227, 48, 51, 234, 14, 48, 3, 246, 248decmac 11006 . . . . . . 7  |-  ( (;; 1 4 0  x.  1 )  +  (; 5
6  +  0 ) )  = ;; 1 9 6
250239oveq1i 6287 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  1 )  =  ( 1  +  1 )
251250, 135, 1843eqtri 2495 . . . . . . 7  |-  ( ( 1  x.  1 )  +  1 )  = ; 0
2
25216, 14, 228, 14, 47, 233, 14, 18, 3, 249, 251decmac 11006 . . . . . 6  |-  ( (;;; 1 4 0 1  x.  1 )  +  ( 1  + ;; 5 6 0 ) )  = ;;; 1 9 6 2
25334mulid2i 9590 . . . . . . . . . . 11  |-  ( 1  x.  4 )  =  4
254253, 65oveq12i 6289 . . . . . . . . . 10  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
255 4p1e5 10653 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
256254, 255eqtri 2491 . . . . . . . . 9  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  5
257187oveq1i 6287 . . . . . . . . . 10  |-  ( ( 4  x.  4 )  +  0 )  =  (; 1 6  +  0 )
258257, 59eqtri 2491 . . . . . . . . 9  |-  ( ( 4  x.  4 )  +  0 )  = ; 1
6
25914, 2, 3, 3, 52, 94, 2, 48, 14, 256, 258decmac 11006 . . . . . . . 8  |-  ( (; 1
4  x.  4 )  +  ( 0  +  0 ) )  = ; 5
6
26073oveq1i 6287 . . . . . . . . 9  |-  ( ( 0  x.  4 )  +  0 )  =  ( 0  +  0 )
261260, 55, 393eqtri 2495 . . . . . . . 8  |-  ( ( 0  x.  4 )  +  0 )  = ; 0
0
26215, 3, 3, 3, 51, 94, 2, 3, 3, 259, 261decmac 11006 . . . . . . 7  |-  ( (;; 1 4 0  x.  4 )  +  ( 0  +  0 ) )  = ;; 5 6 0
263253oveq1i 6287 . . . . . . . 8  |-  ( ( 1  x.  4 )  +  4 )  =  ( 4  +  4 )
264 4p4e8 10663 . . . . . . . 8  |-  ( 4  +  4 )  =  8
265263, 264, 2063eqtri 2495 . . . . . . 7  |-  ( ( 1  x.  4 )  +  4 )  = ; 0
8
26616, 14, 3, 2, 47, 106, 2, 111, 3, 262, 265decmac 11006 . . . . . 6  |-  ( (;;; 1 4 0 1  x.  4 )  +  4 )  = ;;; 5 6 0 8
26714, 2, 14, 2, 52, 226, 17, 111, 229, 252, 266decma2c 11007 . . . . 5  |-  ( (;;; 1 4 0 1  x. ; 1
4 )  +  (; 1
4  +  0 ) )  = ;;;; 1 9 6 2 8
26817nn0cni 10798 . . . . . . . 8  |- ;;; 1 4 0 1  e.  CC
269268mul01i 9760 . . . . . . 7  |-  (;;; 1 4 0 1  x.  0 )  =  0
270269oveq1i 6287 . . . . . 6  |-  ( (;;; 1 4 0 1  x.  0 )  +  0 )  =  ( 0  +  0 )
271270, 55, 393eqtri 2495 . . . . 5  |-  ( (;;; 1 4 0 1  x.  0 )  +  0 )  = ; 0 0
27215, 3, 15, 3, 51, 51, 17, 3, 3, 267, 271decma2c 11007 . . . 4  |-  ( (;;; 1 4 0 1  x. ;; 1 4 0 )  + ;; 1 4 0 )  = ;;;;; 1 9 6 2 8 0
273268mulid1i 9589 . . . 4  |-  (;;; 1 4 0 1  x.  1 )  = ;;; 1 4 0 1
27417, 16, 14, 47, 14, 16, 272, 273decmul2c 11015 . . 3  |-  (;;; 1 4 0 1  x. ;;; 1 4 0 1 )  = ;;;;;; 1 9 6 2 8 0 1
275224, 274eqtr4i 2494 . 2  |-  ( (;; 4 9 0  x.  N )  + ;;; 2 3 1 1 )  =  (;;; 1 4 0 1  x. ;;; 1 4 0 1 )
2768, 9, 5, 13, 17, 22, 157, 166, 275mod2xi 14405 1  |-  ( ( 2 ^;; 8 0 0 )  mod 
N )  =  (;;; 2 3 1 1  mod 
N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374  (class class class)co 6277   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488   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:  4001lem3  14474  4001lem4  14475
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