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Theorem 4001lem2 13140
Description: Lemma for 2503prm 13138. 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 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 9nn0 9989 . . . . 5  |-  9  e.  NN0
112, 10deccl 10138 . . . 4  |- ; 4 9  e.  NN0
1211, 3deccl 10138 . . 3  |- ;; 4 9 0  e.  NN0
1312nn0zi 10048 . 2  |- ;; 4 9 0  e.  ZZ
14 1nn0 9981 . . . . 5  |-  1  e.  NN0
1514, 2deccl 10138 . . . 4  |- ; 1 4  e.  NN0
1615, 3deccl 10138 . . 3  |- ;; 1 4 0  e.  NN0
1716, 14deccl 10138 . 2  |- ;;; 1 4 0 1  e.  NN0
18 2nn0 9982 . . . . 5  |-  2  e.  NN0
19 3nn0 9983 . . . . 5  |-  3  e.  NN0
2018, 19deccl 10138 . . . 4  |- ; 2 3  e.  NN0
2120, 14deccl 10138 . . 3  |- ;; 2 3 1  e.  NN0
2221, 14deccl 10138 . 2  |- ;;; 2 3 1 1  e.  NN0
2318, 3deccl 10138 . . . 4  |- ; 2 0  e.  NN0
2423, 3deccl 10138 . . 3  |- ;; 2 0 0  e.  NN0
2523, 19deccl 10138 . . . 4  |- ;; 2 0 3  e.  NN0
2625nn0zi 10048 . . 3  |- ;; 2 0 3  e.  ZZ
2710, 3deccl 10138 . . . 4  |- ; 9 0  e.  NN0
2827, 18deccl 10138 . . 3  |- ;; 9 0 2  e.  NN0
2914001lem1 13139 . . 3  |-  ( ( 2 ^;; 2 0 0 )  mod 
N )  =  (;; 9 0 2  mod 
N )
30 eqid 2283 . . . 4  |- ;; 2 0 0  = ;; 2 0 0
31 eqid 2283 . . . . . . 7  |- ; 2 0  = ; 2 0
32 2t2e4 9871 . . . . . . . . 9  |-  ( 2  x.  2 )  =  4
3332oveq1i 5868 . . . . . . . 8  |-  ( ( 2  x.  2 )  +  0 )  =  ( 4  +  0 )
34 4cn 9820 . . . . . . . . 9  |-  4  e.  CC
3534addid1i 8999 . . . . . . . 8  |-  ( 4  +  0 )  =  4
3633, 35eqtri 2303 . . . . . . 7  |-  ( ( 2  x.  2 )  +  0 )  =  4
37 2cn 9816 . . . . . . . . 9  |-  2  e.  CC
3837mul01i 9002 . . . . . . . 8  |-  ( 2  x.  0 )  =  0
393dec0h 10140 . . . . . . . 8  |-  0  = ; 0 0
4038, 39eqtri 2303 . . . . . . 7  |-  ( 2  x.  0 )  = ; 0
0
4118, 18, 3, 31, 3, 3, 36, 40decmul2c 10172 . . . . . 6  |-  ( 2  x. ; 2 0 )  = ; 4
0
4241oveq1i 5868 . . . . 5  |-  ( ( 2  x. ; 2 0 )  +  0 )  =  (; 4
0  +  0 )
434nn0cni 9977 . . . . . 6  |- ; 4 0  e.  CC
4443addid1i 8999 . . . . 5  |-  (; 4 0  +  0 )  = ; 4 0
4542, 44eqtri 2303 . . . 4  |-  ( ( 2  x. ; 2 0 )  +  0 )  = ; 4 0
4618, 23, 3, 30, 3, 3, 45, 40decmul2c 10172 . . 3  |-  ( 2  x. ;; 2 0 0 )  = ;; 4 0 0
47 eqid 2283 . . . . 5  |- ;;; 1 4 0 1  = ;;; 1 4 0 1
48 6nn0 9986 . . . . . . 7  |-  6  e.  NN0
4914, 48deccl 10138 . . . . . 6  |- ; 1 6  e.  NN0
50 eqid 2283 . . . . . 6  |- ;; 4 0 0  = ;; 4 0 0
51 eqid 2283 . . . . . . 7  |- ;; 1 4 0  = ;; 1 4 0
52 eqid 2283 . . . . . . . 8  |- ; 1 4  = ; 1 4
53 4p2e6 9857 . . . . . . . 8  |-  ( 4  +  2 )  =  6
5414, 2, 18, 52, 53decaddi 10168 . . . . . . 7  |-  (; 1 4  +  2 )  = ; 1 6
55 00id 8987 . . . . . . 7  |-  ( 0  +  0 )  =  0
5615, 3, 18, 3, 51, 31, 54, 55decadd 10165 . . . . . 6  |-  (;; 1 4 0  + ; 2 0 )  = ;; 1 6 0
57 eqid 2283 . . . . . . 7  |- ; 4 0  = ; 4 0
5849nn0cni 9977 . . . . . . . 8  |- ; 1 6  e.  CC
5958addid1i 8999 . . . . . . 7  |-  (; 1 6  +  0 )  = ; 1 6
60 eqid 2283 . . . . . . . 8  |- ;; 2 0 3  = ;; 2 0 3
61 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
6261addid1i 8999 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
6314dec0h 10140 . . . . . . . . 9  |-  1  = ; 0 1
6462, 63eqtri 2303 . . . . . . . 8  |-  ( 1  +  0 )  = ; 0
1
6561addid2i 9000 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
6665, 63eqtri 2303 . . . . . . . . 9  |-  ( 0  +  1 )  = ; 0
1
67 4t2e8 9874 . . . . . . . . . . . 12  |-  ( 4  x.  2 )  =  8
6834, 37, 67mulcomli 8844 . . . . . . . . . . 11  |-  ( 2  x.  4 )  =  8
6968, 55oveq12i 5870 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  ( 0  +  0 ) )  =  ( 8  +  0 )
70 8nn 9883 . . . . . . . . . . . 12  |-  8  e.  NN
7170nncni 9756 . . . . . . . . . . 11  |-  8  e.  CC
7271addid1i 8999 . . . . . . . . . 10  |-  ( 8  +  0 )  =  8
7369, 72eqtri 2303 . . . . . . . . 9  |-  ( ( 2  x.  4 )  +  ( 0  +  0 ) )  =  8
7434mul02i 9001 . . . . . . . . . . 11  |-  ( 0  x.  4 )  =  0
7574oveq1i 5868 . . . . . . . . . 10  |-  ( ( 0  x.  4 )  +  1 )  =  ( 0  +  1 )
7675, 65, 633eqtri 2307 . . . . . . . . 9  |-  ( ( 0  x.  4 )  +  1 )  = ; 0
1
7718, 3, 3, 14, 31, 66, 2, 14, 3, 73, 76decmac 10163 . . . . . . . 8  |-  ( (; 2
0  x.  4 )  +  ( 0  +  1 ) )  = ; 8
1
78 2p1e3 9847 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
79 3cn 9818 . . . . . . . . . 10  |-  3  e.  CC
80 4t3e12 10196 . . . . . . . . . 10  |-  ( 4  x.  3 )  = ; 1
2
8134, 79, 80mulcomli 8844 . . . . . . . . 9  |-  ( 3  x.  4 )  = ; 1
2
8214, 18, 78, 81decsuc 10147 . . . . . . . 8  |-  ( ( 3  x.  4 )  +  1 )  = ; 1
3
8323, 19, 3, 14, 60, 64, 2, 19, 14, 77, 82decmac 10163 . . . . . . 7  |-  ( (;; 2 0 3  x.  4 )  +  ( 1  +  0 ) )  = ;; 8 1 3
8425nn0cni 9977 . . . . . . . . . 10  |- ;; 2 0 3  e.  CC
8584mul01i 9002 . . . . . . . . 9  |-  (;; 2 0 3  x.  0 )  =  0
8685oveq1i 5868 . . . . . . . 8  |-  ( (;; 2 0 3  x.  0 )  +  6 )  =  ( 0  +  6 )
87 6nn 9881 . . . . . . . . . 10  |-  6  e.  NN
8887nncni 9756 . . . . . . . . 9  |-  6  e.  CC
8988addid2i 9000 . . . . . . . 8  |-  ( 0  +  6 )  =  6
9048dec0h 10140 . . . . . . . 8  |-  6  = ; 0 6
9186, 89, 903eqtri 2307 . . . . . . 7  |-  ( (;; 2 0 3  x.  0 )  +  6 )  = ; 0 6
922, 3, 14, 48, 57, 59, 25, 48, 3, 83, 91decma2c 10164 . . . . . 6  |-  ( (;; 2 0 3  x. ; 4
0 )  +  (; 1
6  +  0 ) )  = ;;; 8 1 3 6
9385oveq1i 5868 . . . . . . 7  |-  ( (;; 2 0 3  x.  0 )  +  0 )  =  ( 0  +  0 )
9493, 55, 393eqtri 2307 . . . . . 6  |-  ( (;; 2 0 3  x.  0 )  +  0 )  = ; 0 0
954, 3, 49, 3, 50, 56, 25, 3, 3, 92, 94decma2c 10164 . . . . 5  |-  ( (;; 2 0 3  x. ;; 4 0 0 )  +  (;; 1 4 0  + ; 2 0 ) )  = ;;;; 8 1 3 6 0
9655, 39eqtri 2303 . . . . . . 7  |-  ( 0  +  0 )  = ; 0
0
9737mulid1i 8839 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
9897, 55oveq12i 5870 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
9937addid1i 8999 . . . . . . . 8  |-  ( 2  +  0 )  =  2
10098, 99eqtri 2303 . . . . . . 7  |-  ( ( 2  x.  1 )  +  ( 0  +  0 ) )  =  2
10161mul02i 9001 . . . . . . . . 9  |-  ( 0  x.  1 )  =  0
102101oveq1i 5868 . . . . . . . 8  |-  ( ( 0  x.  1 )  +  0 )  =  ( 0  +  0 )
103102, 55, 393eqtri 2307 . . . . . . 7  |-  ( ( 0  x.  1 )  +  0 )  = ; 0
0
10418, 3, 3, 3, 31, 96, 14, 3, 3, 100, 103decmac 10163 . . . . . 6  |-  ( (; 2
0  x.  1 )  +  ( 0  +  0 ) )  = ; 2
0
10579mulid1i 8839 . . . . . . . 8  |-  ( 3  x.  1 )  =  3
106105oveq1i 5868 . . . . . . 7  |-  ( ( 3  x.  1 )  +  1 )  =  ( 3  +  1 )
107 3p1e4 9848 . . . . . . 7  |-  ( 3  +  1 )  =  4
1082dec0h 10140 . . . . . . 7  |-  4  = ; 0 4
109106, 107, 1083eqtri 2307 . . . . . 6  |-  ( ( 3  x.  1 )  +  1 )  = ; 0
4
11023, 19, 3, 14, 60, 63, 14, 2, 3, 104, 109decmac 10163 . . . . 5  |-  ( (;; 2 0 3  x.  1 )  +  1 )  = ;; 2 0 4
1115, 14, 16, 14, 1, 47, 25, 2, 23, 95, 110decma2c 10164 . . . 4  |-  ( (;; 2 0 3  x.  N )  + ;;; 1 4 0 1 )  = ;;;;; 8 1 3 6 0 4
112 eqid 2283 . . . . 5  |- ;; 9 0 2  = ;; 9 0 2
113 8nn0 9988 . . . . . . 7  |-  8  e.  NN0
11414, 113deccl 10138 . . . . . 6  |- ; 1 8  e.  NN0
115114, 3deccl 10138 . . . . 5  |- ;; 1 8 0  e.  NN0
116 eqid 2283 . . . . . 6  |- ; 9 0  = ; 9 0
117 eqid 2283 . . . . . 6  |- ;; 1 8 0  = ;; 1 8 0
118114nn0cni 9977 . . . . . . . 8  |- ; 1 8  e.  CC
119118addid1i 8999 . . . . . . 7  |-  (; 1 8  +  0 )  = ; 1 8
12037, 61, 78addcomli 9004 . . . . . . . . 9  |-  ( 1  +  2 )  =  3
12119dec0h 10140 . . . . . . . . 9  |-  3  = ; 0 3
122120, 121eqtri 2303 . . . . . . . 8  |-  ( 1  +  2 )  = ; 0
3
123 9t9e81 10226 . . . . . . . . . 10  |-  ( 9  x.  9 )  = ; 8
1
124123oveq1i 5868 . . . . . . . . 9  |-  ( ( 9  x.  9 )  +  0 )  =  (; 8 1  +  0 )
125113, 14deccl 10138 . . . . . . . . . . 11  |- ; 8 1  e.  NN0
126125nn0cni 9977 . . . . . . . . . 10  |- ; 8 1  e.  CC
127126addid1i 8999 . . . . . . . . 9  |-  (; 8 1  +  0 )  = ; 8 1
128124, 127eqtri 2303 . . . . . . . 8  |-  ( ( 9  x.  9 )  +  0 )  = ; 8
1
129 9nn 9884 . . . . . . . . . . . 12  |-  9  e.  NN
130129nncni 9756 . . . . . . . . . . 11  |-  9  e.  CC
131130mul02i 9001 . . . . . . . . . 10  |-  ( 0  x.  9 )  =  0
132131oveq1i 5868 . . . . . . . . 9  |-  ( ( 0  x.  9 )  +  3 )  =  ( 0  +  3 )
13379addid2i 9000 . . . . . . . . 9  |-  ( 0  +  3 )  =  3
134132, 133eqtri 2303 . . . . . . . 8  |-  ( ( 0  x.  9 )  +  3 )  =  3
13510, 3, 3, 19, 116, 122, 10, 128, 134decma 10162 . . . . . . 7  |-  ( (; 9
0  x.  9 )  +  ( 1  +  2 ) )  = ;; 8 1 3
136 9t2e18 10219 . . . . . . . . 9  |-  ( 9  x.  2 )  = ; 1
8
137130, 37, 136mulcomli 8844 . . . . . . . 8  |-  ( 2  x.  9 )  = ; 1
8
138 1p1e2 9840 . . . . . . . 8  |-  ( 1  +  1 )  =  2
139 8p8e16 10185 . . . . . . . 8  |-  ( 8  +  8 )  = ; 1
6
14014, 113, 113, 137, 138, 48, 139decaddci 10169 . . . . . . 7  |-  ( ( 2  x.  9 )  +  8 )  = ; 2
6
14127, 18, 14, 113, 112, 119, 10, 48, 18, 135, 140decmac 10163 . . . . . 6  |-  ( (;; 9 0 2  x.  9 )  +  (; 1
8  +  0 ) )  = ;;; 8 1 3 6
14228nn0cni 9977 . . . . . . . . 9  |- ;; 9 0 2  e.  CC
143142mul01i 9002 . . . . . . . 8  |-  (;; 9 0 2  x.  0 )  =  0
144143oveq1i 5868 . . . . . . 7  |-  ( (;; 9 0 2  x.  0 )  +  0 )  =  ( 0  +  0 )
145144, 55, 393eqtri 2307 . . . . . 6  |-  ( (;; 9 0 2  x.  0 )  +  0 )  = ; 0 0
14610, 3, 114, 3, 116, 117, 28, 3, 3, 141, 145decma2c 10164 . . . . 5  |-  ( (;; 9 0 2  x. ; 9
0 )  + ;; 1 8 0 )  = ;;;; 8 1 3 6 0
147136oveq1i 5868 . . . . . . . . . 10  |-  ( ( 9  x.  2 )  +  0 )  =  (; 1 8  +  0 )
148147, 119eqtri 2303 . . . . . . . . 9  |-  ( ( 9  x.  2 )  +  0 )  = ; 1
8
14937mul02i 9001 . . . . . . . . . 10  |-  ( 0  x.  2 )  =  0
150149, 39eqtri 2303 . . . . . . . . 9  |-  ( 0  x.  2 )  = ; 0
0
15118, 10, 3, 116, 3, 3, 148, 150decmul1c 10171 . . . . . . . 8  |-  (; 9 0  x.  2 )  = ;; 1 8 0
152151oveq1i 5868 . . . . . . 7  |-  ( (; 9
0  x.  2 )  +  0 )  =  (;; 1 8 0  +  0 )
153115nn0cni 9977 . . . . . . . 8  |- ;; 1 8 0  e.  CC
154153addid1i 8999 . . . . . . 7  |-  (;; 1 8 0  +  0 )  = ;; 1 8 0
155152, 154eqtri 2303 . . . . . 6  |-  ( (; 9
0  x.  2 )  +  0 )  = ;; 1 8 0
15632, 108eqtri 2303 . . . . . 6  |-  ( 2  x.  2 )  = ; 0
4
15718, 27, 18, 112, 2, 3, 155, 156decmul1c 10171 . . . . 5  |-  (;; 9 0 2  x.  2 )  = ;;; 1 8 0 4
15828, 27, 18, 112, 2, 115, 146, 157decmul2c 10172 . . . 4  |-  (;; 9 0 2  x. ;; 9 0 2 )  = ;;;;; 8 1 3 6 0 4
159111, 158eqtr4i 2306 . . 3  |-  ( (;; 2 0 3  x.  N )  + ;;; 1 4 0 1 )  =  (;; 9 0 2  x. ;; 9 0 2 )
1608, 9, 24, 26, 28, 17, 29, 46, 159mod2xi 13084 . 2  |-  ( ( 2 ^;; 4 0 0 )  mod 
N )  =  (;;; 1 4 0 1  mod 
N )
16168oveq1i 5868 . . . . . . 7  |-  ( ( 2  x.  4 )  +  0 )  =  ( 8  +  0 )
162161, 72eqtri 2303 . . . . . 6  |-  ( ( 2  x.  4 )  +  0 )  =  8
16318, 2, 3, 57, 3, 3, 162, 40decmul2c 10172 . . . . 5  |-  ( 2  x. ; 4 0 )  = ; 8
0
164163oveq1i 5868 . . . 4  |-  ( ( 2  x. ; 4 0 )  +  0 )  =  (; 8
0  +  0 )
165113, 3deccl 10138 . . . . . 6  |- ; 8 0  e.  NN0
166165nn0cni 9977 . . . . 5  |- ; 8 0  e.  CC
167166addid1i 8999 . . . 4  |-  (; 8 0  +  0 )  = ; 8 0
168164, 167eqtri 2303 . . 3  |-  ( ( 2  x. ; 4 0 )  +  0 )  = ; 8 0
16918, 4, 3, 50, 3, 3, 168, 40decmul2c 10172 . 2  |-  ( 2  x. ;; 4 0 0 )  = ;; 8 0 0
170 eqid 2283 . . . 4  |- ;;; 2 3 1 1  = ;;; 2 3 1 1
17118, 113deccl 10138 . . . . 5  |- ; 2 8  e.  NN0
172 eqid 2283 . . . . . 6  |- ;; 2 3 1  = ;; 2 3 1
173 eqid 2283 . . . . . 6  |- ; 4 9  = ; 4 9
174 7nn0 9987 . . . . . . 7  |-  7  e.  NN0
175 7p1e8 9852 . . . . . . 7  |-  ( 7  +  1 )  =  8
176 eqid 2283 . . . . . . . 8  |- ; 2 3  = ; 2 3
177 4p3e7 9858 . . . . . . . . 9  |-  ( 4  +  3 )  =  7
17834, 79, 177addcomli 9004 . . . . . . . 8  |-  ( 3  +  4 )  =  7
17918, 19, 2, 176, 178decaddi 10168 . . . . . . 7  |-  (; 2 3  +  4 )  = ; 2 7
18018, 174, 175, 179decsuc 10147 . . . . . 6  |-  ( (; 2
3  +  4 )  +  1 )  = ; 2
8
181 9p1e10 9854 . . . . . . 7  |-  ( 9  +  1 )  =  10
182130, 61, 181addcomli 9004 . . . . . 6  |-  ( 1  +  9 )  =  10
18320, 14, 2, 10, 172, 173, 180, 182decaddc2 10167 . . . . 5  |-  (;; 2 3 1  + ; 4 9 )  = ;; 2 8 0
184171nn0cni 9977 . . . . . . 7  |- ; 2 8  e.  CC
185184addid1i 8999 . . . . . 6  |-  (; 2 8  +  0 )  = ; 2 8
186 eqid 2283 . . . . . . 7  |- ;; 4 9 0  = ;; 4 9 0
18718dec0h 10140 . . . . . . . 8  |-  2  = ; 0 2
18899, 187eqtri 2303 . . . . . . 7  |-  ( 2  +  0 )  = ; 0
2
189133oveq2i 5869 . . . . . . . . 9  |-  ( ( 4  x.  4 )  +  ( 0  +  3 ) )  =  ( ( 4  x.  4 )  +  3 )
190 4t4e16 10197 . . . . . . . . . 10  |-  ( 4  x.  4 )  = ; 1
6
191 6p3e9 9865 . . . . . . . . . 10  |-  ( 6  +  3 )  =  9
19214, 48, 19, 190, 191decaddi 10168 . . . . . . . . 9  |-  ( ( 4  x.  4 )  +  3 )  = ; 1
9
193189, 192eqtri 2303 . . . . . . . 8  |-  ( ( 4  x.  4 )  +  ( 0  +  3 ) )  = ; 1
9
194 9t4e36 10221 . . . . . . . . . 10  |-  ( 9  x.  4 )  = ; 3
6
195194oveq1i 5868 . . . . . . . . 9  |-  ( ( 9  x.  4 )  +  0 )  =  (; 3 6  +  0 )
19619, 48deccl 10138 . . . . . . . . . . 11  |- ; 3 6  e.  NN0
197196nn0cni 9977 . . . . . . . . . 10  |- ; 3 6  e.  CC
198197addid1i 8999 . . . . . . . . 9  |-  (; 3 6  +  0 )  = ; 3 6
199195, 198eqtri 2303 . . . . . . . 8  |-  ( ( 9  x.  4 )  +  0 )  = ; 3
6
2002, 10, 3, 3, 173, 96, 2, 48, 19, 193, 199decmac 10163 . . . . . . 7  |-  ( (; 4
9  x.  4 )  +  ( 0  +  0 ) )  = ;; 1 9 6
20174oveq1i 5868 . . . . . . . 8  |-  ( ( 0  x.  4 )  +  2 )  =  ( 0  +  2 )
20237addid2i 9000 . . . . . . . 8  |-  ( 0  +  2 )  =  2
203201, 202, 1873eqtri 2307 . . . . . . 7  |-  ( ( 0  x.  4 )  +  2 )  = ; 0
2
20411, 3, 3, 18, 186, 188, 2, 18, 3, 200, 203decmac 10163 . . . . . 6  |-  ( (;; 4 9 0  x.  4 )  +  ( 2  +  0 ) )  = ;;; 1 9 6 2
20512nn0cni 9977 . . . . . . . . 9  |- ;; 4 9 0  e.  CC
206205mul01i 9002 . . . . . . . 8  |-  (;; 4 9 0  x.  0 )  =  0
207206oveq1i 5868 . . . . . . 7  |-  ( (;; 4 9 0  x.  0 )  +  8 )  =  ( 0  +  8 )
20871addid2i 9000 . . . . . . 7  |-  ( 0  +  8 )  =  8
209113dec0h 10140 . . . . . . 7  |-  8  = ; 0 8
210207, 208, 2093eqtri 2307 . . . . . 6  |-  ( (;; 4 9 0  x.  0 )  +  8 )  = ; 0 8
2112, 3, 18, 113, 57, 185, 12, 113, 3, 204, 210decma2c 10164 . . . . 5  |-  ( (;; 4 9 0  x. ; 4
0 )  +  (; 2
8  +  0 ) )  = ;;;; 1 9 6 2 8
212206oveq1i 5868 . . . . . 6  |-  ( (;; 4 9 0  x.  0 )  +  0 )  =  ( 0  +  0 )
213212, 55, 393eqtri 2307 . . . . 5  |-  ( (;; 4 9 0  x.  0 )  +  0 )  = ; 0 0
2144, 3, 171, 3, 50, 183, 12, 3, 3, 211, 213decma2c 10164 . . . 4  |-  ( (;; 4 9 0  x. ;; 4 0 0 )  +  (;; 2 3 1  + ; 4 9 ) )  = ;;;;; 1 9 6 2 8 0
21534mulid1i 8839 . . . . . . . 8  |-  ( 4  x.  1 )  =  4
216215, 55oveq12i 5870 . . . . . . 7  |-  ( ( 4  x.  1 )  +  ( 0  +  0 ) )  =  ( 4  +  0 )
217216, 35eqtri 2303 . . . . . 6  |-  ( ( 4  x.  1 )  +  ( 0  +  0 ) )  =  4
218130mulid1i 8839 . . . . . . . 8  |-  ( 9  x.  1 )  =  9
219218oveq1i 5868 . . . . . . 7  |-  ( ( 9  x.  1 )  +  0 )  =  ( 9  +  0 )
220130addid1i 8999 . . . . . . 7  |-  ( 9  +  0 )  =  9
22110dec0h 10140 . . . . . . 7  |-  9  = ; 0 9
222219, 220, 2213eqtri 2307 . . . . . 6  |-  ( ( 9  x.  1 )  +  0 )  = ; 0
9
2232, 10, 3, 3, 173, 96, 14, 10, 3, 217, 222decmac 10163 . . . . 5  |-  ( (; 4
9  x.  1 )  +  ( 0  +  0 ) )  = ; 4
9
224101oveq1i 5868 . . . . . 6  |-  ( ( 0  x.  1 )  +  1 )  =  ( 0  +  1 )
225224, 65, 633eqtri 2307 . . . . 5  |-  ( ( 0  x.  1 )  +  1 )  = ; 0
1
22611, 3, 3, 14, 186, 63, 14, 14, 3, 223, 225decmac 10163 . . . 4  |-  ( (;; 4 9 0  x.  1 )  +  1 )  = ;; 4 9 1
2275, 14, 21, 14, 1, 170, 12, 14, 11, 214, 226decma2c 10164 . . 3  |-  ( (;; 4 9 0  x.  N )  + ;;; 2 3 1 1 )  = ;;;;;; 1 9 6 2 8 0 1
22815nn0cni 9977 . . . . . . 7  |- ; 1 4  e.  CC
229228addid1i 8999 . . . . . 6  |-  (; 1 4  +  0 )  = ; 1 4
230 5nn0 9985 . . . . . . . 8  |-  5  e.  NN0
231230, 48deccl 10138 . . . . . . 7  |- ; 5 6  e.  NN0
232231, 3deccl 10138 . . . . . 6  |- ;; 5 6 0  e.  NN0
233 eqid 2283 . . . . . . . 8  |- ;; 5 6 0  = ;; 5 6 0
234231nn0cni 9977 . . . . . . . . 9  |- ; 5 6  e.  CC
235234addid2i 9000 . . . . . . . 8  |-  ( 0  + ; 5 6 )  = ; 5
6
2363, 14, 231, 3, 63, 233, 235, 62decadd 10165 . . . . . . 7  |-  ( 1  + ;; 5 6 0 )  = ;; 5 6 1
237234addid1i 8999 . . . . . . . 8  |-  (; 5 6  +  0 )  = ; 5 6
238 5nn 9880 . . . . . . . . . . . 12  |-  5  e.  NN
239238nncni 9756 . . . . . . . . . . 11  |-  5  e.  CC
240239addid1i 8999 . . . . . . . . . 10  |-  ( 5  +  0 )  =  5
241230dec0h 10140 . . . . . . . . . 10  |-  5  = ; 0 5
242240, 241eqtri 2303 . . . . . . . . 9  |-  ( 5  +  0 )  = ; 0
5
24361mulid1i 8839 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
244243, 55oveq12i 5870 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
245244, 62eqtri 2303 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
246215oveq1i 5868 . . . . . . . . . 10  |-  ( ( 4  x.  1 )  +  5 )  =  ( 4  +  5 )
247 5p4e9 9862 . . . . . . . . . . 11  |-  ( 5  +  4 )  =  9
248239, 34, 247addcomli 9004 . . . . . . . . . 10  |-  ( 4  +  5 )  =  9
249246, 248, 2213eqtri 2307 . . . . . . . . 9  |-  ( ( 4  x.  1 )  +  5 )  = ; 0
9
25014, 2, 3, 230, 52, 242, 14, 10, 3, 245, 249decmac 10163 . . . . . . . 8  |-  ( (; 1
4  x.  1 )  +  ( 5  +  0 ) )  = ; 1
9
251101oveq1i 5868 . . . . . . . . 9  |-  ( ( 0  x.  1 )  +  6 )  =  ( 0  +  6 )
252251, 89, 903eqtri 2307 . . . . . . . 8  |-  ( ( 0  x.  1 )  +  6 )  = ; 0
6
25315, 3, 230, 48, 51, 237, 14, 48, 3, 250, 252decmac 10163 . . . . . . 7  |-  ( (;; 1 4 0  x.  1 )  +  (; 5
6  +  0 ) )  = ;; 1 9 6
254243oveq1i 5868 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  1 )  =  ( 1  +  1 )
255254, 138, 1873eqtri 2307 . . . . . . 7  |-  ( ( 1  x.  1 )  +  1 )  = ; 0
2
25616, 14, 231, 14, 47, 236, 14, 18, 3, 253, 255decmac 10163 . . . . . 6  |-  ( (;;; 1 4 0 1  x.  1 )  +  ( 1  + ;; 5 6 0 ) )  = ;;; 1 9 6 2
25734mulid2i 8840 . . . . . . . . . . 11  |-  ( 1  x.  4 )  =  4
258257, 65oveq12i 5870 . . . . . . . . . 10  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
259 4p1e5 9849 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
260258, 259eqtri 2303 . . . . . . . . 9  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  5
261190oveq1i 5868 . . . . . . . . . 10  |-  ( ( 4  x.  4 )  +  0 )  =  (; 1 6  +  0 )
262261, 59eqtri 2303 . . . . . . . . 9  |-  ( ( 4  x.  4 )  +  0 )  = ; 1
6
26314, 2, 3, 3, 52, 96, 2, 48, 14, 260, 262decmac 10163 . . . . . . . 8  |-  ( (; 1
4  x.  4 )  +  ( 0  +  0 ) )  = ; 5
6
26474oveq1i 5868 . . . . . . . . 9  |-  ( ( 0  x.  4 )  +  0 )  =  ( 0  +  0 )
265264, 55, 393eqtri 2307 . . . . . . . 8  |-  ( ( 0  x.  4 )  +  0 )  = ; 0
0
26615, 3, 3, 3, 51, 96, 2, 3, 3, 263, 265decmac 10163 . . . . . . 7  |-  ( (;; 1 4 0  x.  4 )  +  ( 0  +  0 ) )  = ;; 5 6 0
267257oveq1i 5868 . . . . . . . 8  |-  ( ( 1  x.  4 )  +  4 )  =  ( 4  +  4 )
268 4p4e8 9859 . . . . . . . 8  |-  ( 4  +  4 )  =  8
269267, 268, 2093eqtri 2307 . . . . . . 7  |-  ( ( 1  x.  4 )  +  4 )  = ; 0
8
27016, 14, 3, 2, 47, 108, 2, 113, 3, 266, 269decmac 10163 . . . . . 6  |-  ( (;;; 1 4 0 1  x.  4 )  +  4 )  = ;;; 5 6 0 8
27114, 2, 14, 2, 52, 229, 17, 113, 232, 256, 270decma2c 10164 . . . . 5  |-  ( (;;; 1 4 0 1  x. ; 1
4 )  +  (; 1
4  +  0 ) )  = ;;;; 1 9 6 2 8
27217nn0cni 9977 . . . . . . . 8  |- ;;; 1 4 0 1  e.  CC
273272mul01i 9002 . . . . . . 7  |-  (;;; 1 4 0 1  x.  0 )  =  0
274273oveq1i 5868 . . . . . 6  |-  ( (;;; 1 4 0 1  x.  0 )  +  0 )  =  ( 0  +  0 )
275274, 55, 393eqtri 2307 . . . . 5  |-  ( (;;; 1 4 0 1  x.  0 )  +  0 )  = ; 0 0
27615, 3, 15, 3, 51, 51, 17, 3, 3, 271, 275decma2c 10164 . . . 4  |-  ( (;;; 1 4 0 1  x. ;; 1 4 0 )  + ;; 1 4 0 )  = ;;;;; 1 9 6 2 8 0
277272mulid1i 8839 . . . 4  |-  (;;; 1 4 0 1  x.  1 )  = ;;; 1 4 0 1
27817, 16, 14, 47, 14, 16, 276, 277decmul2c 10172 . . 3  |-  (;;; 1 4 0 1  x. ;;; 1 4 0 1 )  = ;;;;;; 1 9 6 2 8 0 1
279227, 278eqtr4i 2306 . 2  |-  ( (;; 4 9 0  x.  N )  + ;;; 2 3 1 1 )  =  (;;; 1 4 0 1  x. ;;; 1 4 0 1 )
2808, 9, 5, 13, 17, 22, 160, 169, 279mod2xi 13084 1  |-  ( ( 2 ^;; 8 0 0 )  mod 
N )  =  (;;; 2 3 1 1  mod 
N )
Colors of variables: wff set class
Syntax hints:    = wceq 1623  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   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:  4001lem3  13141  4001lem4  13142
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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