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Theorem 4001lem1 13462
Description: Lemma for 4001prm 13466. Calculate a power mod. In decimal, we calculate  2 ^ 1 2  =  4 0 9 6  =  N  +  9 5,  2 ^ 2 4  =  ( 2 ^ 1 2 ) ^ 2  ==  9
5 ^ 2  =  2 N  +  1 0 2 3,  2 ^ 2 5  =  2 ^ 2 4  x.  2  ==  1 0 2 3  x.  2  =  2 0 4 6,  2 ^ 5 0  =  ( 2 ^ 2 5 ) ^ 2  ==  2
0 4 6 ^ 2  =  1 0 4 6 N  + 
1 0 7 0,  2 ^ 1 0 0  =  ( 2 ^ 5 0 ) ^ 2  ==  1 0 7 0 ^ 2  =  2 8 6 N  + 
6 1 4 and  2 ^ 2 0 0  =  ( 2 ^ 1 0 0 ) ^ 2  ==  6 1 4 ^ 2  =  9 4 N  +  9 0 2  ==  9 0 2. (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
4001lem1  |-  ( ( 2 ^;; 2 0 0 )  mod 
N )  =  (;; 9 0 2  mod 
N )

Proof of Theorem 4001lem1
StepHypRef Expression
1 4001prm.1 . . 3  |-  N  = ;;; 4 0 0 1
2 4nn0 10242 . . . . . 6  |-  4  e.  NN0
3 0nn0 10238 . . . . . 6  |-  0  e.  NN0
42, 3deccl 10398 . . . . 5  |- ; 4 0  e.  NN0
54, 3deccl 10398 . . . 4  |- ;; 4 0 0  e.  NN0
6 1nn 10013 . . . 4  |-  1  e.  NN
75, 6decnncl 10397 . . 3  |- ;;; 4 0 0 1  e.  NN
81, 7eqeltri 2508 . 2  |-  N  e.  NN
9 2nn 10135 . 2  |-  2  e.  NN
10 10nn0 10248 . . 3  |-  10  e.  NN0
1110, 3deccl 10398 . 2  |- ; 10 0  e.  NN0
12 9nn0 10247 . . . 4  |-  9  e.  NN0
1312, 2deccl 10398 . . 3  |- ; 9 4  e.  NN0
1413nn0zi 10308 . 2  |- ; 9 4  e.  ZZ
15 6nn0 10244 . . . 4  |-  6  e.  NN0
16 1nn0 10239 . . . 4  |-  1  e.  NN0
1715, 16deccl 10398 . . 3  |- ; 6 1  e.  NN0
1817, 2deccl 10398 . 2  |- ;; 6 1 4  e.  NN0
1912, 3deccl 10398 . . 3  |- ; 9 0  e.  NN0
20 2nn0 10240 . . 3  |-  2  e.  NN0
2119, 20deccl 10398 . 2  |- ;; 9 0 2  e.  NN0
22 5nn0 10243 . . . 4  |-  5  e.  NN0
2322, 3deccl 10398 . . 3  |- ; 5 0  e.  NN0
24 8nn0 10246 . . . . . 6  |-  8  e.  NN0
2520, 24deccl 10398 . . . . 5  |- ; 2 8  e.  NN0
2625, 15deccl 10398 . . . 4  |- ;; 2 8 6  e.  NN0
2726nn0zi 10308 . . 3  |- ;; 2 8 6  e.  ZZ
28 7nn0 10245 . . . . 5  |-  7  e.  NN0
2910, 28deccl 10398 . . . 4  |- ; 10 7  e.  NN0
3029, 3deccl 10398 . . 3  |- ;; 10 7 0  e.  NN0
3120, 22deccl 10398 . . . 4  |- ; 2 5  e.  NN0
3210, 2deccl 10398 . . . . . 6  |- ; 10 4  e.  NN0
3332, 15deccl 10398 . . . . 5  |- ;; 10 4 6  e.  NN0
3433nn0zi 10308 . . . 4  |- ;; 10 4 6  e.  ZZ
3520, 3deccl 10398 . . . . . 6  |- ; 2 0  e.  NN0
3635, 2deccl 10398 . . . . 5  |- ;; 2 0 4  e.  NN0
3736, 15deccl 10398 . . . 4  |- ;;; 2 0 4 6  e.  NN0
3820, 2deccl 10398 . . . . 5  |- ; 2 4  e.  NN0
39 0z 10295 . . . . 5  |-  0  e.  ZZ
4010, 20deccl 10398 . . . . . 6  |- ; 10 2  e.  NN0
41 3nn0 10241 . . . . . 6  |-  3  e.  NN0
4240, 41deccl 10398 . . . . 5  |- ;; 10 2 3  e.  NN0
4316, 20deccl 10398 . . . . . 6  |- ; 1 2  e.  NN0
44 2z 10314 . . . . . 6  |-  2  e.  ZZ
4512, 22deccl 10398 . . . . . 6  |- ; 9 5  e.  NN0
46 1z 10313 . . . . . . 7  |-  1  e.  ZZ
4715, 2deccl 10398 . . . . . . 7  |- ; 6 4  e.  NN0
48 2exp6 13424 . . . . . . . 8  |-  ( 2 ^ 6 )  = ; 6
4
4948oveq1i 6093 . . . . . . 7  |-  ( ( 2 ^ 6 )  mod  N )  =  (; 6 4  mod  N
)
50 6nn 10139 . . . . . . . . 9  |-  6  e.  NN
5150nncni 10012 . . . . . . . 8  |-  6  e.  CC
52 2cn 10072 . . . . . . . 8  |-  2  e.  CC
53 6t2e12 10461 . . . . . . . 8  |-  ( 6  x.  2 )  = ; 1
2
5451, 52, 53mulcomli 9099 . . . . . . 7  |-  ( 2  x.  6 )  = ; 1
2
55 eqid 2438 . . . . . . . . 9  |- ; 9 5  = ; 9 5
56 eqid 2438 . . . . . . . . . 10  |- ;; 4 0 0  = ;; 4 0 0
57 9nn 10142 . . . . . . . . . . . . 13  |-  9  e.  NN
5857nncni 10012 . . . . . . . . . . . 12  |-  9  e.  CC
5958addid1i 9255 . . . . . . . . . . 11  |-  ( 9  +  0 )  =  9
6012dec0h 10400 . . . . . . . . . . 11  |-  9  = ; 0 9
6159, 60eqtri 2458 . . . . . . . . . 10  |-  ( 9  +  0 )  = ; 0
9
62 eqid 2438 . . . . . . . . . . 11  |- ; 4 0  = ; 4 0
63 00id 9243 . . . . . . . . . . . 12  |-  ( 0  +  0 )  =  0
643dec0h 10400 . . . . . . . . . . . 12  |-  0  = ; 0 0
6563, 64eqtri 2458 . . . . . . . . . . 11  |-  ( 0  +  0 )  = ; 0
0
66 4cn 10076 . . . . . . . . . . . . . 14  |-  4  e.  CC
6766mulid2i 9095 . . . . . . . . . . . . 13  |-  ( 1  x.  4 )  =  4
6867, 63oveq12i 6095 . . . . . . . . . . . 12  |-  ( ( 1  x.  4 )  +  ( 0  +  0 ) )  =  ( 4  +  0 )
6966addid1i 9255 . . . . . . . . . . . 12  |-  ( 4  +  0 )  =  4
7068, 69eqtri 2458 . . . . . . . . . . 11  |-  ( ( 1  x.  4 )  +  ( 0  +  0 ) )  =  4
71 ax-1cn 9050 . . . . . . . . . . . . . 14  |-  1  e.  CC
7271mul01i 9258 . . . . . . . . . . . . 13  |-  ( 1  x.  0 )  =  0
7372oveq1i 6093 . . . . . . . . . . . 12  |-  ( ( 1  x.  0 )  +  0 )  =  ( 0  +  0 )
7473, 65eqtri 2458 . . . . . . . . . . 11  |-  ( ( 1  x.  0 )  +  0 )  = ; 0
0
752, 3, 3, 3, 62, 65, 16, 3, 3, 70, 74decma2c 10424 . . . . . . . . . 10  |-  ( ( 1  x. ; 4 0 )  +  ( 0  +  0 ) )  = ; 4 0
7672oveq1i 6093 . . . . . . . . . . 11  |-  ( ( 1  x.  0 )  +  9 )  =  ( 0  +  9 )
7758addid2i 9256 . . . . . . . . . . . 12  |-  ( 0  +  9 )  =  9
7877, 60eqtri 2458 . . . . . . . . . . 11  |-  ( 0  +  9 )  = ; 0
9
7976, 78eqtri 2458 . . . . . . . . . 10  |-  ( ( 1  x.  0 )  +  9 )  = ; 0
9
804, 3, 3, 12, 56, 61, 16, 12, 3, 75, 79decma2c 10424 . . . . . . . . 9  |-  ( ( 1  x. ;; 4 0 0 )  +  ( 9  +  0 ) )  = ;; 4 0 9
8171mulid1i 9094 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
8281oveq1i 6093 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  5 )  =  ( 1  +  5 )
83 5nn 10138 . . . . . . . . . . . . 13  |-  5  e.  NN
8483nncni 10012 . . . . . . . . . . . 12  |-  5  e.  CC
85 5p1e6 10108 . . . . . . . . . . . 12  |-  ( 5  +  1 )  =  6
8684, 71, 85addcomli 9260 . . . . . . . . . . 11  |-  ( 1  +  5 )  =  6
8715dec0h 10400 . . . . . . . . . . 11  |-  6  = ; 0 6
8886, 87eqtri 2458 . . . . . . . . . 10  |-  ( 1  +  5 )  = ; 0
6
8982, 88eqtri 2458 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  5 )  = ; 0
6
905, 16, 12, 22, 1, 55, 16, 15, 3, 80, 89decma2c 10424 . . . . . . . 8  |-  ( ( 1  x.  N )  + ; 9 5 )  = ;;; 4 0 9 6
91 eqid 2438 . . . . . . . . 9  |- ; 6 4  = ; 6 4
92 eqid 2438 . . . . . . . . . 10  |- ; 2 5  = ; 2 5
93 2p2e4 10100 . . . . . . . . . . . 12  |-  ( 2  +  2 )  =  4
9493oveq2i 6094 . . . . . . . . . . 11  |-  ( ( 6  x.  6 )  +  ( 2  +  2 ) )  =  ( ( 6  x.  6 )  +  4 )
95 6t6e36 10465 . . . . . . . . . . . 12  |-  ( 6  x.  6 )  = ; 3
6
96 3p1e4 10106 . . . . . . . . . . . 12  |-  ( 3  +  1 )  =  4
97 6p4e10 10124 . . . . . . . . . . . 12  |-  ( 6  +  4 )  =  10
9841, 15, 2, 95, 96, 97decaddci2 10430 . . . . . . . . . . 11  |-  ( ( 6  x.  6 )  +  4 )  = ; 4
0
9994, 98eqtri 2458 . . . . . . . . . 10  |-  ( ( 6  x.  6 )  +  ( 2  +  2 ) )  = ; 4
0
100 6t4e24 10463 . . . . . . . . . . . 12  |-  ( 6  x.  4 )  = ; 2
4
10151, 66, 100mulcomli 9099 . . . . . . . . . . 11  |-  ( 4  x.  6 )  = ; 2
4
102 5p4e9 10120 . . . . . . . . . . . 12  |-  ( 5  +  4 )  =  9
10384, 66, 102addcomli 9260 . . . . . . . . . . 11  |-  ( 4  +  5 )  =  9
10420, 2, 22, 101, 103decaddi 10428 . . . . . . . . . 10  |-  ( ( 4  x.  6 )  +  5 )  = ; 2
9
10515, 2, 20, 22, 91, 92, 15, 12, 20, 99, 104decmac 10423 . . . . . . . . 9  |-  ( (; 6
4  x.  6 )  + ; 2 5 )  = ;; 4 0 9
106 4p1e5 10107 . . . . . . . . . . 11  |-  ( 4  +  1 )  =  5
10720, 2, 106, 100decsuc 10407 . . . . . . . . . 10  |-  ( ( 6  x.  4 )  +  1 )  = ; 2
5
108 4t4e16 10457 . . . . . . . . . 10  |-  ( 4  x.  4 )  = ; 1
6
1092, 15, 2, 91, 15, 16, 107, 108decmul1c 10431 . . . . . . . . 9  |-  (; 6 4  x.  4 )  = ;; 2 5 6
11047, 15, 2, 91, 15, 31, 105, 109decmul2c 10432 . . . . . . . 8  |-  (; 6 4  x. ; 6 4 )  = ;;; 4 0 9 6
11190, 110eqtr4i 2461 . . . . . . 7  |-  ( ( 1  x.  N )  + ; 9 5 )  =  (; 6 4  x. ; 6 4 )
1128, 9, 15, 46, 47, 45, 49, 54, 111mod2xi 13407 . . . . . 6  |-  ( ( 2 ^; 1 2 )  mod 
N )  =  (; 9
5  mod  N )
113 eqid 2438 . . . . . . 7  |- ; 1 2  = ; 1 2
11452mulid1i 9094 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
115114oveq1i 6093 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  0 )  =  ( 2  +  0 )
11652addid1i 9255 . . . . . . . 8  |-  ( 2  +  0 )  =  2
117115, 116eqtri 2458 . . . . . . 7  |-  ( ( 2  x.  1 )  +  0 )  =  2
118 2t2e4 10129 . . . . . . . 8  |-  ( 2  x.  2 )  =  4
1192dec0h 10400 . . . . . . . 8  |-  4  = ; 0 4
120118, 119eqtri 2458 . . . . . . 7  |-  ( 2  x.  2 )  = ; 0
4
12120, 16, 20, 113, 2, 3, 117, 120decmul2c 10432 . . . . . 6  |-  ( 2  x. ; 1 2 )  = ; 2
4
122 eqid 2438 . . . . . . . 8  |- ;; 10 2 3  = ;; 10 2 3
12340nn0cni 10235 . . . . . . . . . 10  |- ; 10 2  e.  CC
124123addid1i 9255 . . . . . . . . 9  |-  (; 10 2  +  0 )  = ; 10 2
125 dec10p 10413 . . . . . . . . . 10  |-  ( 10  +  0 )  = ; 1
0
126 4t2e8 10132 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
12766, 52, 126mulcomli 9099 . . . . . . . . . . . 12  |-  ( 2  x.  4 )  =  8
12871addid1i 9255 . . . . . . . . . . . 12  |-  ( 1  +  0 )  =  1
129127, 128oveq12i 6095 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  ( 1  +  0 ) )  =  ( 8  +  1 )
130 8p1e9 10111 . . . . . . . . . . 11  |-  ( 8  +  1 )  =  9
131129, 130eqtri 2458 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  ( 1  +  0 ) )  =  9
13252mul01i 9258 . . . . . . . . . . . 12  |-  ( 2  x.  0 )  =  0
133132oveq1i 6093 . . . . . . . . . . 11  |-  ( ( 2  x.  0 )  +  0 )  =  ( 0  +  0 )
134133, 65eqtri 2458 . . . . . . . . . 10  |-  ( ( 2  x.  0 )  +  0 )  = ; 0
0
1352, 3, 16, 3, 62, 125, 20, 3, 3, 131, 134decma2c 10424 . . . . . . . . 9  |-  ( ( 2  x. ; 4 0 )  +  ( 10  +  0 ) )  = ; 9 0
136132oveq1i 6093 . . . . . . . . . 10  |-  ( ( 2  x.  0 )  +  2 )  =  ( 0  +  2 )
13752addid2i 9256 . . . . . . . . . . 11  |-  ( 0  +  2 )  =  2
13820dec0h 10400 . . . . . . . . . . 11  |-  2  = ; 0 2
139137, 138eqtri 2458 . . . . . . . . . 10  |-  ( 0  +  2 )  = ; 0
2
140136, 139eqtri 2458 . . . . . . . . 9  |-  ( ( 2  x.  0 )  +  2 )  = ; 0
2
1414, 3, 10, 20, 56, 124, 20, 20, 3, 135, 140decma2c 10424 . . . . . . . 8  |-  ( ( 2  x. ;; 4 0 0 )  +  (; 10 2  +  0
) )  = ;; 9 0 2
142114oveq1i 6093 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  3 )  =  ( 2  +  3 )
143 3cn 10074 . . . . . . . . . . 11  |-  3  e.  CC
144 3p2e5 10113 . . . . . . . . . . 11  |-  ( 3  +  2 )  =  5
145143, 52, 144addcomli 9260 . . . . . . . . . 10  |-  ( 2  +  3 )  =  5
14622dec0h 10400 . . . . . . . . . 10  |-  5  = ; 0 5
147145, 146eqtri 2458 . . . . . . . . 9  |-  ( 2  +  3 )  = ; 0
5
148142, 147eqtri 2458 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  3 )  = ; 0
5
1495, 16, 40, 41, 1, 122, 20, 22, 3, 141, 148decma2c 10424 . . . . . . 7  |-  ( ( 2  x.  N )  + ;; 10 2 3 )  = ;;; 9 0 2 5
1502, 28deccl 10398 . . . . . . . 8  |- ; 4 7  e.  NN0
151 eqid 2438 . . . . . . . . 9  |- ; 4 7  = ; 4 7
152103oveq2i 6094 . . . . . . . . . 10  |-  ( ( 9  x.  9 )  +  ( 4  +  5 ) )  =  ( ( 9  x.  9 )  +  9 )
153 9t9e81 10486 . . . . . . . . . . 11  |-  ( 9  x.  9 )  = ; 8
1
154 9p1e10 10112 . . . . . . . . . . . 12  |-  ( 9  +  1 )  =  10
15558, 71, 154addcomli 9260 . . . . . . . . . . 11  |-  ( 1  +  9 )  =  10
15624, 16, 12, 153, 130, 155decaddci2 10430 . . . . . . . . . 10  |-  ( ( 9  x.  9 )  +  9 )  = ; 9
0
157152, 156eqtri 2458 . . . . . . . . 9  |-  ( ( 9  x.  9 )  +  ( 4  +  5 ) )  = ; 9
0
158 9t5e45 10482 . . . . . . . . . . 11  |-  ( 9  x.  5 )  = ; 4
5
15958, 84, 158mulcomli 9099 . . . . . . . . . 10  |-  ( 5  x.  9 )  = ; 4
5
160 7nn 10140 . . . . . . . . . . . 12  |-  7  e.  NN
161160nncni 10012 . . . . . . . . . . 11  |-  7  e.  CC
162 7p5e12 10437 . . . . . . . . . . 11  |-  ( 7  +  5 )  = ; 1
2
163161, 84, 162addcomli 9260 . . . . . . . . . 10  |-  ( 5  +  7 )  = ; 1
2
1642, 22, 28, 159, 106, 20, 163decaddci 10429 . . . . . . . . 9  |-  ( ( 5  x.  9 )  +  7 )  = ; 5
2
16512, 22, 2, 28, 55, 151, 12, 20, 22, 157, 164decmac 10423 . . . . . . . 8  |-  ( (; 9
5  x.  9 )  + ; 4 7 )  = ;; 9 0 2
166 5p2e7 10118 . . . . . . . . . 10  |-  ( 5  +  2 )  =  7
1672, 22, 20, 158, 166decaddi 10428 . . . . . . . . 9  |-  ( ( 9  x.  5 )  +  2 )  = ; 4
7
168 5t5e25 10460 . . . . . . . . 9  |-  ( 5  x.  5 )  = ; 2
5
16922, 12, 22, 55, 22, 20, 167, 168decmul1c 10431 . . . . . . . 8  |-  (; 9 5  x.  5 )  = ;; 4 7 5
17045, 12, 22, 55, 22, 150, 165, 169decmul2c 10432 . . . . . . 7  |-  (; 9 5  x. ; 9 5 )  = ;;; 9 0 2 5
171149, 170eqtr4i 2461 . . . . . 6  |-  ( ( 2  x.  N )  + ;; 10 2 3 )  =  (; 9
5  x. ; 9 5 )
1728, 9, 43, 44, 45, 42, 112, 121, 171mod2xi 13407 . . . . 5  |-  ( ( 2 ^; 2 4 )  mod 
N )  =  (;; 10 2 3  mod 
N )
173 eqid 2438 . . . . . 6  |- ; 2 4  = ; 2 4
17420, 2, 106, 173decsuc 10407 . . . . 5  |-  (; 2 4  +  1 )  = ; 2 5
17537nn0cni 10235 . . . . . . 7  |- ;;; 2 0 4 6  e.  CC
176175addid2i 9256 . . . . . 6  |-  ( 0  + ;;; 2 0 4 6 )  = ;;; 2 0 4 6
1778nncni 10012 . . . . . . . 8  |-  N  e.  CC
178177mul02i 9257 . . . . . . 7  |-  ( 0  x.  N )  =  0
179178oveq1i 6093 . . . . . 6  |-  ( ( 0  x.  N )  + ;;; 2 0 4 6 )  =  ( 0  + ;;; 2 0 4 6 )
180 eqid 2438 . . . . . . . . . 10  |- ; 10 2  = ; 10 2
18120dec0u 10399 . . . . . . . . . . . 12  |-  ( 10  x.  2 )  = ; 2
0
182181oveq1i 6093 . . . . . . . . . . 11  |-  ( ( 10  x.  2 )  +  0 )  =  (; 2 0  +  0 )
18335nn0cni 10235 . . . . . . . . . . . 12  |- ; 2 0  e.  CC
184183addid1i 9255 . . . . . . . . . . 11  |-  (; 2 0  +  0 )  = ; 2 0
185182, 184eqtri 2458 . . . . . . . . . 10  |-  ( ( 10  x.  2 )  +  0 )  = ; 2
0
18620, 10, 20, 180, 2, 3, 185, 120decmul1c 10431 . . . . . . . . 9  |-  (; 10 2  x.  2 )  = ;; 2 0 4
187186oveq1i 6093 . . . . . . . 8  |-  ( (; 10 2  x.  2 )  +  0 )  =  (;; 2 0 4  +  0 )
18836nn0cni 10235 . . . . . . . . 9  |- ;; 2 0 4  e.  CC
189188addid1i 9255 . . . . . . . 8  |-  (;; 2 0 4  +  0 )  = ;; 2 0 4
190187, 189eqtri 2458 . . . . . . 7  |-  ( (; 10 2  x.  2 )  +  0 )  = ;; 2 0 4
191 3t2e6 10130 . . . . . . . 8  |-  ( 3  x.  2 )  =  6
192191, 87eqtri 2458 . . . . . . 7  |-  ( 3  x.  2 )  = ; 0
6
19320, 40, 41, 122, 15, 3, 190, 192decmul1c 10431 . . . . . 6  |-  (;; 10 2 3  x.  2 )  = ;;; 2 0 4 6
194176, 179, 1933eqtr4i 2468 . . . . 5  |-  ( ( 0  x.  N )  + ;;; 2 0 4 6 )  =  (;; 10 2 3  x.  2 )
1958, 9, 38, 39, 42, 37, 172, 174, 194modxp1i 13408 . . . 4  |-  ( ( 2 ^; 2 5 )  mod 
N )  =  (;;; 2 0 4 6  mod 
N )
196118oveq1i 6093 . . . . . 6  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
197196, 106eqtri 2458 . . . . 5  |-  ( ( 2  x.  2 )  +  1 )  =  5
198 5t2e10 10133 . . . . . . 7  |-  ( 5  x.  2 )  =  10
19984, 52, 198mulcomli 9099 . . . . . 6  |-  ( 2  x.  5 )  =  10
200 dec10 10414 . . . . . 6  |-  10  = ; 1 0
201199, 200eqtri 2458 . . . . 5  |-  ( 2  x.  5 )  = ; 1
0
20220, 20, 22, 92, 3, 16, 197, 201decmul2c 10432 . . . 4  |-  ( 2  x. ; 2 5 )  = ; 5
0
203 eqid 2438 . . . . . 6  |- ;; 10 7 0  = ;; 10 7 0
20420, 16deccl 10398 . . . . . . 7  |- ; 2 1  e.  NN0
205 eqid 2438 . . . . . . . 8  |- ; 10 7  = ; 10 7
206 eqid 2438 . . . . . . . 8  |- ; 10 4  = ; 10 4
207 0p1e1 10095 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
208 10p10e20 10454 . . . . . . . . 9  |-  ( 10  +  10 )  = ; 2
0
20920, 3, 207, 208decsuc 10407 . . . . . . . 8  |-  ( ( 10  +  10 )  +  1 )  = ; 2
1
210 7p4e11 10436 . . . . . . . 8  |-  ( 7  +  4 )  = ; 1
1
21110, 28, 10, 2, 205, 206, 209, 16, 210decaddc 10426 . . . . . . 7  |-  (; 10 7  + ; 10 4 )  = ;; 2 1 1
212204nn0cni 10235 . . . . . . . . 9  |- ; 2 1  e.  CC
213212addid1i 9255 . . . . . . . 8  |-  (; 2 1  +  0 )  = ; 2 1
214 eqid 2438 . . . . . . . . 9  |- ;; 10 4 6  = ;; 10 4 6
215116, 138eqtri 2458 . . . . . . . . 9  |-  ( 2  +  0 )  = ; 0
2
21616dec0h 10400 . . . . . . . . . . . 12  |-  1  = ; 0 1
217207, 216eqtri 2458 . . . . . . . . . . 11  |-  ( 0  +  1 )  = ; 0
1
21866mul02i 9257 . . . . . . . . . . . . 13  |-  ( 0  x.  4 )  =  0
219218oveq1i 6093 . . . . . . . . . . . 12  |-  ( ( 0  x.  4 )  +  1 )  =  ( 0  +  1 )
220219, 217eqtri 2458 . . . . . . . . . . 11  |-  ( ( 0  x.  4 )  +  1 )  = ; 0
1
22116, 3, 3, 16, 200, 217, 2, 16, 3, 70, 220decmac 10423 . . . . . . . . . 10  |-  ( ( 10  x.  4 )  +  ( 0  +  1 ) )  = ; 4
1
222 6p2e8 10122 . . . . . . . . . . 11  |-  ( 6  +  2 )  =  8
22316, 15, 20, 108, 222decaddi 10428 . . . . . . . . . 10  |-  ( ( 4  x.  4 )  +  2 )  = ; 1
8
22410, 2, 3, 20, 206, 139, 2, 24, 16, 221, 223decmac 10423 . . . . . . . . 9  |-  ( (; 10 4  x.  4 )  +  ( 0  +  2 ) )  = ;; 4 1 8
225 4p2e6 10115 . . . . . . . . . 10  |-  ( 4  +  2 )  =  6
22620, 2, 20, 100, 225decaddi 10428 . . . . . . . . 9  |-  ( ( 6  x.  4 )  +  2 )  = ; 2
6
22732, 15, 3, 20, 214, 215, 2, 15, 20, 224, 226decmac 10423 . . . . . . . 8  |-  ( (;; 10 4 6  x.  4 )  +  ( 2  +  0 ) )  = ;;; 4 1 8 6
22833nn0cni 10235 . . . . . . . . . . 11  |- ;; 10 4 6  e.  CC
229228mul01i 9258 . . . . . . . . . 10  |-  (;; 10 4 6  x.  0 )  =  0
230229oveq1i 6093 . . . . . . . . 9  |-  ( (;; 10 4 6  x.  0 )  +  1 )  =  ( 0  +  1 )
231230, 217eqtri 2458 . . . . . . . 8  |-  ( (;; 10 4 6  x.  0 )  +  1 )  = ; 0 1
2322, 3, 20, 16, 62, 213, 33, 16, 3, 227, 231decma2c 10424 . . . . . . 7  |-  ( (;; 10 4 6  x. ; 4
0 )  +  (; 2
1  +  0 ) )  = ;;;; 4 1 8 6 1
2334, 3, 204, 16, 56, 211, 33, 16, 3, 232, 231decma2c 10424 . . . . . 6  |-  ( (;; 10 4 6  x. ;; 4 0 0 )  +  (; 10 7  + ; 10 4 ) )  = ;;;;; 4 1 8 6 1 1
234228mulid1i 9094 . . . . . . . 8  |-  (;; 10 4 6  x.  1 )  = ;; 10 4 6
235234oveq1i 6093 . . . . . . 7  |-  ( (;; 10 4 6  x.  1 )  +  0 )  =  (;; 10 4 6  +  0 )
236228addid1i 9255 . . . . . . 7  |-  (;; 10 4 6  +  0 )  = ;; 10 4 6
237235, 236eqtri 2458 . . . . . 6  |-  ( (;; 10 4 6  x.  1 )  +  0 )  = ;; 10 4 6
2385, 16, 29, 3, 1, 203, 33, 15, 32, 233, 237decma2c 10424 . . . . 5  |-  ( (;; 10 4 6  x.  N )  + ;; 10 7 0 )  = ;;;;;; 4 1 8 6 1 1 6
239 eqid 2438 . . . . . 6  |- ;;; 2 0 4 6  = ;;; 2 0 4 6
24043, 20deccl 10398 . . . . . . 7  |- ;; 1 2 2  e.  NN0
241240, 28deccl 10398 . . . . . 6  |- ;;; 1 2 2 7  e.  NN0
242 eqid 2438 . . . . . . 7  |- ;; 2 0 4  = ;; 2 0 4
243 eqid 2438 . . . . . . 7  |- ;;; 1 2 2 7  = ;;; 1 2 2 7
24424, 16deccl 10398 . . . . . . . 8  |- ; 8 1  e.  NN0
245244, 12deccl 10398 . . . . . . 7  |- ;; 8 1 9  e.  NN0
246 eqid 2438 . . . . . . . 8  |- ; 2 0  = ; 2 0
247 eqid 2438 . . . . . . . . 9  |- ;; 1 2 2  = ;; 1 2 2
248 eqid 2438 . . . . . . . . 9  |- ;; 8 1 9  = ;; 8 1 9
249 eqid 2438 . . . . . . . . . . 11  |- ; 8 1  = ; 8 1
250 8nn 10141 . . . . . . . . . . . . 13  |-  8  e.  NN
251250nncni 10012 . . . . . . . . . . . 12  |-  8  e.  CC
252251, 71, 130addcomli 9260 . . . . . . . . . . 11  |-  ( 1  +  8 )  =  9
253 2p1e3 10105 . . . . . . . . . . 11  |-  ( 2  +  1 )  =  3
25416, 20, 24, 16, 113, 249, 252, 253decadd 10425 . . . . . . . . . 10  |-  (; 1 2  + ; 8 1 )  = ; 9
3
25512, 41, 96, 254decsuc 10407 . . . . . . . . 9  |-  ( (; 1
2  + ; 8 1 )  +  1 )  = ; 9 4
256 9p2e11 10446 . . . . . . . . . 10  |-  ( 9  +  2 )  = ; 1
1
25758, 52, 256addcomli 9260 . . . . . . . . 9  |-  ( 2  +  9 )  = ; 1
1
25843, 20, 244, 12, 247, 248, 255, 16, 257decaddc 10426 . . . . . . . 8  |-  (;; 1 2 2  + ;; 8 1 9 )  = ;; 9 4 1
25913nn0cni 10235 . . . . . . . . . 10  |- ; 9 4  e.  CC
260259addid1i 9255 . . . . . . . . 9  |-  (; 9 4  +  0 )  = ; 9 4
261154, 200eqtri 2458 . . . . . . . . . 10  |-  ( 9  +  1 )  = ; 1
0
262128, 216eqtri 2458 . . . . . . . . . . 11  |-  ( 1  +  0 )  = ; 0
1
263118, 63oveq12i 6095 . . . . . . . . . . . 12  |-  ( ( 2  x.  2 )  +  ( 0  +  0 ) )  =  ( 4  +  0 )
264263, 69eqtri 2458 . . . . . . . . . . 11  |-  ( ( 2  x.  2 )  +  ( 0  +  0 ) )  =  4
26552mul02i 9257 . . . . . . . . . . . . 13  |-  ( 0  x.  2 )  =  0
266265oveq1i 6093 . . . . . . . . . . . 12  |-  ( ( 0  x.  2 )  +  1 )  =  ( 0  +  1 )
267266, 217eqtri 2458 . . . . . . . . . . 11  |-  ( ( 0  x.  2 )  +  1 )  = ; 0
1
26820, 3, 3, 16, 246, 262, 20, 16, 3, 264, 267decmac 10423 . . . . . . . . . 10  |-  ( (; 2
0  x.  2 )  +  ( 1  +  0 ) )  = ; 4
1
269126oveq1i 6093 . . . . . . . . . . 11  |-  ( ( 4  x.  2 )  +  0 )  =  ( 8  +  0 )
270251addid1i 9255 . . . . . . . . . . . 12  |-  ( 8  +  0 )  =  8
27124dec0h 10400 . . . . . . . . . . . 12  |-  8  = ; 0 8
272270, 271eqtri 2458 . . . . . . . . . . 11  |-  ( 8  +  0 )  = ; 0
8
273269, 272eqtri 2458 . . . . . . . . . 10  |-  ( ( 4  x.  2 )  +  0 )  = ; 0
8
27435, 2, 16, 3, 242, 261, 20, 24, 3, 268, 273decmac 10423 . . . . . . . . 9  |-  ( (;; 2 0 4  x.  2 )  +  ( 9  +  1 ) )  = ;; 4 1 8
27566, 52, 225addcomli 9260 . . . . . . . . . 10  |-  ( 2  +  4 )  =  6
27616, 20, 2, 53, 275decaddi 10428 . . . . . . . . 9  |-  ( ( 6  x.  2 )  +  4 )  = ; 1
6
27736, 15, 12, 2, 239, 260, 20, 15, 16, 274, 276decmac 10423 . . . . . . . 8  |-  ( (;;; 2 0 4 6  x.  2 )  +  (; 9
4  +  0 ) )  = ;;; 4 1 8 6
278175mul01i 9258 . . . . . . . . . 10  |-  (;;; 2 0 4 6  x.  0 )  =  0
279278oveq1i 6093 . . . . . . . . 9  |-  ( (;;; 2 0 4 6  x.  0 )  +  1 )  =  ( 0  +  1 )
280279, 217eqtri 2458 . . . . . . . 8  |-  ( (;;; 2 0 4 6  x.  0 )  +  1 )  = ; 0 1
28120, 3, 13, 16, 246, 258, 37, 16, 3, 277, 280decma2c 10424 . . . . . . 7  |-  ( (;;; 2 0 4 6  x. ; 2
0 )  +  (;; 1 2 2  + ;; 8 1 9 ) )  = ;;;; 4 1 8 6 1
28228dec0h 10400 . . . . . . . 8  |-  7  = ; 0 7
283143addid2i 9256 . . . . . . . . . 10  |-  ( 0  +  3 )  =  3
28441dec0h 10400 . . . . . . . . . 10  |-  3  = ; 0 3
285283, 284eqtri 2458 . . . . . . . . 9  |-  ( 0  +  3 )  = ; 0
3
286127, 63oveq12i 6095 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  ( 0  +  0 ) )  =  ( 8  +  0 )
287286, 270eqtri 2458 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  ( 0  +  0 ) )  =  8
28820, 3, 3, 16, 246, 217, 2, 16, 3, 287, 220decmac 10423 . . . . . . . . 9  |-  ( (; 2
0  x.  4 )  +  ( 0  +  1 ) )  = ; 8
1
289 6p3e9 10123 . . . . . . . . . 10  |-  ( 6  +  3 )  =  9
29016, 15, 41, 108, 289decaddi 10428 . . . . . . . . 9  |-  ( ( 4  x.  4 )  +  3 )  = ; 1
9
29135, 2, 3, 41, 242, 285, 2, 12, 16, 288, 290decmac 10423 . . . . . . . 8  |-  ( (;; 2 0 4  x.  4 )  +  ( 0  +  3 ) )  = ;; 8 1 9
292161, 66, 210addcomli 9260 . . . . . . . . 9  |-  ( 4  +  7 )  = ; 1
1
29320, 2, 28, 100, 253, 16, 292decaddci 10429 . . . . . . . 8  |-  ( ( 6  x.  4 )  +  7 )  = ; 3
1
29436, 15, 3, 28, 239, 282, 2, 16, 41, 291, 293decmac 10423 . . . . . . 7  |-  ( (;;; 2 0 4 6  x.  4 )  +  7 )  = ;;; 8 1 9 1
29535, 2, 240, 28, 242, 243, 37, 16, 245, 281, 294decma2c 10424 . . . . . 6  |-  ( (;;; 2 0 4 6  x. ;; 2 0 4 )  + ;;; 1 2 2 7 )  = ;;;;; 4 1 8 6 1 1
29654oveq1i 6093 . . . . . . . . . 10  |-  ( ( 2  x.  6 )  +  0 )  =  (; 1 2  +  0 )
29743nn0cni 10235 . . . . . . . . . . 11  |- ; 1 2  e.  CC
298297addid1i 9255 . . . . . . . . . 10  |-  (; 1 2  +  0 )  = ; 1 2
299296, 298eqtri 2458 . . . . . . . . 9  |-  ( ( 2  x.  6 )  +  0 )  = ; 1
2
30051mul02i 9257 . . . . . . . . . . 11  |-  ( 0  x.  6 )  =  0
301300oveq1i 6093 . . . . . . . . . 10  |-  ( ( 0  x.  6 )  +  2 )  =  ( 0  +  2 )
302301, 137eqtri 2458 . . . . . . . . 9  |-  ( ( 0  x.  6 )  +  2 )  =  2
30320, 3, 3, 20, 246, 139, 15, 299, 302decma 10422 . . . . . . . 8  |-  ( (; 2
0  x.  6 )  +  ( 0  +  2 ) )  = ;; 1 2 2
304 4p3e7 10116 . . . . . . . . 9  |-  ( 4  +  3 )  =  7
30520, 2, 41, 101, 304decaddi 10428 . . . . . . . 8  |-  ( ( 4  x.  6 )  +  3 )  = ; 2
7
30635, 2, 3, 41, 242, 284, 15, 28, 20, 303, 305decmac 10423 . . . . . . 7  |-  ( (;; 2 0 4  x.  6 )  +  3 )  = ;;; 1 2 2 7
30715, 36, 15, 239, 15, 41, 306, 95decmul1c 10431 . . . . . 6  |-  (;;; 2 0 4 6  x.  6 )  = ;;;; 1 2 2 7 6
30837, 36, 15, 239, 15, 241, 295, 307decmul2c 10432 . . . . 5  |-  (;;; 2 0 4 6  x. ;;; 2 0 4 6 )  = ;;;;;; 4 1 8 6 1 1 6
309238, 308eqtr4i 2461 . . . 4  |-  ( (;; 10 4 6  x.  N )  + ;; 10 7 0 )  =  (;;; 2 0 4 6  x. ;;; 2 0 4 6 )
3108, 9, 31, 34, 37, 30, 195, 202, 309mod2xi 13407 . . 3  |-  ( ( 2 ^; 5 0 )  mod 
N )  =  (;; 10 7 0  mod 
N )
311 eqid 2438 . . . 4  |- ; 5 0  = ; 5 0
312199oveq1i 6093 . . . . 5  |-  ( ( 2  x.  5 )  +  0 )  =  ( 10  +  0 )
313 10nn 10143 . . . . . . 7  |-  10  e.  NN
314313nncni 10012 . . . . . 6  |-  10  e.  CC
315314addid1i 9255 . . . . 5  |-  ( 10  +  0 )  =  10
316312, 315eqtri 2458 . . . 4  |-  ( ( 2  x.  5 )  +  0 )  =  10
317132, 64eqtri 2458 . . . 4  |-  ( 2  x.  0 )  = ; 0
0
31820, 22, 3, 311, 3, 3, 316, 317decmul2c 10432 . . 3  |-  ( 2  x. ; 5 0 )  = ; 10 0
319 eqid 2438 . . . . 5  |- ;; 6 1 4  = ;; 6 1 4
32020, 12deccl 10398 . . . . 5  |- ; 2 9  e.  NN0
321 eqid 2438 . . . . . . 7  |- ; 6 1  = ; 6 1
322 eqid 2438 . . . . . . 7  |- ; 2 9  = ; 2 9
323222oveq1i 6093 . . . . . . . 8  |-  ( ( 6  +  2 )  +  1 )  =  ( 8  +  1 )
324323, 130eqtri 2458 . . . . . . 7  |-  ( ( 6  +  2 )  +  1 )  =  9
32515, 16, 20, 12, 321, 322, 324, 155decaddc2 10427 . . . . . 6  |-  (; 6 1  + ; 2 9 )  = ; 9
0
326 eqid 2438 . . . . . . . 8  |- ;; 2 8 6  = ;; 2 8 6
327 eqid 2438 . . . . . . . . 9  |- ; 2 8  = ; 2 8
328127, 283oveq12i 6095 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  ( 0  +  3 ) )  =  ( 8  +  3 )
329 8p3e11 10440 . . . . . . . . . 10  |-  ( 8  +  3 )  = ; 1
1
330328, 329eqtri 2458 . . . . . . . . 9  |-  ( ( 2  x.  4 )  +  ( 0  +  3 ) )  = ; 1
1
331 8t4e32 10474 . . . . . . . . . 10  |-  ( 8  x.  4 )  = ; 3
2
33241, 20, 20, 331, 93decaddi 10428 . . . . . . . . 9  |-  ( ( 8  x.  4 )  +  2 )  = ; 3
4
33320, 24, 3, 20, 327, 139, 2, 2, 41, 330, 332decmac 10423 . . . . . . . 8  |-  ( (; 2
8  x.  4 )  +  ( 0  +  2 ) )  = ;; 1 1 4
334100oveq1i 6093 . . . . . . . . 9  |-  ( ( 6  x.  4 )  +  0 )  =  (; 2 4  +  0 )
33538nn0cni 10235 . . . . . . . . . 10  |- ; 2 4  e.  CC
336335addid1i 9255 . . . . . . . . 9  |-  (; 2 4  +  0 )  = ; 2 4
337334, 336eqtri 2458 . . . . . . . 8  |-  ( ( 6  x.  4 )  +  0 )  = ; 2
4
33825, 15, 3, 3, 326, 65, 2, 2, 20, 333, 337decmac 10423 . . . . . . 7  |-  ( (;; 2 8 6  x.  4 )  +  ( 0  +  0 ) )  = ;;; 1 1 4 4
33926nn0cni 10235 . . . . . . . . . 10  |- ;; 2 8 6  e.  CC
340339mul01i 9258 . . . . . . . . 9  |-  (;; 2 8 6  x.  0 )  =  0
341340oveq1i 6093 . . . . . . . 8  |-  ( (;; 2 8 6  x.  0 )  +  9 )  =  ( 0  +  9 )
342341, 78eqtri 2458 . . . . . . 7  |-  ( (;; 2 8 6  x.  0 )  +  9 )  = ; 0 9
3432, 3, 3, 12, 62, 61, 26, 12, 3, 338, 342decma2c 10424 . . . . . 6  |-  ( (;; 2 8 6  x. ; 4
0 )  +  ( 9  +  0 ) )  = ;;;; 1 1 4 4 9
344340oveq1i 6093 . . . . . . 7  |-  ( (;; 2 8 6  x.  0 )  +  0 )  =  ( 0  +  0 )
345344, 65eqtri 2458 . . . . . 6  |-  ( (;; 2 8 6  x.  0 )  +  0 )  = ; 0 0
3464, 3, 12, 3, 56, 325, 26, 3, 3, 343, 345decma2c 10424 . . . . 5  |-  ( (;; 2 8 6  x. ;; 4 0 0 )  +  (; 6 1  + ; 2 9 ) )  = ;;;;; 1 1 4 4 9 0
347114, 63oveq12i 6095 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
348347, 116eqtri 2458 . . . . . . 7  |-  ( ( 2  x.  1 )  +  ( 0  +  0 ) )  =  2
349251mulid1i 9094 . . . . . . . . 9  |-  ( 8  x.  1 )  =  8
350349oveq1i 6093 . . . . . . . 8  |-  ( ( 8  x.  1 )  +  1 )  =  ( 8  +  1 )
351130, 60eqtri 2458 . . . . . . . 8  |-  ( 8  +  1 )  = ; 0
9
352350, 351eqtri 2458 . . . . . . 7  |-  ( ( 8  x.  1 )  +  1 )  = ; 0
9
35320, 24, 3, 16, 327, 217, 16, 12, 3, 348, 352decmac 10423 . . . . . 6  |-  ( (; 2
8  x.  1 )  +  ( 0  +  1 ) )  = ; 2
9
35451mulid1i 9094 . . . . . . . 8  |-  ( 6  x.  1 )  =  6
355354oveq1i 6093 . . . . . . 7  |-  ( ( 6  x.  1 )  +  4 )  =  ( 6  +  4 )
35697, 200eqtri 2458 . . . . . . 7  |-  ( 6  +  4 )  = ; 1
0
357355, 356eqtri 2458 . . . . . 6  |-  ( ( 6  x.  1 )  +  4 )  = ; 1
0
35825, 15, 3, 2, 326, 119, 16, 3, 16, 353, 357decmac 10423 . . . . 5  |-  ( (;; 2 8 6  x.  1 )  +  4 )  = ;; 2 9 0
3595, 16, 17, 2, 1, 319, 26, 3, 320, 346, 358decma2c 10424 . . . 4  |-  ( (;; 2 8 6  x.  N )  + ;; 6 1 4 )  = ;;;;;; 1 1 4 4 9 0 0
36016, 16deccl 10398 . . . . . . . . 9  |- ; 1 1  e.  NN0
361360, 2deccl 10398 . . . . . . . 8  |- ;; 1 1 4  e.  NN0
362361, 2deccl 10398 . . . . . . 7  |- ;;; 1 1 4 4  e.  NN0
363362, 12deccl 10398 . . . . . 6  |- ;;;; 1 1 4 4 9  e.  NN0
36428, 2deccl 10398 . . . . . . . 8  |- ; 7 4  e.  NN0
365364, 12deccl 10398 . . . . . . 7  |- ;; 7 4 9  e.  NN0
366 eqid 2438 . . . . . . . 8  |- ;; 7 4 9  = ;; 7 4 9
367364nn0cni 10235 . . . . . . . . . 10  |- ; 7 4  e.  CC
368367addid1i 9255 . . . . . . . . 9  |-  (; 7 4  +  0 )  = ; 7 4
369161addid1i 9255 . . . . . . . . . . 11  |-  ( 7  +  0 )  =  7
370369, 282eqtri 2458 . . . . . . . . . 10  |-  ( 7  +  0 )  = ; 0
7
37181, 63oveq12i 6095 . . . . . . . . . . . 12  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
372371, 128eqtri 2458 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
37371mul02i 9257 . . . . . . . . . . . . 13  |-  ( 0  x.  1 )  =  0
374373oveq1i 6093 . . . . . . . . . . . 12  |-  ( ( 0  x.  1 )  +  1 )  =  ( 0  +  1 )
375374, 217eqtri 2458 . . . . . . . . . . 11  |-  ( ( 0  x.  1 )  +  1 )  = ; 0
1
37616, 3, 3, 16, 200, 217, 16, 16, 3, 372, 375decmac 10423 . . . . . . . . . 10  |-  ( ( 10  x.  1 )  +  ( 0  +  1 ) )  = ; 1
1
377161mulid1i 9094 . . . . . . . . . . . 12  |-  ( 7  x.  1 )  =  7
378377oveq1i 6093 . . . . . . . . . . 11  |-  ( ( 7  x.  1 )  +  7 )  =  ( 7  +  7 )
379 7p7e14 10439 . . . . . . . . . . 11  |-  ( 7  +  7 )  = ; 1
4
380378, 379eqtri 2458 . . . . . . . . . 10  |-  ( ( 7  x.  1 )  +  7 )  = ; 1
4
38110, 28, 3, 28, 205, 370, 16, 2, 16, 376, 380decmac 10423 . . . . . . . . 9  |-  ( (; 10 7  x.  1 )  +  ( 7  +  0 ) )  = ;; 1 1 4
382373oveq1i 6093 . . . . . . . . . 10  |-  ( ( 0  x.  1 )  +  4 )  =  ( 0  +  4 )
38366addid2i 9256 . . . . . . . . . . 11  |-  ( 0  +  4 )  =  4
384383, 119eqtri 2458 . . . . . . . . . 10  |-  ( 0  +  4 )  = ; 0
4
385382, 384eqtri 2458 . . . . . . . . 9  |-  ( ( 0  x.  1 )  +  4 )  = ; 0
4
38629, 3, 28, 2, 203, 368, 16, 2, 3, 381, 385decmac 10423 . . . . . . . 8  |-  ( (;; 10 7 0  x.  1 )  +  (; 7
4  +  0 ) )  = ;;; 1 1 4 4
38730nn0cni 10235 . . . . . . . . . . 11  |- ;; 10 7 0  e.  CC
388387mul01i 9258 . . . . . . . . . 10  |-  (;; 10 7 0  x.  0 )  =  0
389388oveq1i 6093 . . . . . . . . 9  |-  ( (;; 10 7 0  x.  0 )  +  9 )  =  ( 0  +  9 )
390389, 78eqtri 2458 . . . . . . . 8  |-  ( (;; 10 7 0  x.  0 )  +  9 )  = ; 0 9
39116, 3, 364, 12, 200, 366, 30, 12, 3, 386, 390decma2c 10424 . . . . . . 7  |-  ( (;; 10 7 0  x.  10 )  + ;; 7 4 9 )  = ;;;; 1 1 4 4 9
392161mulid2i 9095 . . . . . . . . . . . . . 14  |-  ( 1  x.  7 )  =  7
393392, 63oveq12i 6095 . . . . . . . . . . . . 13  |-  ( ( 1  x.  7 )  +  ( 0  +  0 ) )  =  ( 7  +  0 )
394393, 369eqtri 2458 . . . . . . . . . . . 12  |-  ( ( 1  x.  7 )  +  ( 0  +  0 ) )  =  7
395161mul02i 9257 . . . . . . . . . . . . . 14  |-  ( 0  x.  7 )  =  0
396395oveq1i 6093 . . . . . . . . . . . . 13  |-  ( ( 0  x.  7 )  +  4 )  =  ( 0  +  4 )
397396, 384eqtri 2458 . . . . . . . . . . . 12  |-  ( ( 0  x.  7 )  +  4 )  = ; 0
4
39816, 3, 3, 2, 200, 119, 28, 2, 3, 394, 397decmac 10423 . . . . . . . . . . 11  |-  ( ( 10  x.  7 )  +  4 )  = ; 7
4
399 7t7e49 10471 . . . . . . . . . . 11  |-  ( 7  x.  7 )  = ; 4
9
40028, 10, 28, 205, 12, 2, 398, 399decmul1c 10431 . . . . . . . . . 10  |-  (; 10 7  x.  7 )  = ;; 7 4 9
401400oveq1i 6093 . . . . . . . . 9  |-  ( (; 10 7  x.  7 )  +  0 )  =  (;; 7 4 9  +  0 )
402365nn0cni 10235 . . . . . . . . . 10  |- ;; 7 4 9  e.  CC
403402addid1i 9255 . . . . . . . . 9  |-  (;; 7 4 9  +  0 )  = ;; 7 4 9
404401, 403eqtri 2458 . . . . . . . 8  |-  ( (; 10 7  x.  7 )  +  0 )  = ;; 7 4 9
405395, 64eqtri 2458 . . . . . . . 8  |-  ( 0  x.  7 )  = ; 0
0
40628, 29, 3, 203, 3, 3, 404, 405decmul1c 10431 . . . . . . 7  |-  (;; 10 7 0  x.  7 )  = ;;; 7 4 9 0
40730, 10, 28, 205, 3, 365, 391, 406decmul2c 10432 . . . . . 6  |-  (;; 10 7 0  x. ; 10 7 )  = ;;;;; 1 1 4 4 9 0
408363, 3, 3, 407, 63decaddi 10428 . . . . 5  |-  ( (;; 10 7 0  x. ; 10 7 )  +  0 )  = ;;;;; 1 1 4 4 9 0
409388, 64eqtri 2458 . . . . 5  |-  (;; 10 7 0  x.  0 )  = ; 0 0
41030, 29, 3, 203, 3, 3, 408, 409decmul2c 10432 . . . 4  |-  (;; 10 7 0  x. ;; 10 7 0 )  = ;;;;;; 1 1 4 4 9 0 0
411359, 410eqtr4i 2461 . . 3  |-  ( (;; 2 8 6  x.  N )  + ;; 6 1 4 )  =  (;; 10 7 0  x. ;; 10 7 0 )
4128, 9, 23, 27, 30, 18, 310, 318, 411mod2xi 13407 . 2  |-  ( ( 2 ^; 10 0 )  mod 
N )  =  (;; 6 1 4  mod 
N )
413 eqid 2438 . . 3  |- ; 10 0  = ; 10 0
41420, 16, 3, 200, 3, 3, 117, 317decmul2c 10432 . . . . 5  |-  ( 2  x.  10 )  = ; 2
0
415414oveq1i 6093 . . . 4  |-  ( ( 2  x.  10 )  +  0 )  =  (; 2 0  +  0 )
416415, 184eqtri 2458 . . 3  |-  ( ( 2  x.  10 )  +  0 )  = ; 2
0
41720, 10, 3, 413, 3, 3, 416, 317decmul2c 10432 . 2  |-  ( 2  x. ; 10 0 )  = ;; 2 0 0
418 eqid 2438 . . . 4  |- ;; 9 0 2  = ;; 9 0 2
419 eqid 2438 . . . . . 6  |- ; 9 0  = ; 9 0
42012, 3, 12, 419, 77decaddi 10428 . . . . 5  |-  (; 9 0  +  9 )  = ; 9 9
421 eqid 2438 . . . . . . 7  |- ; 9 4  = ; 9 4
422207oveq2i 6094 . . . . . . . 8  |-  ( ( 9  x.  4 )  +  ( 0  +  1 ) )  =  ( ( 9  x.  4 )  +  1 )
423 6p1e7 10109 . . . . . . . . 9  |-  ( 6  +  1 )  =  7
424 9t4e36 10481 . . . . . . . . 9  |-  ( 9  x.  4 )  = ; 3
6
42541, 15, 423, 424decsuc 10407 . . . . . . . 8  |-  ( ( 9  x.  4 )  +  1 )  = ; 3
7
426422, 425eqtri 2458 . . . . . . 7  |-  ( ( 9  x.  4 )  +  ( 0  +  1 ) )  = ; 3
7
427108oveq1i 6093 . . . . . . . 8  |-  ( ( 4  x.  4 )  +  0 )  =  (; 1 6  +  0 )
42816, 15deccl 10398 . . . . . . . . . 10  |- ; 1 6  e.  NN0
429428nn0cni 10235 . . . . . . . . 9  |- ; 1 6  e.  CC
430429addid1i 9255 . . . . . . . 8  |-  (; 1 6  +  0 )  = ; 1 6
431427, 430eqtri 2458 . . . . . . 7  |-  ( ( 4  x.  4 )  +  0 )  = ; 1
6
43212, 2, 3, 3, 421, 65, 2, 15, 16, 426, 431decmac 10423 . . . . . 6  |-  ( (; 9
4  x.  4 )  +  ( 0  +  0 ) )  = ;; 3 7 6
433259mul01i 9258 . . . . . . . 8  |-  (; 9 4  x.  0 )  =  0
434433oveq1i 6093 . . . . . . 7  |-  ( (; 9
4  x.  0 )  +  9 )  =  ( 0  +  9 )
435434, 78eqtri 2458 . . . . . 6  |-  ( (; 9
4  x.  0 )  +  9 )  = ; 0
9
4362, 3, 3, 12, 62, 61, 13, 12, 3, 432, 435decma2c 10424 . . . . 5  |-  ( (; 9
4  x. ; 4 0 )  +  ( 9  +  0 ) )  = ;;; 3 7 6 9
4374, 3, 12, 12, 56, 420, 13, 12, 3, 436, 435decma2c 10424 . . . 4  |-  ( (; 9
4  x. ;; 4 0 0 )  +  (; 9 0  +  9 ) )  = ;;;; 3 7 6 9 9
43858mulid1i 9094 . . . . . . 7  |-  ( 9  x.  1 )  =  9
439438, 63oveq12i 6095 . . . . . 6  |-  ( ( 9  x.  1 )  +  ( 0  +  0 ) )  =  ( 9  +  0 )
440439, 59eqtri 2458 . . . . 5  |-  ( ( 9  x.  1 )  +  ( 0  +  0 ) )  =  9
44166mulid1i 9094 . . . . . . 7  |-  ( 4  x.  1 )  =  4
442441oveq1i 6093 . . . . . 6  |-  ( ( 4  x.  1 )  +  2 )  =  ( 4  +  2 )
443225, 87eqtri 2458 . . . . . 6  |-  ( 4  +  2 )  = ; 0
6
444442, 443eqtri 2458 . . . . 5  |-  ( ( 4  x.  1 )  +  2 )  = ; 0
6
44512, 2, 3, 20, 421, 138, 16, 15, 3, 440, 444decmac 10423 . . . 4  |-  ( (; 9
4  x.  1 )  +  2 )  = ; 9
6
4465, 16, 19, 20, 1, 418, 13, 15, 12, 437, 445decma2c 10424 . . 3  |-  ( (; 9
4  x.  N )  + ;; 9 0 2 )  = ;;;;; 3 7 6 9 9 6
44738, 22deccl 10398 . . . 4  |- ;; 2 4 5  e.  NN0
448 eqid 2438 . . . . 5  |- ;; 2 4 5  = ;; 2 4 5
44951, 52, 222addcomli 9260 . . . . . . 7  |-  ( 2  +  6 )  =  8
45020, 2, 15, 16, 173, 321, 449, 106decadd 10425 . . . . . 6  |-  (; 2 4  + ; 6 1 )  = ; 8
5
451 8p2e10 10127 . . . . . . . 8  |-  ( 8  +  2 )  =  10
452451, 200eqtri 2458 . . . . . . 7  |-  ( 8  +  2 )  = ; 1
0
45341, 15, 423, 95decsuc 10407 . . . . . . 7  |-  ( ( 6  x.  6 )  +  1 )  = ; 3
7
45451mulid2i 9095 . . . . . . . . 9  |-  ( 1  x.  6 )  =  6
455454oveq1i 6093 . . . . . . . 8  |-  ( ( 1  x.  6 )  +  0 )  =  ( 6  +  0 )
45651addid1i 9255 . . . . . . . 8  |-  ( 6  +  0 )  =  6
457455, 456eqtri 2458 . . . . . . 7  |-  ( ( 1  x.  6 )  +  0 )  =  6
45815, 16, 16, 3, 321, 452, 15, 453, 457decma 10422 . . . . . 6  |-  ( (; 6
1  x.  6 )  +  ( 8  +  2 ) )  = ;; 3 7 6
45917, 2, 24, 22, 319, 450, 15, 12, 20, 458, 104decmac 10423 . . . . 5  |-  ( (;; 6 1 4  x.  6 )  +  (; 2
4  + ; 6 1 ) )  = ;;; 3 7 6 9
460354, 63oveq12i 6095 . . . . . . . 8  |-  ( ( 6  x.  1 )  +  ( 0  +  0 ) )  =  ( 6  +  0 )
461460, 456eqtri 2458 . . . . . . 7  |-  ( ( 6  x.  1 )  +  ( 0  +  0 ) )  =  6
46281oveq1i 6093 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  0 )  =  ( 1  +  0 )
463462, 262eqtri 2458 . . . . . . 7  |-  ( ( 1  x.  1 )  +  0 )  = ; 0
1
46415, 16, 3, 3, 321, 65, 16, 16, 3, 461, 463decmac 10423 . . . . . 6  |-  ( (; 6
1  x.  1 )  +  ( 0  +  0 ) )  = ; 6
1
465441oveq1i 6093 . . . . . . 7  |-  ( ( 4  x.  1 )  +  5 )  =  ( 4  +  5 )
466103, 60eqtri 2458 . . . . . . 7  |-  ( 4  +  5 )  = ; 0
9
467465, 466eqtri 2458 . . . . . 6  |-  ( ( 4  x.  1 )  +  5 )  = ; 0
9
46817, 2, 3, 22, 319, 146, 16, 12, 3, 464, 467decmac 10423 . . . . 5  |-  ( (;; 6 1 4  x.  1 )  +  5 )  = ;; 6 1 9
46915, 16, 38, 22, 321, 448, 18, 12, 17, 459, 468decma2c 10424 . . . 4  |-  ( (;; 6 1 4  x. ; 6
1 )  + ;; 2 4 5 )  = ;;;; 3 7 6 9 9
47067oveq1i 6093 . . . . . . 7  |-  ( ( 1  x.  4 )  +  1 )  =  ( 4  +  1 )
471470, 106eqtri 2458 . . . . . 6  |-  ( ( 1  x.  4 )  +  1 )  =  5
47215, 16, 3, 16, 321, 216, 2, 337, 471decma 10422 . . . . 5  |-  ( (; 6
1  x.  4 )  +  1 )  = ;; 2 4 5
4732, 17, 2, 319, 15, 16, 472, 108decmul1c 10431 . . . 4  |-  (;; 6 1 4  x.  4 )  = ;;; 2 4 5 6
47418, 17, 2, 319, 15, 447, 469, 473decmul2c 10432 . . 3  |-  (;; 6 1 4  x. ;; 6 1 4 )  = ;;;;; 3 7 6 9 9 6
475446, 474eqtr4i 2461 . 2  |-  ( (; 9
4  x.  N )  + ;; 9 0 2 )  =  (;; 6 1 4  x. ;; 6 1 4 )
4768, 9, 11, 14, 18, 21, 412, 417, 475mod2xi 13407 1  |-  ( ( 2 ^;; 2 0 0 )  mod 
N )  =  (;; 9 0 2  mod 
N )
Colors of variables: wff set class
Syntax hints:    = wceq 1653  (class class class)co 6083   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997   NNcn 10002   2c2 10051   3c3 10052   4c4 10053   5c5 10054   6c6 10055   7c7 10056   8c8 10057   9c9 10058   10c10 10059  ;cdc 10384    mod cmo 11252   ^cexp 11384
This theorem is referenced by:  4001lem2  13463  4001lem3  13464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-rp 10615  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385
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