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Theorem 1259lem4 13148
Description: Lemma for 1259prm 13150. Calculate a power mod. In decimal, we calculate  2 ^ 3 0 6  =  ( 2 ^ 7 6 ) ^ 4  x.  4  ==  5 ^ 4  x.  4  =  2 N  -  1 8,  2 ^ 6 1 2  =  ( 2 ^ 3 0 6 ) ^ 2  ==  1 8 ^ 2  =  3 2 4,  2 ^ 6 2 9  =  2 ^ 6 1 2  x.  2 ^ 1 7  ==  3 2 4  x.  1 3 6  =  3 5 N  -  1 and finally  2 ^ ( N  -  1 )  =  ( 2 ^ 6 2 9 ) ^ 2  ==  1 ^ 2  =  1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
Hypothesis
Ref Expression
1259prm.1  |-  N  = ;;; 1 2 5 9
Assertion
Ref Expression
1259lem4  |-  ( ( 2 ^ ( N  -  1 ) )  mod  N )  =  ( 1  mod  N
)

Proof of Theorem 1259lem4
StepHypRef Expression
1 2nn 9893 . 2  |-  2  e.  NN
2 6nn0 10002 . . . 4  |-  6  e.  NN0
3 2nn0 9998 . . . 4  |-  2  e.  NN0
42, 3deccl 10154 . . 3  |- ; 6 2  e.  NN0
5 9nn0 10005 . . 3  |-  9  e.  NN0
64, 5deccl 10154 . 2  |- ;; 6 2 9  e.  NN0
7 0z 10051 . 2  |-  0  e.  ZZ
8 1nn 9773 . 2  |-  1  e.  NN
9 1nn0 9997 . 2  |-  1  e.  NN0
10 1259prm.1 . . . . . 6  |-  N  = ;;; 1 2 5 9
119, 3deccl 10154 . . . . . . . 8  |- ; 1 2  e.  NN0
12 5nn0 10001 . . . . . . . 8  |-  5  e.  NN0
1311, 12deccl 10154 . . . . . . 7  |- ;; 1 2 5  e.  NN0
14 8nn0 10004 . . . . . . 7  |-  8  e.  NN0
15 8p1e9 9869 . . . . . . 7  |-  ( 8  +  1 )  =  9
16 eqid 2296 . . . . . . 7  |- ;;; 1 2 5 8  = ;;; 1 2 5 8
1713, 14, 15, 16decsuc 10163 . . . . . 6  |-  (;;; 1 2 5 8  +  1 )  = ;;; 1 2 5 9
1810, 17eqtr4i 2319 . . . . 5  |-  N  =  (;;; 1 2 5 8  +  1 )
1918oveq1i 5884 . . . 4  |-  ( N  -  1 )  =  ( (;;; 1 2 5 8  +  1 )  - 
1 )
2013, 14deccl 10154 . . . . . 6  |- ;;; 1 2 5 8  e.  NN0
2120nn0cni 9993 . . . . 5  |- ;;; 1 2 5 8  e.  CC
22 ax-1cn 8811 . . . . 5  |-  1  e.  CC
23 pncan 9073 . . . . 5  |-  ( (;;; 1 2 5 8  e.  CC  /\  1  e.  CC )  ->  (
(;;; 1 2 5 8  +  1 )  - 
1 )  = ;;; 1 2 5 8 )
2421, 22, 23mp2an 653 . . . 4  |-  ( (;;; 1 2 5 8  +  1 )  -  1 )  = ;;; 1 2 5 8
2519, 24eqtri 2316 . . 3  |-  ( N  -  1 )  = ;;; 1 2 5 8
2625, 20eqeltri 2366 . 2  |-  ( N  -  1 )  e. 
NN0
27 9nn 9900 . . . . 5  |-  9  e.  NN
2813, 27decnncl 10153 . . . 4  |- ;;; 1 2 5 9  e.  NN
2910, 28eqeltri 2366 . . 3  |-  N  e.  NN
302, 9deccl 10154 . . . 4  |- ; 6 1  e.  NN0
3130, 3deccl 10154 . . 3  |- ;; 6 1 2  e.  NN0
32 3nn0 9999 . . . . 5  |-  3  e.  NN0
33 4nn0 10000 . . . . 5  |-  4  e.  NN0
3432, 33deccl 10154 . . . 4  |- ; 3 4  e.  NN0
3534nn0zi 10064 . . 3  |- ; 3 4  e.  ZZ
3632, 3deccl 10154 . . . 4  |- ; 3 2  e.  NN0
3736, 33deccl 10154 . . 3  |- ;; 3 2 4  e.  NN0
38 7nn0 10003 . . . 4  |-  7  e.  NN0
399, 38deccl 10154 . . 3  |- ; 1 7  e.  NN0
409, 32deccl 10154 . . . 4  |- ; 1 3  e.  NN0
4140, 2deccl 10154 . . 3  |- ;; 1 3 6  e.  NN0
42 0nn0 9996 . . . . . 6  |-  0  e.  NN0
4332, 42deccl 10154 . . . . 5  |- ; 3 0  e.  NN0
4443, 2deccl 10154 . . . 4  |- ;; 3 0 6  e.  NN0
45 8nn 9899 . . . . 5  |-  8  e.  NN
469, 45decnncl 10153 . . . 4  |- ; 1 8  e.  NN
4711, 33deccl 10154 . . . . 5  |- ;; 1 2 4  e.  NN0
4847, 9deccl 10154 . . . 4  |- ;;; 1 2 4 1  e.  NN0
499, 12deccl 10154 . . . . . 6  |- ; 1 5  e.  NN0
5049, 32deccl 10154 . . . . 5  |- ;; 1 5 3  e.  NN0
51 1z 10069 . . . . 5  |-  1  e.  ZZ
5212, 42deccl 10154 . . . . 5  |- ; 5 0  e.  NN0
5349, 3deccl 10154 . . . . . 6  |- ;; 1 5 2  e.  NN0
543, 12deccl 10154 . . . . . 6  |- ; 2 5  e.  NN0
5538, 2deccl 10154 . . . . . . 7  |- ; 7 6  e.  NN0
56101259lem3 13147 . . . . . . 7  |-  ( ( 2 ^; 7 6 )  mod 
N )  =  ( 5  mod  N )
57 eqid 2296 . . . . . . . 8  |- ; 7 6  = ; 7 6
58 4p1e5 9865 . . . . . . . . 9  |-  ( 4  +  1 )  =  5
59 7nn 9898 . . . . . . . . . . 11  |-  7  e.  NN
6059nncni 9772 . . . . . . . . . 10  |-  7  e.  CC
61 2cn 9832 . . . . . . . . . 10  |-  2  e.  CC
62 7t2e14 10222 . . . . . . . . . 10  |-  ( 7  x.  2 )  = ; 1
4
6360, 61, 62mulcomli 8860 . . . . . . . . 9  |-  ( 2  x.  7 )  = ; 1
4
649, 33, 58, 63decsuc 10163 . . . . . . . 8  |-  ( ( 2  x.  7 )  +  1 )  = ; 1
5
65 6nn 9897 . . . . . . . . . 10  |-  6  e.  NN
6665nncni 9772 . . . . . . . . 9  |-  6  e.  CC
67 6t2e12 10217 . . . . . . . . 9  |-  ( 6  x.  2 )  = ; 1
2
6866, 61, 67mulcomli 8860 . . . . . . . 8  |-  ( 2  x.  6 )  = ; 1
2
693, 38, 2, 57, 3, 9, 64, 68decmul2c 10188 . . . . . . 7  |-  ( 2  x. ; 7 6 )  = ;; 1 5 2
7054nn0cni 9993 . . . . . . . . 9  |- ; 2 5  e.  CC
7170addid2i 9016 . . . . . . . 8  |-  ( 0  + ; 2 5 )  = ; 2
5
7229nncni 9772 . . . . . . . . . 10  |-  N  e.  CC
7372mul02i 9017 . . . . . . . . 9  |-  ( 0  x.  N )  =  0
7473oveq1i 5884 . . . . . . . 8  |-  ( ( 0  x.  N )  + ; 2 5 )  =  ( 0  + ; 2 5 )
75 5t5e25 10216 . . . . . . . 8  |-  ( 5  x.  5 )  = ; 2
5
7671, 74, 753eqtr4i 2326 . . . . . . 7  |-  ( ( 0  x.  N )  + ; 2 5 )  =  ( 5  x.  5 )
7729, 1, 55, 7, 12, 54, 56, 69, 76mod2xi 13100 . . . . . 6  |-  ( ( 2 ^;; 1 5 2 )  mod 
N )  =  (; 2
5  mod  N )
78 2p1e3 9863 . . . . . . 7  |-  ( 2  +  1 )  =  3
79 eqid 2296 . . . . . . 7  |- ;; 1 5 2  = ;; 1 5 2
8049, 3, 78, 79decsuc 10163 . . . . . 6  |-  (;; 1 5 2  +  1 )  = ;; 1 5 3
8152nn0cni 9993 . . . . . . . 8  |- ; 5 0  e.  CC
8281addid2i 9016 . . . . . . 7  |-  ( 0  + ; 5 0 )  = ; 5
0
8373oveq1i 5884 . . . . . . 7  |-  ( ( 0  x.  N )  + ; 5 0 )  =  ( 0  + ; 5 0 )
84 eqid 2296 . . . . . . . 8  |- ; 2 5  = ; 2 5
85 2t2e4 9887 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
8685oveq1i 5884 . . . . . . . . 9  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
8786, 58eqtri 2316 . . . . . . . 8  |-  ( ( 2  x.  2 )  +  1 )  =  5
88 5t2e10 9891 . . . . . . . . 9  |-  ( 5  x.  2 )  =  10
89 dec10 10170 . . . . . . . . 9  |-  10  = ; 1 0
9088, 89eqtri 2316 . . . . . . . 8  |-  ( 5  x.  2 )  = ; 1
0
913, 3, 12, 84, 42, 9, 87, 90decmul1c 10187 . . . . . . 7  |-  (; 2 5  x.  2 )  = ; 5 0
9282, 83, 913eqtr4i 2326 . . . . . 6  |-  ( ( 0  x.  N )  + ; 5 0 )  =  (; 2 5  x.  2 )
9329, 1, 53, 7, 54, 52, 77, 80, 92modxp1i 13101 . . . . 5  |-  ( ( 2 ^;; 1 5 3 )  mod 
N )  =  (; 5
0  mod  N )
94 eqid 2296 . . . . . 6  |- ;; 1 5 3  = ;; 1 5 3
95 eqid 2296 . . . . . . . . 9  |- ; 1 5  = ; 1 5
9661mulid1i 8855 . . . . . . . . . . 11  |-  ( 2  x.  1 )  =  2
9796oveq1i 5884 . . . . . . . . . 10  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
9897, 78eqtri 2316 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  1 )  =  3
99 5nn 9896 . . . . . . . . . . . 12  |-  5  e.  NN
10099nncni 9772 . . . . . . . . . . 11  |-  5  e.  CC
101100, 61, 88mulcomli 8860 . . . . . . . . . 10  |-  ( 2  x.  5 )  =  10
102101, 89eqtri 2316 . . . . . . . . 9  |-  ( 2  x.  5 )  = ; 1
0
1033, 9, 12, 95, 42, 9, 98, 102decmul2c 10188 . . . . . . . 8  |-  ( 2  x. ; 1 5 )  = ; 3
0
104103oveq1i 5884 . . . . . . 7  |-  ( ( 2  x. ; 1 5 )  +  0 )  =  (; 3
0  +  0 )
10543nn0cni 9993 . . . . . . . 8  |- ; 3 0  e.  CC
106105addid1i 9015 . . . . . . 7  |-  (; 3 0  +  0 )  = ; 3 0
107104, 106eqtri 2316 . . . . . 6  |-  ( ( 2  x. ; 1 5 )  +  0 )  = ; 3 0
108 3cn 9834 . . . . . . . 8  |-  3  e.  CC
109 3t2e6 9888 . . . . . . . 8  |-  ( 3  x.  2 )  =  6
110108, 61, 109mulcomli 8860 . . . . . . 7  |-  ( 2  x.  3 )  =  6
1112dec0h 10156 . . . . . . 7  |-  6  = ; 0 6
112110, 111eqtri 2316 . . . . . 6  |-  ( 2  x.  3 )  = ; 0
6
1133, 49, 32, 94, 2, 42, 107, 112decmul2c 10188 . . . . 5  |-  ( 2  x. ;; 1 5 3 )  = ;; 3 0 6
11472mulid2i 8856 . . . . . . . 8  |-  ( 1  x.  N )  =  N
115114, 10eqtri 2316 . . . . . . 7  |-  ( 1  x.  N )  = ;;; 1 2 5 9
116 eqid 2296 . . . . . . 7  |- ;;; 1 2 4 1  = ;;; 1 2 4 1
1173, 33deccl 10154 . . . . . . . 8  |- ; 2 4  e.  NN0
118 eqid 2296 . . . . . . . . 9  |- ; 2 4  = ; 2 4
1193, 33, 58, 118decsuc 10163 . . . . . . . 8  |-  (; 2 4  +  1 )  = ; 2 5
120 eqid 2296 . . . . . . . . 9  |- ;; 1 2 5  = ;; 1 2 5
121 eqid 2296 . . . . . . . . 9  |- ;; 1 2 4  = ;; 1 2 4
122 eqid 2296 . . . . . . . . . 10  |- ; 1 2  = ; 1 2
123 1p1e2 9856 . . . . . . . . . 10  |-  ( 1  +  1 )  =  2
124 2p2e4 9858 . . . . . . . . . 10  |-  ( 2  +  2 )  =  4
1259, 3, 9, 3, 122, 122, 123, 124decadd 10181 . . . . . . . . 9  |-  (; 1 2  + ; 1 2 )  = ; 2
4
126 5p4e9 9878 . . . . . . . . 9  |-  ( 5  +  4 )  =  9
12711, 12, 11, 33, 120, 121, 125, 126decadd 10181 . . . . . . . 8  |-  (;; 1 2 5  + ;; 1 2 4 )  = ;; 2 4 9
128117, 119, 127decsucc 10167 . . . . . . 7  |-  ( (;; 1 2 5  + ;; 1 2 4 )  +  1 )  = ;; 2 5 0
129 9p1e10 9870 . . . . . . 7  |-  ( 9  +  1 )  =  10
13013, 5, 47, 9, 115, 116, 128, 129decaddc2 10183 . . . . . 6  |-  ( ( 1  x.  N )  + ;;; 1 2 4 1 )  = ;;; 2 5 0 0
131 eqid 2296 . . . . . . 7  |- ; 5 0  = ; 5 0
13275oveq1i 5884 . . . . . . . . . . 11  |-  ( ( 5  x.  5 )  +  0 )  =  (; 2 5  +  0 )
13370addid1i 9015 . . . . . . . . . . 11  |-  (; 2 5  +  0 )  = ; 2 5
134132, 133eqtri 2316 . . . . . . . . . 10  |-  ( ( 5  x.  5 )  +  0 )  = ; 2
5
135100mul02i 9017 . . . . . . . . . . 11  |-  ( 0  x.  5 )  =  0
13642dec0h 10156 . . . . . . . . . . 11  |-  0  = ; 0 0
137135, 136eqtri 2316 . . . . . . . . . 10  |-  ( 0  x.  5 )  = ; 0
0
13812, 12, 42, 131, 42, 42, 134, 137decmul1c 10187 . . . . . . . . 9  |-  (; 5 0  x.  5 )  = ;; 2 5 0
139138oveq1i 5884 . . . . . . . 8  |-  ( (; 5
0  x.  5 )  +  0 )  =  (;; 2 5 0  +  0 )
14054, 42deccl 10154 . . . . . . . . . 10  |- ;; 2 5 0  e.  NN0
141140nn0cni 9993 . . . . . . . . 9  |- ;; 2 5 0  e.  CC
142141addid1i 9015 . . . . . . . 8  |-  (;; 2 5 0  +  0 )  = ;; 2 5 0
143139, 142eqtri 2316 . . . . . . 7  |-  ( (; 5
0  x.  5 )  +  0 )  = ;; 2 5 0
14481mul01i 9018 . . . . . . . 8  |-  (; 5 0  x.  0 )  =  0
145144, 136eqtri 2316 . . . . . . 7  |-  (; 5 0  x.  0 )  = ; 0 0
14652, 12, 42, 131, 42, 42, 143, 145decmul2c 10188 . . . . . 6  |-  (; 5 0  x. ; 5 0 )  = ;;; 2 5 0 0
147130, 146eqtr4i 2319 . . . . 5  |-  ( ( 1  x.  N )  + ;;; 1 2 4 1 )  =  (; 5 0  x. ; 5 0 )
14829, 1, 50, 51, 52, 48, 93, 113, 147mod2xi 13100 . . . 4  |-  ( ( 2 ^;; 3 0 6 )  mod 
N )  =  (;;; 1 2 4 1  mod 
N )
149 eqid 2296 . . . . 5  |- ;; 3 0 6  = ;; 3 0 6
150 eqid 2296 . . . . . 6  |- ; 3 0  = ; 3 0
1519dec0h 10156 . . . . . 6  |-  1  = ; 0 1
152 00id 9003 . . . . . . . 8  |-  ( 0  +  0 )  =  0
153110, 152oveq12i 5886 . . . . . . 7  |-  ( ( 2  x.  3 )  +  ( 0  +  0 ) )  =  ( 6  +  0 )
15466addid1i 9015 . . . . . . 7  |-  ( 6  +  0 )  =  6
155153, 154eqtri 2316 . . . . . 6  |-  ( ( 2  x.  3 )  +  ( 0  +  0 ) )  =  6
15661mul01i 9018 . . . . . . . 8  |-  ( 2  x.  0 )  =  0
157156oveq1i 5884 . . . . . . 7  |-  ( ( 2  x.  0 )  +  1 )  =  ( 0  +  1 )
158 0p1e1 9855 . . . . . . 7  |-  ( 0  +  1 )  =  1
159157, 158, 1513eqtri 2320 . . . . . 6  |-  ( ( 2  x.  0 )  +  1 )  = ; 0
1
16032, 42, 42, 9, 150, 151, 3, 9, 42, 155, 159decma2c 10180 . . . . 5  |-  ( ( 2  x. ; 3 0 )  +  1 )  = ; 6 1
1613, 43, 2, 149, 3, 9, 160, 68decmul2c 10188 . . . 4  |-  ( 2  x. ;; 3 0 6 )  = ;; 6 1 2
162 eqid 2296 . . . . . 6  |- ; 1 8  = ; 1 8
16311, 33, 58, 121decsuc 10163 . . . . . 6  |-  (;; 1 2 4  +  1 )  = ;; 1 2 5
16445nncni 9772 . . . . . . 7  |-  8  e.  CC
165164, 22, 15addcomli 9020 . . . . . 6  |-  ( 1  +  8 )  =  9
16647, 9, 9, 14, 116, 162, 163, 165decadd 10181 . . . . 5  |-  (;;; 1 2 4 1  + ; 1 8 )  = ;;; 1 2 5 9
167166, 10eqtr4i 2319 . . . 4  |-  (;;; 1 2 4 1  + ; 1 8 )  =  N
16837nn0cni 9993 . . . . . 6  |- ;; 3 2 4  e.  CC
169168addid2i 9016 . . . . 5  |-  ( 0  + ;; 3 2 4 )  = ;; 3 2 4
17073oveq1i 5884 . . . . 5  |-  ( ( 0  x.  N )  + ;; 3 2 4 )  =  ( 0  + ;; 3 2 4 )
1719, 14deccl 10154 . . . . . 6  |- ; 1 8  e.  NN0
1729, 33deccl 10154 . . . . . 6  |- ; 1 4  e.  NN0
173 eqid 2296 . . . . . . 7  |- ; 1 4  = ; 1 4
17422mulid1i 8855 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
175174, 123oveq12i 5886 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  ( 1  +  2 )
17661, 22, 78addcomli 9020 . . . . . . . 8  |-  ( 1  +  2 )  =  3
177175, 176eqtri 2316 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  3
178164mulid1i 8855 . . . . . . . . 9  |-  ( 8  x.  1 )  =  8
179178oveq1i 5884 . . . . . . . 8  |-  ( ( 8  x.  1 )  +  4 )  =  ( 8  +  4 )
180 8p4e12 10197 . . . . . . . 8  |-  ( 8  +  4 )  = ; 1
2
181179, 180eqtri 2316 . . . . . . 7  |-  ( ( 8  x.  1 )  +  4 )  = ; 1
2
1829, 14, 9, 33, 162, 173, 9, 3, 9, 177, 181decmac 10179 . . . . . 6  |-  ( (; 1
8  x.  1 )  + ; 1 4 )  = ; 3
2
183164mulid2i 8856 . . . . . . . . 9  |-  ( 1  x.  8 )  =  8
184183oveq1i 5884 . . . . . . . 8  |-  ( ( 1  x.  8 )  +  6 )  =  ( 8  +  6 )
185 8p6e14 10199 . . . . . . . 8  |-  ( 8  +  6 )  = ; 1
4
186184, 185eqtri 2316 . . . . . . 7  |-  ( ( 1  x.  8 )  +  6 )  = ; 1
4
187 8t8e64 10234 . . . . . . 7  |-  ( 8  x.  8 )  = ; 6
4
18814, 9, 14, 162, 33, 2, 186, 187decmul1c 10187 . . . . . 6  |-  (; 1 8  x.  8 )  = ;; 1 4 4
189171, 9, 14, 162, 33, 172, 182, 188decmul2c 10188 . . . . 5  |-  (; 1 8  x. ; 1 8 )  = ;; 3 2 4
190169, 170, 1893eqtr4i 2326 . . . 4  |-  ( ( 0  x.  N )  + ;; 3 2 4 )  =  (; 1
8  x. ; 1 8 )
1911, 44, 7, 46, 37, 48, 148, 161, 167, 190mod2xnegi 13102 . . 3  |-  ( ( 2 ^;; 6 1 2 )  mod 
N )  =  (;; 3 2 4  mod 
N )
192101259lem1 13145 . . 3  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
193 eqid 2296 . . . 4  |- ;; 6 1 2  = ;; 6 1 2
194 eqid 2296 . . . 4  |- ; 1 7  = ; 1 7
195 eqid 2296 . . . . 5  |- ; 6 1  = ; 6 1
1962, 9, 123, 195decsuc 10163 . . . 4  |-  (; 6 1  +  1 )  = ; 6 2
197 7p2e9 9883 . . . . 5  |-  ( 7  +  2 )  =  9
19860, 61, 197addcomli 9020 . . . 4  |-  ( 2  +  7 )  =  9
19930, 3, 9, 38, 193, 194, 196, 198decadd 10181 . . 3  |-  (;; 6 1 2  + ; 1 7 )  = ;; 6 2 9
20032, 9deccl 10154 . . . . 5  |- ; 3 1  e.  NN0
201 eqid 2296 . . . . . . 7  |- ; 3 1  = ; 3 1
202 3p2e5 9871 . . . . . . . . 9  |-  ( 3  +  2 )  =  5
203108, 61, 202addcomli 9020 . . . . . . . 8  |-  ( 2  +  3 )  =  5
2049, 3, 32, 122, 203decaddi 10184 . . . . . . 7  |-  (; 1 2  +  3 )  = ; 1 5
205 5p1e6 9866 . . . . . . 7  |-  ( 5  +  1 )  =  6
20611, 12, 32, 9, 120, 201, 204, 205decadd 10181 . . . . . 6  |-  (;; 1 2 5  + ; 3 1 )  = ;; 1 5 6
207123oveq1i 5884 . . . . . . . . 9  |-  ( ( 1  +  1 )  +  1 )  =  ( 2  +  1 )
208207, 78eqtri 2316 . . . . . . . 8  |-  ( ( 1  +  1 )  +  1 )  =  3
209 7p5e12 10193 . . . . . . . . 9  |-  ( 7  +  5 )  = ; 1
2
21060, 100, 209addcomli 9020 . . . . . . . 8  |-  ( 5  +  7 )  = ; 1
2
2119, 12, 9, 38, 95, 194, 208, 3, 210decaddc 10182 . . . . . . 7  |-  (; 1 5  + ; 1 7 )  = ; 3
2
212 eqid 2296 . . . . . . . 8  |- ; 3 4  = ; 3 4
213 7p3e10 9884 . . . . . . . . . 10  |-  ( 7  +  3 )  =  10
21460, 108, 213addcomli 9020 . . . . . . . . 9  |-  ( 3  +  7 )  =  10
215214, 89eqtri 2316 . . . . . . . 8  |-  ( 3  +  7 )  = ; 1
0
216108mulid1i 8855 . . . . . . . . . 10  |-  ( 3  x.  1 )  =  3
21722addid1i 9015 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
218216, 217oveq12i 5886 . . . . . . . . 9  |-  ( ( 3  x.  1 )  +  ( 1  +  0 ) )  =  ( 3  +  1 )
219 3p1e4 9864 . . . . . . . . 9  |-  ( 3  +  1 )  =  4
220218, 219eqtri 2316 . . . . . . . 8  |-  ( ( 3  x.  1 )  +  ( 1  +  0 ) )  =  4
221 4cn 9836 . . . . . . . . . . 11  |-  4  e.  CC
222221mulid1i 8855 . . . . . . . . . 10  |-  ( 4  x.  1 )  =  4
223222oveq1i 5884 . . . . . . . . 9  |-  ( ( 4  x.  1 )  +  0 )  =  ( 4  +  0 )
224221addid1i 9015 . . . . . . . . 9  |-  ( 4  +  0 )  =  4
22533dec0h 10156 . . . . . . . . 9  |-  4  = ; 0 4
226223, 224, 2253eqtri 2320 . . . . . . . 8  |-  ( ( 4  x.  1 )  +  0 )  = ; 0
4
22732, 33, 9, 42, 212, 215, 9, 33, 42, 220, 226decmac 10179 . . . . . . 7  |-  ( (; 3
4  x.  1 )  +  ( 3  +  7 ) )  = ; 4
4
2283dec0h 10156 . . . . . . . 8  |-  2  = ; 0 2
229109, 158oveq12i 5886 . . . . . . . . 9  |-  ( ( 3  x.  2 )  +  ( 0  +  1 ) )  =  ( 6  +  1 )
230 6p1e7 9867 . . . . . . . . 9  |-  ( 6  +  1 )  =  7
231229, 230eqtri 2316 . . . . . . . 8  |-  ( ( 3  x.  2 )  +  ( 0  +  1 ) )  =  7
232 4t2e8 9890 . . . . . . . . . 10  |-  ( 4  x.  2 )  =  8
233232oveq1i 5884 . . . . . . . . 9  |-  ( ( 4  x.  2 )  +  2 )  =  ( 8  +  2 )
234 8p2e10 9885 . . . . . . . . 9  |-  ( 8  +  2 )  =  10
235233, 234, 893eqtri 2320 . . . . . . . 8  |-  ( ( 4  x.  2 )  +  2 )  = ; 1
0
23632, 33, 42, 3, 212, 228, 3, 42, 9, 231, 235decmac 10179 . . . . . . 7  |-  ( (; 3
4  x.  2 )  +  2 )  = ; 7
0
2379, 3, 32, 3, 122, 211, 34, 42, 38, 227, 236decma2c 10180 . . . . . 6  |-  ( (; 3
4  x. ; 1 2 )  +  (; 1 5  + ; 1 7 ) )  = ;; 4 4 0
23861addid2i 9016 . . . . . . . . 9  |-  ( 0  +  2 )  =  2
239238oveq2i 5885 . . . . . . . 8  |-  ( ( 3  x.  5 )  +  ( 0  +  2 ) )  =  ( ( 3  x.  5 )  +  2 )
240 5t3e15 10214 . . . . . . . . . 10  |-  ( 5  x.  3 )  = ; 1
5
241100, 108, 240mulcomli 8860 . . . . . . . . 9  |-  ( 3  x.  5 )  = ; 1
5
242 5p2e7 9876 . . . . . . . . 9  |-  ( 5  +  2 )  =  7
2439, 12, 3, 241, 242decaddi 10184 . . . . . . . 8  |-  ( ( 3  x.  5 )  +  2 )  = ; 1
7
244239, 243eqtri 2316 . . . . . . 7  |-  ( ( 3  x.  5 )  +  ( 0  +  2 ) )  = ; 1
7
245 5t4e20 10215 . . . . . . . . 9  |-  ( 5  x.  4 )  = ; 2
0
246100, 221, 245mulcomli 8860 . . . . . . . 8  |-  ( 4  x.  5 )  = ; 2
0
24766addid2i 9016 . . . . . . . 8  |-  ( 0  +  6 )  =  6
2483, 42, 2, 246, 247decaddi 10184 . . . . . . 7  |-  ( ( 4  x.  5 )  +  6 )  = ; 2
6
24932, 33, 42, 2, 212, 111, 12, 2, 3, 244, 248decmac 10179 . . . . . 6  |-  ( (; 3
4  x.  5 )  +  6 )  = ;; 1 7 6
25011, 12, 49, 2, 120, 206, 34, 2, 39, 237, 249decma2c 10180 . . . . 5  |-  ( (; 3
4  x. ;; 1 2 5 )  +  (;; 1 2 5  + ; 3 1 ) )  = ;;; 4 4 0 6
25114dec0h 10156 . . . . . 6  |-  8  = ; 0 8
252221addid2i 9016 . . . . . . . 8  |-  ( 0  +  4 )  =  4
253252oveq2i 5885 . . . . . . 7  |-  ( ( 3  x.  9 )  +  ( 0  +  4 ) )  =  ( ( 3  x.  9 )  +  4 )
25427nncni 9772 . . . . . . . . 9  |-  9  e.  CC
255 9t3e27 10236 . . . . . . . . 9  |-  ( 9  x.  3 )  = ; 2
7
256254, 108, 255mulcomli 8860 . . . . . . . 8  |-  ( 3  x.  9 )  = ; 2
7
257 7p4e11 10192 . . . . . . . 8  |-  ( 7  +  4 )  = ; 1
1
2583, 38, 33, 256, 78, 9, 257decaddci 10185 . . . . . . 7  |-  ( ( 3  x.  9 )  +  4 )  = ; 3
1
259253, 258eqtri 2316 . . . . . 6  |-  ( ( 3  x.  9 )  +  ( 0  +  4 ) )  = ; 3
1
260 9t4e36 10237 . . . . . . . 8  |-  ( 9  x.  4 )  = ; 3
6
261254, 221, 260mulcomli 8860 . . . . . . 7  |-  ( 4  x.  9 )  = ; 3
6
262164, 66, 185addcomli 9020 . . . . . . 7  |-  ( 6  +  8 )  = ; 1
4
26332, 2, 14, 261, 219, 33, 262decaddci 10185 . . . . . 6  |-  ( ( 4  x.  9 )  +  8 )  = ; 4
4
26432, 33, 42, 14, 212, 251, 5, 33, 33, 259, 263decmac 10179 . . . . 5  |-  ( (; 3
4  x.  9 )  +  8 )  = ;; 3 1 4
26513, 5, 13, 14, 10, 25, 34, 33, 200, 250, 264decma2c 10180 . . . 4  |-  ( (; 3
4  x.  N )  +  ( N  - 
1 ) )  = ;;;; 4 4 0 6 4
266 eqid 2296 . . . . 5  |- ;; 1 3 6  = ;; 1 3 6
2679, 5deccl 10154 . . . . . 6  |- ; 1 9  e.  NN0
268267, 33deccl 10154 . . . . 5  |- ;; 1 9 4  e.  NN0
269 eqid 2296 . . . . . 6  |- ; 1 3  = ; 1 3
270 eqid 2296 . . . . . 6  |- ;; 1 9 4  = ;; 1 9 4
2715, 38deccl 10154 . . . . . 6  |- ; 9 7  e.  NN0
2729, 9deccl 10154 . . . . . . 7  |- ; 1 1  e.  NN0
273 eqid 2296 . . . . . . 7  |- ;; 3 2 4  = ;; 3 2 4
274 eqid 2296 . . . . . . . 8  |- ; 1 9  = ; 1 9
275 eqid 2296 . . . . . . . 8  |- ; 9 7  = ; 9 7
276254, 22, 129addcomli 9020 . . . . . . . . . 10  |-  ( 1  +  9 )  =  10
277276, 89eqtri 2316 . . . . . . . . 9  |-  ( 1  +  9 )  = ; 1
0
2789, 42, 158, 277decsuc 10163 . . . . . . . 8  |-  ( ( 1  +  9 )  +  1 )  = ; 1
1
279 9p7e16 10207 . . . . . . . 8  |-  ( 9  +  7 )  = ; 1
6
2809, 5, 5, 38, 274, 275, 278, 2, 279decaddc 10182 . . . . . . 7  |-  (; 1 9  + ; 9 7 )  = ;; 1 1 6
281 eqid 2296 . . . . . . . 8  |- ; 3 2  = ; 3 2
282 eqid 2296 . . . . . . . . 9  |- ; 1 1  = ; 1 1
2839, 9, 123, 282decsuc 10163 . . . . . . . 8  |-  (; 1 1  +  1 )  = ; 1 2
28496oveq1i 5884 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  2 )  =  ( 2  +  2 )
285284, 124, 2253eqtri 2320 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  2 )  = ; 0
4
28632, 3, 9, 3, 281, 283, 9, 33, 42, 220, 285decmac 10179 . . . . . . 7  |-  ( (; 3
2  x.  1 )  +  (; 1 1  +  1 ) )  = ; 4 4
287222oveq1i 5884 . . . . . . . 8  |-  ( ( 4  x.  1 )  +  6 )  =  ( 4  +  6 )
288 6p4e10 9882 . . . . . . . . 9  |-  ( 6  +  4 )  =  10
28966, 221, 288addcomli 9020 . . . . . . . 8  |-  ( 4  +  6 )  =  10
290287, 289, 893eqtri 2320 . . . . . . 7  |-  ( ( 4  x.  1 )  +  6 )  = ; 1
0
29136, 33, 272, 2, 273, 280, 9, 42, 9, 286, 290decmac 10179 . . . . . 6  |-  ( (;; 3 2 4  x.  1 )  +  (; 1
9  + ; 9 7 ) )  = ;; 4 4 0
292158, 151eqtri 2316 . . . . . . . 8  |-  ( 0  +  1 )  = ; 0
1
293 3t3e9 9889 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
294293, 152oveq12i 5886 . . . . . . . . 9  |-  ( ( 3  x.  3 )  +  ( 0  +  0 ) )  =  ( 9  +  0 )
295254addid1i 9015 . . . . . . . . 9  |-  ( 9  +  0 )  =  9
296294, 295eqtri 2316 . . . . . . . 8  |-  ( ( 3  x.  3 )  +  ( 0  +  0 ) )  =  9
297110oveq1i 5884 . . . . . . . . 9  |-  ( ( 2  x.  3 )  +  1 )  =  ( 6  +  1 )
29838dec0h 10156 . . . . . . . . 9  |-  7  = ; 0 7
299297, 230, 2983eqtri 2320 . . . . . . . 8  |-  ( ( 2  x.  3 )  +  1 )  = ; 0
7
30032, 3, 42, 9, 281, 292, 32, 38, 42, 296, 299decmac 10179 . . . . . . 7  |-  ( (; 3
2  x.  3 )  +  ( 0  +  1 ) )  = ; 9
7
301 4t3e12 10212 . . . . . . . 8  |-  ( 4  x.  3 )  = ; 1
2
302 4p2e6 9873 . . . . . . . . 9  |-  ( 4  +  2 )  =  6
303221, 61, 302addcomli 9020 . . . . . . . 8  |-  ( 2  +  4 )  =  6
3049, 3, 33, 301, 303decaddi 10184 . . . . . . 7  |-  ( ( 4  x.  3 )  +  4 )  = ; 1
6
30536, 33, 42, 33, 273, 225, 32, 2, 9, 300, 304decmac 10179 . . . . . 6  |-  ( (;; 3 2 4  x.  3 )  +  4 )  = ;; 9 7 6
3069, 32, 267, 33, 269, 270, 37, 2, 271, 291, 305decma2c 10180 . . . . 5  |-  ( (;; 3 2 4  x. ; 1
3 )  + ;; 1 9 4 )  = ;;; 4 4 0 6
307158oveq2i 5885 . . . . . . . 8  |-  ( ( 3  x.  6 )  +  ( 0  +  1 ) )  =  ( ( 3  x.  6 )  +  1 )
308 6t3e18 10218 . . . . . . . . . 10  |-  ( 6  x.  3 )  = ; 1
8
30966, 108, 308mulcomli 8860 . . . . . . . . 9  |-  ( 3  x.  6 )  = ; 1
8
3109, 14, 15, 309decsuc 10163 . . . . . . . 8  |-  ( ( 3  x.  6 )  +  1 )  = ; 1
9
311307, 310eqtri 2316 . . . . . . 7  |-  ( ( 3  x.  6 )  +  ( 0  +  1 ) )  = ; 1
9
3129, 3, 3, 68, 124decaddi 10184 . . . . . . 7  |-  ( ( 2  x.  6 )  +  2 )  = ; 1
4
31332, 3, 42, 3, 281, 228, 2, 33, 9, 311, 312decmac 10179 . . . . . 6  |-  ( (; 3
2  x.  6 )  +  2 )  = ;; 1 9 4
314 6t4e24 10219 . . . . . . 7  |-  ( 6  x.  4 )  = ; 2
4
31566, 221, 314mulcomli 8860 . . . . . 6  |-  ( 4  x.  6 )  = ; 2
4
3162, 36, 33, 273, 33, 3, 313, 315decmul1c 10187 . . . . 5  |-  (;; 3 2 4  x.  6 )  = ;;; 1 9 4 4
31737, 40, 2, 266, 33, 268, 306, 316decmul2c 10188 . . . 4  |-  (;; 3 2 4  x. ;; 1 3 6 )  = ;;;; 4 4 0 6 4
318265, 317eqtr4i 2319 . . 3  |-  ( (; 3
4  x.  N )  +  ( N  - 
1 ) )  =  (;; 3 2 4  x. ;; 1 3 6 )
31929, 1, 31, 35, 37, 26, 39, 41, 191, 192, 199, 318modxai 13099 . 2  |-  ( ( 2 ^;; 6 2 9 )  mod 
N )  =  ( ( N  -  1 )  mod  N )
320 eqid 2296 . . . 4  |- ;; 6 2 9  = ;; 6 2 9
321 eqid 2296 . . . . 5  |- ; 6 2  = ; 6 2
322152oveq2i 5885 . . . . . 6  |-  ( ( 2  x.  6 )  +  ( 0  +  0 ) )  =  ( ( 2  x.  6 )  +  0 )
32368oveq1i 5884 . . . . . 6  |-  ( ( 2  x.  6 )  +  0 )  =  (; 1 2  +  0 )
32411nn0cni 9993 . . . . . . 7  |- ; 1 2  e.  CC
325324addid1i 9015 . . . . . 6  |-  (; 1 2  +  0 )  = ; 1 2
326322, 323, 3253eqtri 2320 . . . . 5  |-  ( ( 2  x.  6 )  +  ( 0  +  0 ) )  = ; 1
2
32712dec0h 10156 . . . . . 6  |-  5  = ; 0 5
32886, 58, 3273eqtri 2320 . . . . 5  |-  ( ( 2  x.  2 )  +  1 )  = ; 0
5
3292, 3, 42, 9, 321, 151, 3, 12, 42, 326, 328decma2c 10180 . . . 4  |-  ( ( 2  x. ; 6 2 )  +  1 )  = ;; 1 2 5
330 9t2e18 10235 . . . . 5  |-  ( 9  x.  2 )  = ; 1
8
331254, 61, 330mulcomli 8860 . . . 4  |-  ( 2  x.  9 )  = ; 1
8
3323, 4, 5, 320, 14, 9, 329, 331decmul2c 10188 . . 3  |-  ( 2  x. ;; 6 2 9 )  = ;;; 1 2 5 8
333332, 25eqtr4i 2319 . 2  |-  ( 2  x. ;; 6 2 9 )  =  ( N  -  1 )
334 npcan 9076 . . 3  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
33572, 22, 334mp2an 653 . 2  |-  ( ( N  -  1 )  +  1 )  =  N
33673oveq1i 5884 . . 3  |-  ( ( 0  x.  N )  +  1 )  =  ( 0  +  1 )
337158, 336, 1743eqtr4i 2326 . 2  |-  ( ( 0  x.  N )  +  1 )  =  ( 1  x.  1 )
3381, 6, 7, 8, 9, 26, 319, 333, 335, 337mod2xnegi 13102 1  |-  ( ( 2 ^ ( N  -  1 ) )  mod  N )  =  ( 1  mod  N
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   NNcn 9762   2c2 9811   3c3 9812   4c4 9813   5c5 9814   6c6 9815   7c7 9816   8c8 9817   9c9 9818   10c10 9819   NN0cn0 9981  ;cdc 10140    mod cmo 10989   ^cexp 11120
This theorem is referenced by:  1259prm  13150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121
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