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Theorem 1259lem2 13453
Description: Lemma for 1259prm 13457. Calculate a power mod. In decimal, we calculate  2 ^ 3 4  =  ( 2 ^ 1 7 ) ^ 2  ==  1
3 6 ^ 2  ==  1 4 N  +  8 7 0. (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
1259lem2  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )

Proof of Theorem 1259lem2
StepHypRef Expression
1 1259prm.1 . . 3  |-  N  = ;;; 1 2 5 9
2 1nn0 10239 . . . . . 6  |-  1  e.  NN0
3 2nn0 10240 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10398 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10243 . . . . 5  |-  5  e.  NN0
64, 5deccl 10398 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10142 . . . 4  |-  9  e.  NN
86, 7decnncl 10397 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2508 . 2  |-  N  e.  NN
10 2nn 10135 . 2  |-  2  e.  NN
11 7nn0 10245 . . 3  |-  7  e.  NN0
122, 11deccl 10398 . 2  |- ; 1 7  e.  NN0
13 4nn0 10242 . . . 4  |-  4  e.  NN0
142, 13deccl 10398 . . 3  |- ; 1 4  e.  NN0
1514nn0zi 10308 . 2  |- ; 1 4  e.  ZZ
16 3nn0 10241 . . . 4  |-  3  e.  NN0
172, 16deccl 10398 . . 3  |- ; 1 3  e.  NN0
18 6nn0 10244 . . 3  |-  6  e.  NN0
1917, 18deccl 10398 . 2  |- ;; 1 3 6  e.  NN0
20 8nn0 10246 . . . 4  |-  8  e.  NN0
2120, 11deccl 10398 . . 3  |- ; 8 7  e.  NN0
22 0nn0 10238 . . 3  |-  0  e.  NN0
2321, 22deccl 10398 . 2  |- ;; 8 7 0  e.  NN0
2411259lem1 13452 . 2  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
25 eqid 2438 . . 3  |- ; 1 7  = ; 1 7
26 2cn 10072 . . . . . 6  |-  2  e.  CC
2726mulid1i 9094 . . . . 5  |-  ( 2  x.  1 )  =  2
2827oveq1i 6093 . . . 4  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
29 2p1e3 10105 . . . 4  |-  ( 2  +  1 )  =  3
3028, 29eqtri 2458 . . 3  |-  ( ( 2  x.  1 )  +  1 )  =  3
31 7nn 10140 . . . . 5  |-  7  e.  NN
3231nncni 10012 . . . 4  |-  7  e.  CC
33 7t2e14 10466 . . . 4  |-  ( 7  x.  2 )  = ; 1
4
3432, 26, 33mulcomli 9099 . . 3  |-  ( 2  x.  7 )  = ; 1
4
353, 2, 11, 25, 13, 2, 30, 34decmul2c 10432 . 2  |-  ( 2  x. ; 1 7 )  = ; 3
4
36 9nn0 10247 . . . 4  |-  9  e.  NN0
37 eqid 2438 . . . 4  |- ;; 8 7 0  = ;; 8 7 0
38 eqid 2438 . . . . 5  |- ;; 1 2 5  = ;; 1 2 5
39 eqid 2438 . . . . . 6  |- ; 8 7  = ; 8 7
40 eqid 2438 . . . . . 6  |- ; 1 2  = ; 1 2
41 8p1e9 10111 . . . . . 6  |-  ( 8  +  1 )  =  9
42 7p2e9 10125 . . . . . 6  |-  ( 7  +  2 )  =  9
4320, 11, 2, 3, 39, 40, 41, 42decadd 10425 . . . . 5  |-  (; 8 7  + ; 1 2 )  = ; 9
9
44 9p7e16 10451 . . . . . 6  |-  ( 9  +  7 )  = ; 1
6
45 eqid 2438 . . . . . . 7  |- ; 1 4  = ; 1 4
46 3cn 10074 . . . . . . . . 9  |-  3  e.  CC
47 ax-1cn 9050 . . . . . . . . 9  |-  1  e.  CC
48 3p1e4 10106 . . . . . . . . 9  |-  ( 3  +  1 )  =  4
4946, 47, 48addcomli 9260 . . . . . . . 8  |-  ( 1  +  3 )  =  4
5013dec0h 10400 . . . . . . . 8  |-  4  = ; 0 4
5149, 50eqtri 2458 . . . . . . 7  |-  ( 1  +  3 )  = ; 0
4
5247mulid1i 9094 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
53 00id 9243 . . . . . . . . 9  |-  ( 0  +  0 )  =  0
5452, 53oveq12i 6095 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
5547addid1i 9255 . . . . . . . 8  |-  ( 1  +  0 )  =  1
5654, 55eqtri 2458 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
57 4cn 10076 . . . . . . . . . 10  |-  4  e.  CC
5857mulid1i 9094 . . . . . . . . 9  |-  ( 4  x.  1 )  =  4
5958oveq1i 6093 . . . . . . . 8  |-  ( ( 4  x.  1 )  +  4 )  =  ( 4  +  4 )
60 4p4e8 10117 . . . . . . . 8  |-  ( 4  +  4 )  =  8
6120dec0h 10400 . . . . . . . 8  |-  8  = ; 0 8
6259, 60, 613eqtri 2462 . . . . . . 7  |-  ( ( 4  x.  1 )  +  4 )  = ; 0
8
632, 13, 22, 13, 45, 51, 2, 20, 22, 56, 62decmac 10423 . . . . . 6  |-  ( (; 1
4  x.  1 )  +  ( 1  +  3 ) )  = ; 1
8
6418dec0h 10400 . . . . . . 7  |-  6  = ; 0 6
6526mulid2i 9095 . . . . . . . . 9  |-  ( 1  x.  2 )  =  2
6647addid2i 9256 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
6765, 66oveq12i 6095 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
6867, 29eqtri 2458 . . . . . . 7  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  3
69 4t2e8 10132 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
7069oveq1i 6093 . . . . . . . 8  |-  ( ( 4  x.  2 )  +  6 )  =  ( 8  +  6 )
71 8p6e14 10443 . . . . . . . 8  |-  ( 8  +  6 )  = ; 1
4
7270, 71eqtri 2458 . . . . . . 7  |-  ( ( 4  x.  2 )  +  6 )  = ; 1
4
732, 13, 22, 18, 45, 64, 3, 13, 2, 68, 72decmac 10423 . . . . . 6  |-  ( (; 1
4  x.  2 )  +  6 )  = ; 3
4
742, 3, 2, 18, 40, 44, 14, 13, 16, 63, 73decma2c 10424 . . . . 5  |-  ( (; 1
4  x. ; 1 2 )  +  ( 9  +  7 ) )  = ;; 1 8 4
7536dec0h 10400 . . . . . 6  |-  9  = ; 0 9
76 5nn 10138 . . . . . . . . . 10  |-  5  e.  NN
7776nncni 10012 . . . . . . . . 9  |-  5  e.  CC
7877mulid2i 9095 . . . . . . . 8  |-  ( 1  x.  5 )  =  5
7926addid2i 9256 . . . . . . . 8  |-  ( 0  +  2 )  =  2
8078, 79oveq12i 6095 . . . . . . 7  |-  ( ( 1  x.  5 )  +  ( 0  +  2 ) )  =  ( 5  +  2 )
81 5p2e7 10118 . . . . . . 7  |-  ( 5  +  2 )  =  7
8280, 81eqtri 2458 . . . . . 6  |-  ( ( 1  x.  5 )  +  ( 0  +  2 ) )  =  7
83 5t4e20 10459 . . . . . . . 8  |-  ( 5  x.  4 )  = ; 2
0
8477, 57, 83mulcomli 9099 . . . . . . 7  |-  ( 4  x.  5 )  = ; 2
0
857nncni 10012 . . . . . . . 8  |-  9  e.  CC
8685addid2i 9256 . . . . . . 7  |-  ( 0  +  9 )  =  9
873, 22, 36, 84, 86decaddi 10428 . . . . . 6  |-  ( ( 4  x.  5 )  +  9 )  = ; 2
9
882, 13, 22, 36, 45, 75, 5, 36, 3, 82, 87decmac 10423 . . . . 5  |-  ( (; 1
4  x.  5 )  +  9 )  = ; 7
9
894, 5, 36, 36, 38, 43, 14, 36, 11, 74, 88decma2c 10424 . . . 4  |-  ( (; 1
4  x. ;; 1 2 5 )  +  (; 8 7  + ; 1 2 ) )  = ;;; 1 8 4 9
9085mulid2i 9095 . . . . . . . . 9  |-  ( 1  x.  9 )  =  9
9190oveq1i 6093 . . . . . . . 8  |-  ( ( 1  x.  9 )  +  3 )  =  ( 9  +  3 )
92 9p3e12 10447 . . . . . . . 8  |-  ( 9  +  3 )  = ; 1
2
9391, 92eqtri 2458 . . . . . . 7  |-  ( ( 1  x.  9 )  +  3 )  = ; 1
2
94 9t4e36 10481 . . . . . . . 8  |-  ( 9  x.  4 )  = ; 3
6
9585, 57, 94mulcomli 9099 . . . . . . 7  |-  ( 4  x.  9 )  = ; 3
6
9636, 2, 13, 45, 18, 16, 93, 95decmul1c 10431 . . . . . 6  |-  (; 1 4  x.  9 )  = ;; 1 2 6
9796oveq1i 6093 . . . . 5  |-  ( (; 1
4  x.  9 )  +  0 )  =  (;; 1 2 6  +  0 )
984, 18deccl 10398 . . . . . . 7  |- ;; 1 2 6  e.  NN0
9998nn0cni 10235 . . . . . 6  |- ;; 1 2 6  e.  CC
10099addid1i 9255 . . . . 5  |-  (;; 1 2 6  +  0 )  = ;; 1 2 6
10197, 100eqtri 2458 . . . 4  |-  ( (; 1
4  x.  9 )  +  0 )  = ;; 1 2 6
1026, 36, 21, 22, 1, 37, 14, 18, 4, 89, 101decma2c 10424 . . 3  |-  ( (; 1
4  x.  N )  + ;; 8 7 0 )  = ;;;; 1 8 4 9 6
103 eqid 2438 . . . 4  |- ;; 1 3 6  = ;; 1 3 6
10420, 2deccl 10398 . . . 4  |- ; 8 1  e.  NN0
105 eqid 2438 . . . . 5  |- ; 1 3  = ; 1 3
106 eqid 2438 . . . . 5  |- ; 8 1  = ; 8 1
10713, 22deccl 10398 . . . . 5  |- ; 4 0  e.  NN0
108 eqid 2438 . . . . . . 7  |- ; 4 0  = ; 4 0
10957addid2i 9256 . . . . . . 7  |-  ( 0  +  4 )  =  4
110 8nn 10141 . . . . . . . . 9  |-  8  e.  NN
111110nncni 10012 . . . . . . . 8  |-  8  e.  CC
112111addid1i 9255 . . . . . . 7  |-  ( 8  +  0 )  =  8
11322, 20, 13, 22, 61, 108, 109, 112decadd 10425 . . . . . 6  |-  ( 8  + ; 4 0 )  = ; 4
8
114 4p1e5 10107 . . . . . . . 8  |-  ( 4  +  1 )  =  5
1155dec0h 10400 . . . . . . . 8  |-  5  = ; 0 5
116114, 115eqtri 2458 . . . . . . 7  |-  ( 4  +  1 )  = ; 0
5
11746mulid1i 9094 . . . . . . . . 9  |-  ( 3  x.  1 )  =  3
118117oveq1i 6093 . . . . . . . 8  |-  ( ( 3  x.  1 )  +  5 )  =  ( 3  +  5 )
119 5p3e8 10119 . . . . . . . . 9  |-  ( 5  +  3 )  =  8
12077, 46, 119addcomli 9260 . . . . . . . 8  |-  ( 3  +  5 )  =  8
121118, 120, 613eqtri 2462 . . . . . . 7  |-  ( ( 3  x.  1 )  +  5 )  = ; 0
8
1222, 16, 22, 5, 105, 116, 2, 20, 22, 56, 121decmac 10423 . . . . . 6  |-  ( (; 1
3  x.  1 )  +  ( 4  +  1 ) )  = ; 1
8
123 6nn 10139 . . . . . . . . . 10  |-  6  e.  NN
124123nncni 10012 . . . . . . . . 9  |-  6  e.  CC
125124mulid1i 9094 . . . . . . . 8  |-  ( 6  x.  1 )  =  6
126125oveq1i 6093 . . . . . . 7  |-  ( ( 6  x.  1 )  +  8 )  =  ( 6  +  8 )
127111, 124, 71addcomli 9260 . . . . . . 7  |-  ( 6  +  8 )  = ; 1
4
128126, 127eqtri 2458 . . . . . 6  |-  ( ( 6  x.  1 )  +  8 )  = ; 1
4
12917, 18, 13, 20, 103, 113, 2, 13, 2, 122, 128decmac 10423 . . . . 5  |-  ( (;; 1 3 6  x.  1 )  +  ( 8  + ; 4 0 ) )  = ;; 1 8 4
1302dec0h 10400 . . . . . 6  |-  1  = ; 0 1
13166, 130eqtri 2458 . . . . . . 7  |-  ( 0  +  1 )  = ; 0
1
13246mulid2i 9095 . . . . . . . . 9  |-  ( 1  x.  3 )  =  3
133132, 66oveq12i 6095 . . . . . . . 8  |-  ( ( 1  x.  3 )  +  ( 0  +  1 ) )  =  ( 3  +  1 )
134133, 48eqtri 2458 . . . . . . 7  |-  ( ( 1  x.  3 )  +  ( 0  +  1 ) )  =  4
135 3t3e9 10131 . . . . . . . . 9  |-  ( 3  x.  3 )  =  9
136135oveq1i 6093 . . . . . . . 8  |-  ( ( 3  x.  3 )  +  1 )  =  ( 9  +  1 )
137 9p1e10 10112 . . . . . . . 8  |-  ( 9  +  1 )  =  10
138 dec10 10414 . . . . . . . 8  |-  10  = ; 1 0
139136, 137, 1383eqtri 2462 . . . . . . 7  |-  ( ( 3  x.  3 )  +  1 )  = ; 1
0
1402, 16, 22, 2, 105, 131, 16, 22, 2, 134, 139decmac 10423 . . . . . 6  |-  ( (; 1
3  x.  3 )  +  ( 0  +  1 ) )  = ; 4
0
141 6t3e18 10462 . . . . . . 7  |-  ( 6  x.  3 )  = ; 1
8
1422, 20, 2, 141, 41decaddi 10428 . . . . . 6  |-  ( ( 6  x.  3 )  +  1 )  = ; 1
9
14317, 18, 22, 2, 103, 130, 16, 36, 2, 140, 142decmac 10423 . . . . 5  |-  ( (;; 1 3 6  x.  3 )  +  1 )  = ;; 4 0 9
1442, 16, 20, 2, 105, 106, 19, 36, 107, 129, 143decma2c 10424 . . . 4  |-  ( (;; 1 3 6  x. ; 1
3 )  + ; 8 1 )  = ;;; 1 8 4 9
14516dec0h 10400 . . . . . 6  |-  3  = ; 0 3
146124mulid2i 9095 . . . . . . . 8  |-  ( 1  x.  6 )  =  6
147146, 79oveq12i 6095 . . . . . . 7  |-  ( ( 1  x.  6 )  +  ( 0  +  2 ) )  =  ( 6  +  2 )
148 6p2e8 10122 . . . . . . 7  |-  ( 6  +  2 )  =  8
149147, 148eqtri 2458 . . . . . 6  |-  ( ( 1  x.  6 )  +  ( 0  +  2 ) )  =  8
150124, 46, 141mulcomli 9099 . . . . . . 7  |-  ( 3  x.  6 )  = ; 1
8
151 1p1e2 10096 . . . . . . 7  |-  ( 1  +  1 )  =  2
152 8p3e11 10440 . . . . . . 7  |-  ( 8  +  3 )  = ; 1
1
1532, 20, 16, 150, 151, 2, 152decaddci 10429 . . . . . 6  |-  ( ( 3  x.  6 )  +  3 )  = ; 2
1
1542, 16, 22, 16, 105, 145, 18, 2, 3, 149, 153decmac 10423 . . . . 5  |-  ( (; 1
3  x.  6 )  +  3 )  = ; 8
1
155 6t6e36 10465 . . . . 5  |-  ( 6  x.  6 )  = ; 3
6
15618, 17, 18, 103, 18, 16, 154, 155decmul1c 10431 . . . 4  |-  (;; 1 3 6  x.  6 )  = ;; 8 1 6
15719, 17, 18, 103, 18, 104, 144, 156decmul2c 10432 . . 3  |-  (;; 1 3 6  x. ;; 1 3 6 )  = ;;;; 1 8 4 9 6
158102, 157eqtr4i 2461 . 2  |-  ( (; 1
4  x.  N )  + ;; 8 7 0 )  =  (;; 1 3 6  x. ;; 1 3 6 )
1599, 10, 12, 15, 19, 23, 24, 35, 158mod2xi 13407 1  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  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:  1259lem3  13454  1259lem5  13456
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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