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Theorem 1259lem1 13405
Description: Lemma for 1259prm 13410. Calculate a power mod. In decimal, we calculate  2 ^ 1 6  =  5 2 N  +  6 8  ==  6 8 and  2 ^ 1 7  ==  6 8  x.  2  =  1 3 6 in this lemma. (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
1259lem1  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )

Proof of Theorem 1259lem1
StepHypRef Expression
1 1259prm.1 . . 3  |-  N  = ;;; 1 2 5 9
2 1nn0 10193 . . . . . 6  |-  1  e.  NN0
3 2nn0 10194 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10352 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10197 . . . . 5  |-  5  e.  NN0
64, 5deccl 10352 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10096 . . . 4  |-  9  e.  NN
86, 7decnncl 10351 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2474 . 2  |-  N  e.  NN
10 2nn 10089 . 2  |-  2  e.  NN
11 6nn0 10198 . . 3  |-  6  e.  NN0
122, 11deccl 10352 . 2  |- ; 1 6  e.  NN0
13 0z 10249 . 2  |-  0  e.  ZZ
14 8nn0 10200 . . 3  |-  8  e.  NN0
1511, 14deccl 10352 . 2  |- ; 6 8  e.  NN0
16 3nn0 10195 . . . 4  |-  3  e.  NN0
172, 16deccl 10352 . . 3  |- ; 1 3  e.  NN0
1817, 11deccl 10352 . 2  |- ;; 1 3 6  e.  NN0
195, 3deccl 10352 . . . 4  |- ; 5 2  e.  NN0
2019nn0zi 10262 . . 3  |- ; 5 2  e.  ZZ
213, 14nn0expcli 11362 . . 3  |-  ( 2 ^ 8 )  e. 
NN0
22 eqid 2404 . . 3  |-  ( ( 2 ^ 8 )  mod  N )  =  ( ( 2 ^ 8 )  mod  N
)
2314nn0cni 10189 . . . 4  |-  8  e.  CC
24 2cn 10026 . . . 4  |-  2  e.  CC
25 8t2e16 10426 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
2623, 24, 25mulcomli 9053 . . 3  |-  ( 2  x.  8 )  = ; 1
6
27 9nn0 10201 . . . . 5  |-  9  e.  NN0
28 eqid 2404 . . . . 5  |- ; 6 8  = ; 6 8
29 4nn0 10196 . . . . . 6  |-  4  e.  NN0
30 7nn0 10199 . . . . . 6  |-  7  e.  NN0
3129, 30deccl 10352 . . . . 5  |- ; 4 7  e.  NN0
32 eqid 2404 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
33 0nn0 10192 . . . . . . 7  |-  0  e.  NN0
3411dec0h 10354 . . . . . . 7  |-  6  = ; 0 6
35 eqid 2404 . . . . . . 7  |- ; 4 7  = ; 4 7
36 4cn 10030 . . . . . . . . . 10  |-  4  e.  CC
3736addid2i 9210 . . . . . . . . 9  |-  ( 0  +  4 )  =  4
3837oveq1i 6050 . . . . . . . 8  |-  ( ( 0  +  4 )  +  1 )  =  ( 4  +  1 )
39 4p1e5 10061 . . . . . . . 8  |-  ( 4  +  1 )  =  5
4038, 39eqtri 2424 . . . . . . 7  |-  ( ( 0  +  4 )  +  1 )  =  5
41 7nn 10094 . . . . . . . . 9  |-  7  e.  NN
4241nncni 9966 . . . . . . . 8  |-  7  e.  CC
43 6nn 10093 . . . . . . . . 9  |-  6  e.  NN
4443nncni 9966 . . . . . . . 8  |-  6  e.  CC
45 7p6e13 10392 . . . . . . . 8  |-  ( 7  +  6 )  = ; 1
3
4642, 44, 45addcomli 9214 . . . . . . 7  |-  ( 6  +  7 )  = ; 1
3
4733, 11, 29, 30, 34, 35, 40, 16, 46decaddc 10380 . . . . . 6  |-  ( 6  + ; 4 7 )  = ; 5
3
483, 11deccl 10352 . . . . . 6  |- ; 2 6  e.  NN0
49 eqid 2404 . . . . . . 7  |- ; 1 2  = ; 1 2
505dec0h 10354 . . . . . . . 8  |-  5  = ; 0 5
51 eqid 2404 . . . . . . . 8  |- ; 2 6  = ; 2 6
5224addid2i 9210 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
5352oveq1i 6050 . . . . . . . . 9  |-  ( ( 0  +  2 )  +  1 )  =  ( 2  +  1 )
54 2p1e3 10059 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
5553, 54eqtri 2424 . . . . . . . 8  |-  ( ( 0  +  2 )  +  1 )  =  3
56 5nn 10092 . . . . . . . . . 10  |-  5  e.  NN
5756nncni 9966 . . . . . . . . 9  |-  5  e.  CC
58 6p5e11 10388 . . . . . . . . 9  |-  ( 6  +  5 )  = ; 1
1
5944, 57, 58addcomli 9214 . . . . . . . 8  |-  ( 5  +  6 )  = ; 1
1
6033, 5, 3, 11, 50, 51, 55, 2, 59decaddc 10380 . . . . . . 7  |-  ( 5  + ; 2 6 )  = ; 3
1
61 10nn0 10202 . . . . . . 7  |-  10  e.  NN0
62 eqid 2404 . . . . . . . 8  |- ; 5 2  = ; 5 2
6316dec0h 10354 . . . . . . . . 9  |-  3  = ; 0 3
64 dec10 10368 . . . . . . . . 9  |-  10  = ; 1 0
65 ax-1cn 9004 . . . . . . . . . 10  |-  1  e.  CC
6665addid2i 9210 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
67 3cn 10028 . . . . . . . . . 10  |-  3  e.  CC
6867addid1i 9209 . . . . . . . . 9  |-  ( 3  +  0 )  =  3
6933, 16, 2, 33, 63, 64, 66, 68decadd 10379 . . . . . . . 8  |-  ( 3  +  10 )  = ; 1
3
7057mulid1i 9048 . . . . . . . . . 10  |-  ( 5  x.  1 )  =  5
7165addid1i 9209 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
7270, 71oveq12i 6052 . . . . . . . . 9  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  ( 5  +  1 )
73 5p1e6 10062 . . . . . . . . 9  |-  ( 5  +  1 )  =  6
7472, 73eqtri 2424 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  6
7524mulid1i 9048 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
7675oveq1i 6050 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  3 )  =  ( 2  +  3 )
77 3p2e5 10067 . . . . . . . . . 10  |-  ( 3  +  2 )  =  5
7867, 24, 77addcomli 9214 . . . . . . . . 9  |-  ( 2  +  3 )  =  5
7976, 78, 503eqtri 2428 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  3 )  = ; 0
5
805, 3, 2, 16, 62, 69, 2, 5, 33, 74, 79decmac 10377 . . . . . . 7  |-  ( (; 5
2  x.  1 )  +  ( 3  +  10 ) )  = ; 6
5
812dec0h 10354 . . . . . . . 8  |-  1  = ; 0 1
82 5t2e10 10087 . . . . . . . . . 10  |-  ( 5  x.  2 )  =  10
83 00id 9197 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
8482, 83oveq12i 6052 . . . . . . . . 9  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  ( 10  +  0 )
85 10nn 10097 . . . . . . . . . . 11  |-  10  e.  NN
8685nncni 9966 . . . . . . . . . 10  |-  10  e.  CC
8786addid1i 9209 . . . . . . . . 9  |-  ( 10  +  0 )  =  10
8884, 87eqtri 2424 . . . . . . . 8  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  10
89 2t2e4 10083 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
9089oveq1i 6050 . . . . . . . . 9  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
9190, 39, 503eqtri 2428 . . . . . . . 8  |-  ( ( 2  x.  2 )  +  1 )  = ; 0
5
925, 3, 33, 2, 62, 81, 3, 5, 33, 88, 91decmac 10377 . . . . . . 7  |-  ( (; 5
2  x.  2 )  +  1 )  = ; 10 5
932, 3, 16, 2, 49, 60, 19, 5, 61, 80, 92decma2c 10378 . . . . . 6  |-  ( (; 5
2  x. ; 1 2 )  +  ( 5  + ; 2 6 ) )  = ;; 6 5 5
9466oveq2i 6051 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  =  ( ( 5  x.  5 )  +  1 )
95 5t5e25 10414 . . . . . . . . 9  |-  ( 5  x.  5 )  = ; 2
5
963, 5, 73, 95decsuc 10361 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  1 )  = ; 2
6
9794, 96eqtri 2424 . . . . . . 7  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  = ; 2
6
9857, 24, 82mulcomli 9053 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
9998, 64eqtri 2424 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
10067addid2i 9210 . . . . . . . 8  |-  ( 0  +  3 )  =  3
1012, 33, 16, 99, 100decaddi 10382 . . . . . . 7  |-  ( ( 2  x.  5 )  +  3 )  = ; 1
3
1025, 3, 33, 16, 62, 63, 5, 16, 2, 97, 101decmac 10377 . . . . . 6  |-  ( (; 5
2  x.  5 )  +  3 )  = ;; 2 6 3
1034, 5, 5, 16, 32, 47, 19, 16, 48, 93, 102decma2c 10378 . . . . 5  |-  ( (; 5
2  x. ;; 1 2 5 )  +  ( 6  + ; 4 7 ) )  = ;;; 6 5 5 3
10414dec0h 10354 . . . . . 6  |-  8  = ; 0 8
10552oveq2i 6051 . . . . . . 7  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  =  ( ( 5  x.  9 )  +  2 )
1067nncni 9966 . . . . . . . . 9  |-  9  e.  CC
107 9t5e45 10436 . . . . . . . . 9  |-  ( 9  x.  5 )  = ; 4
5
108106, 57, 107mulcomli 9053 . . . . . . . 8  |-  ( 5  x.  9 )  = ; 4
5
109 5p2e7 10072 . . . . . . . 8  |-  ( 5  +  2 )  =  7
11029, 5, 3, 108, 109decaddi 10382 . . . . . . 7  |-  ( ( 5  x.  9 )  +  2 )  = ; 4
7
111105, 110eqtri 2424 . . . . . 6  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  = ; 4
7
112 9t2e18 10433 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
113106, 24, 112mulcomli 9053 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
114 1p1e2 10050 . . . . . . 7  |-  ( 1  +  1 )  =  2
115 8p8e16 10399 . . . . . . 7  |-  ( 8  +  8 )  = ; 1
6
1162, 14, 14, 113, 114, 11, 115decaddci 10383 . . . . . 6  |-  ( ( 2  x.  9 )  +  8 )  = ; 2
6
1175, 3, 33, 14, 62, 104, 27, 11, 3, 111, 116decmac 10377 . . . . 5  |-  ( (; 5
2  x.  9 )  +  8 )  = ;; 4 7 6
1186, 27, 11, 14, 1, 28, 19, 11, 31, 103, 117decma2c 10378 . . . 4  |-  ( (; 5
2  x.  N )  + ; 6 8 )  = ;;;; 6 5 5 3 6
119 2exp16 13379 . . . 4  |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
120 eqid 2404 . . . . 5  |-  ( 2 ^ 8 )  =  ( 2 ^ 8 )
121 eqid 2404 . . . . 5  |-  ( ( 2 ^ 8 )  x.  ( 2 ^ 8 ) )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1223, 14, 26, 120, 121numexp2x 13370 . . . 4  |-  ( 2 ^; 1 6 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
123118, 119, 1223eqtr2i 2430 . . 3  |-  ( (; 5
2  x.  N )  + ; 6 8 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1249, 10, 14, 20, 21, 15, 22, 26, 123mod2xi 13360 . 2  |-  ( ( 2 ^; 1 6 )  mod 
N )  =  (; 6
8  mod  N )
125 6p1e7 10063 . . 3  |-  ( 6  +  1 )  =  7
126 eqid 2404 . . 3  |- ; 1 6  = ; 1 6
1272, 11, 125, 126decsuc 10361 . 2  |-  (; 1 6  +  1 )  = ; 1 7
12818nn0cni 10189 . . . 4  |- ;; 1 3 6  e.  CC
129128addid2i 9210 . . 3  |-  ( 0  + ;; 1 3 6 )  = ;; 1 3 6
1309nncni 9966 . . . . 5  |-  N  e.  CC
131130mul02i 9211 . . . 4  |-  ( 0  x.  N )  =  0
132131oveq1i 6050 . . 3  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  ( 0  + ;; 1 3 6 )
133 6t2e12 10415 . . . . 5  |-  ( 6  x.  2 )  = ; 1
2
1342, 3, 54, 133decsuc 10361 . . . 4  |-  ( ( 6  x.  2 )  +  1 )  = ; 1
3
1353, 11, 14, 28, 11, 2, 134, 25decmul1c 10385 . . 3  |-  (; 6 8  x.  2 )  = ;; 1 3 6
136129, 132, 1353eqtr4i 2434 . 2  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  (; 6
8  x.  2 )
1379, 10, 12, 13, 15, 18, 124, 127, 136modxp1i 13361 1  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
Colors of variables: wff set class
Syntax hints:    = wceq 1649  (class class class)co 6040   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951   NNcn 9956   2c2 10005   3c3 10006   4c4 10007   5c5 10008   6c6 10009   7c7 10010   8c8 10011   9c9 10012   10c10 10013  ;cdc 10338    mod cmo 11205   ^cexp 11337
This theorem is referenced by:  1259lem2  13406  1259lem4  13408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338
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