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Theorem 1259lem1 13125
Description: Lemma for 1259prm 13130. 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 9977 . . . . . 6  |-  1  e.  NN0
3 2nn0 9978 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10134 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 9981 . . . . 5  |-  5  e.  NN0
64, 5deccl 10134 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 9880 . . . 4  |-  9  e.  NN
86, 7decnncl 10133 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2354 . 2  |-  N  e.  NN
10 2nn 9873 . 2  |-  2  e.  NN
11 6nn0 9982 . . 3  |-  6  e.  NN0
122, 11deccl 10134 . 2  |- ; 1 6  e.  NN0
13 0z 10031 . 2  |-  0  e.  ZZ
14 8nn0 9984 . . 3  |-  8  e.  NN0
1511, 14deccl 10134 . 2  |- ; 6 8  e.  NN0
16 3nn0 9979 . . . 4  |-  3  e.  NN0
172, 16deccl 10134 . . 3  |- ; 1 3  e.  NN0
1817, 11deccl 10134 . 2  |- ;; 1 3 6  e.  NN0
195, 3deccl 10134 . . . 4  |- ; 5 2  e.  NN0
2019nn0zi 10044 . . 3  |- ; 5 2  e.  ZZ
213, 14nn0expcli 11125 . . 3  |-  ( 2 ^ 8 )  e. 
NN0
22 eqid 2284 . . 3  |-  ( ( 2 ^ 8 )  mod  N )  =  ( ( 2 ^ 8 )  mod  N
)
2314nn0cni 9973 . . . 4  |-  8  e.  CC
24 2cn 9812 . . . 4  |-  2  e.  CC
25 8t2e16 10208 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
2623, 24, 25mulcomli 8840 . . 3  |-  ( 2  x.  8 )  = ; 1
6
27 9nn0 9985 . . . . 5  |-  9  e.  NN0
28 eqid 2284 . . . . 5  |- ; 6 8  = ; 6 8
29 4nn0 9980 . . . . . 6  |-  4  e.  NN0
30 7nn0 9983 . . . . . 6  |-  7  e.  NN0
3129, 30deccl 10134 . . . . 5  |- ; 4 7  e.  NN0
32 eqid 2284 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
33 0nn0 9976 . . . . . . 7  |-  0  e.  NN0
3411dec0h 10136 . . . . . . 7  |-  6  = ; 0 6
35 eqid 2284 . . . . . . 7  |- ; 4 7  = ; 4 7
36 4cn 9816 . . . . . . . . . 10  |-  4  e.  CC
3736addid2i 8996 . . . . . . . . 9  |-  ( 0  +  4 )  =  4
3837oveq1i 5830 . . . . . . . 8  |-  ( ( 0  +  4 )  +  1 )  =  ( 4  +  1 )
39 4p1e5 9845 . . . . . . . 8  |-  ( 4  +  1 )  =  5
4038, 39eqtri 2304 . . . . . . 7  |-  ( ( 0  +  4 )  +  1 )  =  5
41 7nn 9878 . . . . . . . . 9  |-  7  e.  NN
4241nncni 9752 . . . . . . . 8  |-  7  e.  CC
43 6nn 9877 . . . . . . . . 9  |-  6  e.  NN
4443nncni 9752 . . . . . . . 8  |-  6  e.  CC
45 7p6e13 10174 . . . . . . . 8  |-  ( 7  +  6 )  = ; 1
3
4642, 44, 45addcomli 9000 . . . . . . 7  |-  ( 6  +  7 )  = ; 1
3
4733, 11, 29, 30, 34, 35, 40, 16, 46decaddc 10162 . . . . . 6  |-  ( 6  + ; 4 7 )  = ; 5
3
483, 11deccl 10134 . . . . . 6  |- ; 2 6  e.  NN0
49 eqid 2284 . . . . . . 7  |- ; 1 2  = ; 1 2
505dec0h 10136 . . . . . . . 8  |-  5  = ; 0 5
51 eqid 2284 . . . . . . . 8  |- ; 2 6  = ; 2 6
5224addid2i 8996 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
5352oveq1i 5830 . . . . . . . . 9  |-  ( ( 0  +  2 )  +  1 )  =  ( 2  +  1 )
54 2p1e3 9843 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
5553, 54eqtri 2304 . . . . . . . 8  |-  ( ( 0  +  2 )  +  1 )  =  3
56 5nn 9876 . . . . . . . . . 10  |-  5  e.  NN
5756nncni 9752 . . . . . . . . 9  |-  5  e.  CC
58 6p5e11 10170 . . . . . . . . 9  |-  ( 6  +  5 )  = ; 1
1
5944, 57, 58addcomli 9000 . . . . . . . 8  |-  ( 5  +  6 )  = ; 1
1
6033, 5, 3, 11, 50, 51, 55, 2, 59decaddc 10162 . . . . . . 7  |-  ( 5  + ; 2 6 )  = ; 3
1
61 10nn0 9986 . . . . . . 7  |-  10  e.  NN0
62 eqid 2284 . . . . . . . 8  |- ; 5 2  = ; 5 2
6316dec0h 10136 . . . . . . . . 9  |-  3  = ; 0 3
64 dec10 10150 . . . . . . . . 9  |-  10  = ; 1 0
65 ax-1cn 8791 . . . . . . . . . 10  |-  1  e.  CC
6665addid2i 8996 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
67 3cn 9814 . . . . . . . . . 10  |-  3  e.  CC
6867addid1i 8995 . . . . . . . . 9  |-  ( 3  +  0 )  =  3
6933, 16, 2, 33, 63, 64, 66, 68decadd 10161 . . . . . . . 8  |-  ( 3  +  10 )  = ; 1
3
7057mulid1i 8835 . . . . . . . . . 10  |-  ( 5  x.  1 )  =  5
7165addid1i 8995 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
7270, 71oveq12i 5832 . . . . . . . . 9  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  ( 5  +  1 )
73 5p1e6 9846 . . . . . . . . 9  |-  ( 5  +  1 )  =  6
7472, 73eqtri 2304 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  6
7524mulid1i 8835 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
7675oveq1i 5830 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  3 )  =  ( 2  +  3 )
77 3p2e5 9851 . . . . . . . . . 10  |-  ( 3  +  2 )  =  5
7867, 24, 77addcomli 9000 . . . . . . . . 9  |-  ( 2  +  3 )  =  5
7976, 78, 503eqtri 2308 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  3 )  = ; 0
5
805, 3, 2, 16, 62, 69, 2, 5, 33, 74, 79decmac 10159 . . . . . . 7  |-  ( (; 5
2  x.  1 )  +  ( 3  +  10 ) )  = ; 6
5
812dec0h 10136 . . . . . . . 8  |-  1  = ; 0 1
82 5t2e10 9871 . . . . . . . . . 10  |-  ( 5  x.  2 )  =  10
83 00id 8983 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
8482, 83oveq12i 5832 . . . . . . . . 9  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  ( 10  +  0 )
85 10nn 9881 . . . . . . . . . . 11  |-  10  e.  NN
8685nncni 9752 . . . . . . . . . 10  |-  10  e.  CC
8786addid1i 8995 . . . . . . . . 9  |-  ( 10  +  0 )  =  10
8884, 87eqtri 2304 . . . . . . . 8  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  10
89 2t2e4 9867 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
9089oveq1i 5830 . . . . . . . . 9  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
9190, 39, 503eqtri 2308 . . . . . . . 8  |-  ( ( 2  x.  2 )  +  1 )  = ; 0
5
925, 3, 33, 2, 62, 81, 3, 5, 33, 88, 91decmac 10159 . . . . . . 7  |-  ( (; 5
2  x.  2 )  +  1 )  = ; 10 5
932, 3, 16, 2, 49, 60, 19, 5, 61, 80, 92decma2c 10160 . . . . . 6  |-  ( (; 5
2  x. ; 1 2 )  +  ( 5  + ; 2 6 ) )  = ;; 6 5 5
9466oveq2i 5831 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  =  ( ( 5  x.  5 )  +  1 )
95 5t5e25 10196 . . . . . . . . 9  |-  ( 5  x.  5 )  = ; 2
5
963, 5, 73, 95decsuc 10143 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  1 )  = ; 2
6
9794, 96eqtri 2304 . . . . . . 7  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  = ; 2
6
9857, 24, 82mulcomli 8840 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
9998, 64eqtri 2304 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
10067addid2i 8996 . . . . . . . 8  |-  ( 0  +  3 )  =  3
1012, 33, 16, 99, 100decaddi 10164 . . . . . . 7  |-  ( ( 2  x.  5 )  +  3 )  = ; 1
3
1025, 3, 33, 16, 62, 63, 5, 16, 2, 97, 101decmac 10159 . . . . . 6  |-  ( (; 5
2  x.  5 )  +  3 )  = ;; 2 6 3
1034, 5, 5, 16, 32, 47, 19, 16, 48, 93, 102decma2c 10160 . . . . 5  |-  ( (; 5
2  x. ;; 1 2 5 )  +  ( 6  + ; 4 7 ) )  = ;;; 6 5 5 3
10414dec0h 10136 . . . . . 6  |-  8  = ; 0 8
10552oveq2i 5831 . . . . . . 7  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  =  ( ( 5  x.  9 )  +  2 )
1067nncni 9752 . . . . . . . . 9  |-  9  e.  CC
107 9t5e45 10218 . . . . . . . . 9  |-  ( 9  x.  5 )  = ; 4
5
108106, 57, 107mulcomli 8840 . . . . . . . 8  |-  ( 5  x.  9 )  = ; 4
5
109 5p2e7 9856 . . . . . . . 8  |-  ( 5  +  2 )  =  7
11029, 5, 3, 108, 109decaddi 10164 . . . . . . 7  |-  ( ( 5  x.  9 )  +  2 )  = ; 4
7
111105, 110eqtri 2304 . . . . . 6  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  = ; 4
7
112 9t2e18 10215 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
113106, 24, 112mulcomli 8840 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
114 1p1e2 9836 . . . . . . 7  |-  ( 1  +  1 )  =  2
115 8p8e16 10181 . . . . . . 7  |-  ( 8  +  8 )  = ; 1
6
1162, 14, 14, 113, 114, 11, 115decaddci 10165 . . . . . 6  |-  ( ( 2  x.  9 )  +  8 )  = ; 2
6
1175, 3, 33, 14, 62, 104, 27, 11, 3, 111, 116decmac 10159 . . . . 5  |-  ( (; 5
2  x.  9 )  +  8 )  = ;; 4 7 6
1186, 27, 11, 14, 1, 28, 19, 11, 31, 103, 117decma2c 10160 . . . 4  |-  ( (; 5
2  x.  N )  + ; 6 8 )  = ;;;; 6 5 5 3 6
119 2exp16 13099 . . . 4  |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
120 eqid 2284 . . . . 5  |-  ( 2 ^ 8 )  =  ( 2 ^ 8 )
121 eqid 2284 . . . . 5  |-  ( ( 2 ^ 8 )  x.  ( 2 ^ 8 ) )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1223, 14, 26, 120, 121numexp2x 13090 . . . 4  |-  ( 2 ^; 1 6 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
123118, 119, 1223eqtr2i 2310 . . 3  |-  ( (; 5
2  x.  N )  + ; 6 8 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1249, 10, 14, 20, 21, 15, 22, 26, 123mod2xi 13080 . 2  |-  ( ( 2 ^; 1 6 )  mod 
N )  =  (; 6
8  mod  N )
125 6p1e7 9847 . . 3  |-  ( 6  +  1 )  =  7
126 eqid 2284 . . 3  |- ; 1 6  = ; 1 6
1272, 11, 125, 126decsuc 10143 . 2  |-  (; 1 6  +  1 )  = ; 1 7
12818nn0cni 9973 . . . 4  |- ;; 1 3 6  e.  CC
129128addid2i 8996 . . 3  |-  ( 0  + ;; 1 3 6 )  = ;; 1 3 6
1309nncni 9752 . . . . 5  |-  N  e.  CC
131130mul02i 8997 . . . 4  |-  ( 0  x.  N )  =  0
132131oveq1i 5830 . . 3  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  ( 0  + ;; 1 3 6 )
133 6t2e12 10197 . . . . 5  |-  ( 6  x.  2 )  = ; 1
2
1342, 3, 54, 133decsuc 10143 . . . 4  |-  ( ( 6  x.  2 )  +  1 )  = ; 1
3
1353, 11, 14, 28, 11, 2, 134, 25decmul1c 10167 . . 3  |-  (; 6 8  x.  2 )  = ;; 1 3 6
136129, 132, 1353eqtr4i 2314 . 2  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  (; 6
8  x.  2 )
1379, 10, 12, 13, 15, 18, 124, 127, 136modxp1i 13081 1  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
Colors of variables: wff set class
Syntax hints:    = wceq 1623  (class class class)co 5820   0cc0 8733   1c1 8734    + caddc 8736    x. cmul 8738   NNcn 9742   2c2 9791   3c3 9792   4c4 9793   5c5 9794   6c6 9795   7c7 9796   8c8 9797   9c9 9798   10c10 9799  ;cdc 10120    mod cmo 10969   ^cexp 11100
This theorem is referenced by:  1259lem2  13126  1259lem4  13128
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-rp 10351  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101
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