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Theorem 1259lem1 13056
Description: Lemma for 1259prm 13061. 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 9913 . . . . . 6  |-  1  e.  NN0
3 2nn0 9914 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10070 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 9917 . . . . 5  |-  5  e.  NN0
64, 5deccl 10070 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 9816 . . . 4  |-  9  e.  NN
86, 7decnncl 10069 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2326 . 2  |-  N  e.  NN
10 2nn 9809 . 2  |-  2  e.  NN
11 6nn0 9918 . . 3  |-  6  e.  NN0
122, 11deccl 10070 . 2  |- ; 1 6  e.  NN0
13 0z 9967 . 2  |-  0  e.  ZZ
14 8nn0 9920 . . 3  |-  8  e.  NN0
1511, 14deccl 10070 . 2  |- ; 6 8  e.  NN0
16 3nn0 9915 . . . 4  |-  3  e.  NN0
172, 16deccl 10070 . . 3  |- ; 1 3  e.  NN0
1817, 11deccl 10070 . 2  |- ;; 1 3 6  e.  NN0
195, 3deccl 10070 . . . 4  |- ; 5 2  e.  NN0
2019nn0zi 9980 . . 3  |- ; 5 2  e.  ZZ
213, 14nn0expcli 11060 . . 3  |-  ( 2 ^ 8 )  e. 
NN0
22 eqid 2256 . . 3  |-  ( ( 2 ^ 8 )  mod  N )  =  ( ( 2 ^ 8 )  mod  N
)
2314nn0cni 9909 . . . 4  |-  8  e.  CC
24 2cn 9749 . . . 4  |-  2  e.  CC
25 8t2e16 10144 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
2623, 24, 25mulcomli 8777 . . 3  |-  ( 2  x.  8 )  = ; 1
6
27 9nn0 9921 . . . . 5  |-  9  e.  NN0
28 eqid 2256 . . . . 5  |- ; 6 8  = ; 6 8
29 4nn0 9916 . . . . . 6  |-  4  e.  NN0
30 7nn0 9919 . . . . . 6  |-  7  e.  NN0
3129, 30deccl 10070 . . . . 5  |- ; 4 7  e.  NN0
32 eqid 2256 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
33 0nn0 9912 . . . . . . 7  |-  0  e.  NN0
3411dec0h 10072 . . . . . . 7  |-  6  = ; 0 6
35 eqid 2256 . . . . . . 7  |- ; 4 7  = ; 4 7
36 4cn 9753 . . . . . . . . . 10  |-  4  e.  CC
3736addid2i 8933 . . . . . . . . 9  |-  ( 0  +  4 )  =  4
3837oveq1i 5767 . . . . . . . 8  |-  ( ( 0  +  4 )  +  1 )  =  ( 4  +  1 )
39 4p1e5 9781 . . . . . . . 8  |-  ( 4  +  1 )  =  5
4038, 39eqtri 2276 . . . . . . 7  |-  ( ( 0  +  4 )  +  1 )  =  5
41 7nn 9814 . . . . . . . . 9  |-  7  e.  NN
4241nncni 9689 . . . . . . . 8  |-  7  e.  CC
43 6nn 9813 . . . . . . . . 9  |-  6  e.  NN
4443nncni 9689 . . . . . . . 8  |-  6  e.  CC
45 7p6e13 10110 . . . . . . . 8  |-  ( 7  +  6 )  = ; 1
3
4642, 44, 45addcomli 8937 . . . . . . 7  |-  ( 6  +  7 )  = ; 1
3
4733, 11, 29, 30, 34, 35, 40, 16, 46decaddc 10098 . . . . . 6  |-  ( 6  + ; 4 7 )  = ; 5
3
483, 11deccl 10070 . . . . . 6  |- ; 2 6  e.  NN0
49 eqid 2256 . . . . . . 7  |- ; 1 2  = ; 1 2
505dec0h 10072 . . . . . . . 8  |-  5  = ; 0 5
51 eqid 2256 . . . . . . . 8  |- ; 2 6  = ; 2 6
5224addid2i 8933 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
5352oveq1i 5767 . . . . . . . . 9  |-  ( ( 0  +  2 )  +  1 )  =  ( 2  +  1 )
54 2p1e3 9779 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
5553, 54eqtri 2276 . . . . . . . 8  |-  ( ( 0  +  2 )  +  1 )  =  3
56 5nn 9812 . . . . . . . . . 10  |-  5  e.  NN
5756nncni 9689 . . . . . . . . 9  |-  5  e.  CC
58 6p5e11 10106 . . . . . . . . 9  |-  ( 6  +  5 )  = ; 1
1
5944, 57, 58addcomli 8937 . . . . . . . 8  |-  ( 5  +  6 )  = ; 1
1
6033, 5, 3, 11, 50, 51, 55, 2, 59decaddc 10098 . . . . . . 7  |-  ( 5  + ; 2 6 )  = ; 3
1
61 10nn0 9922 . . . . . . 7  |-  10  e.  NN0
62 eqid 2256 . . . . . . . 8  |- ; 5 2  = ; 5 2
6316dec0h 10072 . . . . . . . . 9  |-  3  = ; 0 3
64 dec10 10086 . . . . . . . . 9  |-  10  = ; 1 0
65 ax-1cn 8728 . . . . . . . . . 10  |-  1  e.  CC
6665addid2i 8933 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
67 3cn 9751 . . . . . . . . . 10  |-  3  e.  CC
6867addid1i 8932 . . . . . . . . 9  |-  ( 3  +  0 )  =  3
6933, 16, 2, 33, 63, 64, 66, 68decadd 10097 . . . . . . . 8  |-  ( 3  +  10 )  = ; 1
3
7057mulid1i 8772 . . . . . . . . . 10  |-  ( 5  x.  1 )  =  5
7165addid1i 8932 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
7270, 71oveq12i 5769 . . . . . . . . 9  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  ( 5  +  1 )
73 5p1e6 9782 . . . . . . . . 9  |-  ( 5  +  1 )  =  6
7472, 73eqtri 2276 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  6
7524mulid1i 8772 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
7675oveq1i 5767 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  3 )  =  ( 2  +  3 )
77 3p2e5 9787 . . . . . . . . . 10  |-  ( 3  +  2 )  =  5
7867, 24, 77addcomli 8937 . . . . . . . . 9  |-  ( 2  +  3 )  =  5
7976, 78, 503eqtri 2280 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  3 )  = ; 0
5
805, 3, 2, 16, 62, 69, 2, 5, 33, 74, 79decmac 10095 . . . . . . 7  |-  ( (; 5
2  x.  1 )  +  ( 3  +  10 ) )  = ; 6
5
812dec0h 10072 . . . . . . . 8  |-  1  = ; 0 1
82 5t2e10 9807 . . . . . . . . . 10  |-  ( 5  x.  2 )  =  10
83 00id 8920 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
8482, 83oveq12i 5769 . . . . . . . . 9  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  ( 10  +  0 )
85 10nn 9817 . . . . . . . . . . 11  |-  10  e.  NN
8685nncni 9689 . . . . . . . . . 10  |-  10  e.  CC
8786addid1i 8932 . . . . . . . . 9  |-  ( 10  +  0 )  =  10
8884, 87eqtri 2276 . . . . . . . 8  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  10
89 2t2e4 9803 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
9089oveq1i 5767 . . . . . . . . 9  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
9190, 39, 503eqtri 2280 . . . . . . . 8  |-  ( ( 2  x.  2 )  +  1 )  = ; 0
5
925, 3, 33, 2, 62, 81, 3, 5, 33, 88, 91decmac 10095 . . . . . . 7  |-  ( (; 5
2  x.  2 )  +  1 )  = ; 10 5
932, 3, 16, 2, 49, 60, 19, 5, 61, 80, 92decma2c 10096 . . . . . 6  |-  ( (; 5
2  x. ; 1 2 )  +  ( 5  + ; 2 6 ) )  = ;; 6 5 5
9466oveq2i 5768 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  =  ( ( 5  x.  5 )  +  1 )
95 5t5e25 10132 . . . . . . . . 9  |-  ( 5  x.  5 )  = ; 2
5
963, 5, 73, 95decsuc 10079 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  1 )  = ; 2
6
9794, 96eqtri 2276 . . . . . . 7  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  = ; 2
6
9857, 24, 82mulcomli 8777 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
9998, 64eqtri 2276 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
10067addid2i 8933 . . . . . . . 8  |-  ( 0  +  3 )  =  3
1012, 33, 16, 99, 100decaddi 10100 . . . . . . 7  |-  ( ( 2  x.  5 )  +  3 )  = ; 1
3
1025, 3, 33, 16, 62, 63, 5, 16, 2, 97, 101decmac 10095 . . . . . 6  |-  ( (; 5
2  x.  5 )  +  3 )  = ;; 2 6 3
1034, 5, 5, 16, 32, 47, 19, 16, 48, 93, 102decma2c 10096 . . . . 5  |-  ( (; 5
2  x. ;; 1 2 5 )  +  ( 6  + ; 4 7 ) )  = ;;; 6 5 5 3
10414dec0h 10072 . . . . . 6  |-  8  = ; 0 8
10552oveq2i 5768 . . . . . . 7  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  =  ( ( 5  x.  9 )  +  2 )
1067nncni 9689 . . . . . . . . 9  |-  9  e.  CC
107 9t5e45 10154 . . . . . . . . 9  |-  ( 9  x.  5 )  = ; 4
5
108106, 57, 107mulcomli 8777 . . . . . . . 8  |-  ( 5  x.  9 )  = ; 4
5
109 5p2e7 9792 . . . . . . . 8  |-  ( 5  +  2 )  =  7
11029, 5, 3, 108, 109decaddi 10100 . . . . . . 7  |-  ( ( 5  x.  9 )  +  2 )  = ; 4
7
111105, 110eqtri 2276 . . . . . 6  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  = ; 4
7
112 9t2e18 10151 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
113106, 24, 112mulcomli 8777 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
114 1p1e2 9773 . . . . . . 7  |-  ( 1  +  1 )  =  2
115 8p8e16 10117 . . . . . . 7  |-  ( 8  +  8 )  = ; 1
6
1162, 14, 14, 113, 114, 11, 115decaddci 10101 . . . . . 6  |-  ( ( 2  x.  9 )  +  8 )  = ; 2
6
1175, 3, 33, 14, 62, 104, 27, 11, 3, 111, 116decmac 10095 . . . . 5  |-  ( (; 5
2  x.  9 )  +  8 )  = ;; 4 7 6
1186, 27, 11, 14, 1, 28, 19, 11, 31, 103, 117decma2c 10096 . . . 4  |-  ( (; 5
2  x.  N )  + ; 6 8 )  = ;;;; 6 5 5 3 6
119 2exp16 13030 . . . 4  |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
120 eqid 2256 . . . . 5  |-  ( 2 ^ 8 )  =  ( 2 ^ 8 )
121 eqid 2256 . . . . 5  |-  ( ( 2 ^ 8 )  x.  ( 2 ^ 8 ) )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1223, 14, 26, 120, 121numexp2x 13021 . . . 4  |-  ( 2 ^; 1 6 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
123118, 119, 1223eqtr2i 2282 . . 3  |-  ( (; 5
2  x.  N )  + ; 6 8 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1249, 10, 14, 20, 21, 15, 22, 26, 123mod2xi 13011 . 2  |-  ( ( 2 ^; 1 6 )  mod 
N )  =  (; 6
8  mod  N )
125 6p1e7 9783 . . 3  |-  ( 6  +  1 )  =  7
126 eqid 2256 . . 3  |- ; 1 6  = ; 1 6
1272, 11, 125, 126decsuc 10079 . 2  |-  (; 1 6  +  1 )  = ; 1 7
12818nn0cni 9909 . . . 4  |- ;; 1 3 6  e.  CC
129128addid2i 8933 . . 3  |-  ( 0  + ;; 1 3 6 )  = ;; 1 3 6
1309nncni 9689 . . . . 5  |-  N  e.  CC
131130mul02i 8934 . . . 4  |-  ( 0  x.  N )  =  0
132131oveq1i 5767 . . 3  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  ( 0  + ;; 1 3 6 )
133 6t2e12 10133 . . . . 5  |-  ( 6  x.  2 )  = ; 1
2
1342, 3, 54, 133decsuc 10079 . . . 4  |-  ( ( 6  x.  2 )  +  1 )  = ; 1
3
1353, 11, 14, 28, 11, 2, 134, 25decmul1c 10103 . . 3  |-  (; 6 8  x.  2 )  = ;; 1 3 6
136129, 132, 1353eqtr4i 2286 . 2  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  (; 6
8  x.  2 )
1379, 10, 12, 13, 15, 18, 124, 127, 136modxp1i 13012 1  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
Colors of variables: wff set class
Syntax hints:    = wceq 1619  (class class class)co 5757   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675   NNcn 9679   2c2 9728   3c3 9729   4c4 9730   5c5 9731   6c6 9732   7c7 9733   8c8 9734   9c9 9735   10c10 9736  ;cdc 10056    mod cmo 10904   ^cexp 11035
This theorem is referenced by:  1259lem2  13057  1259lem4  13059
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-sup 7127  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-rp 10287  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036
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