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Theorem 1259lem1 14490
Description: Lemma for 1259prm 14495. 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 10817 . . . . . 6  |-  1  e.  NN0
3 2nn0 10818 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10998 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10821 . . . . 5  |-  5  e.  NN0
64, 5deccl 10998 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10706 . . . 4  |-  9  e.  NN
86, 7decnncl 10997 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2527 . 2  |-  N  e.  NN
10 2nn 10699 . 2  |-  2  e.  NN
11 6nn0 10822 . . 3  |-  6  e.  NN0
122, 11deccl 10998 . 2  |- ; 1 6  e.  NN0
13 0z 10881 . 2  |-  0  e.  ZZ
14 8nn0 10824 . . 3  |-  8  e.  NN0
1511, 14deccl 10998 . 2  |- ; 6 8  e.  NN0
16 3nn0 10819 . . . 4  |-  3  e.  NN0
172, 16deccl 10998 . . 3  |- ; 1 3  e.  NN0
1817, 11deccl 10998 . 2  |- ;; 1 3 6  e.  NN0
195, 3deccl 10998 . . . 4  |- ; 5 2  e.  NN0
2019nn0zi 10895 . . 3  |- ; 5 2  e.  ZZ
213, 14nn0expcli 12171 . . 3  |-  ( 2 ^ 8 )  e. 
NN0
22 eqid 2443 . . 3  |-  ( ( 2 ^ 8 )  mod  N )  =  ( ( 2 ^ 8 )  mod  N
)
2314nn0cni 10813 . . . 4  |-  8  e.  CC
24 2cn 10612 . . . 4  |-  2  e.  CC
25 8t2e16 11072 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
2623, 24, 25mulcomli 9606 . . 3  |-  ( 2  x.  8 )  = ; 1
6
27 9nn0 10825 . . . . 5  |-  9  e.  NN0
28 eqid 2443 . . . . 5  |- ; 6 8  = ; 6 8
29 4nn0 10820 . . . . . 6  |-  4  e.  NN0
30 7nn0 10823 . . . . . 6  |-  7  e.  NN0
3129, 30deccl 10998 . . . . 5  |- ; 4 7  e.  NN0
32 eqid 2443 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
33 0nn0 10816 . . . . . . 7  |-  0  e.  NN0
3411dec0h 11000 . . . . . . 7  |-  6  = ; 0 6
35 eqid 2443 . . . . . . 7  |- ; 4 7  = ; 4 7
36 4cn 10619 . . . . . . . . . 10  |-  4  e.  CC
3736addid2i 9771 . . . . . . . . 9  |-  ( 0  +  4 )  =  4
3837oveq1i 6291 . . . . . . . 8  |-  ( ( 0  +  4 )  +  1 )  =  ( 4  +  1 )
39 4p1e5 10668 . . . . . . . 8  |-  ( 4  +  1 )  =  5
4038, 39eqtri 2472 . . . . . . 7  |-  ( ( 0  +  4 )  +  1 )  =  5
41 7cn 10625 . . . . . . . 8  |-  7  e.  CC
42 6cn 10623 . . . . . . . 8  |-  6  e.  CC
43 7p6e13 11038 . . . . . . . 8  |-  ( 7  +  6 )  = ; 1
3
4441, 42, 43addcomli 9775 . . . . . . 7  |-  ( 6  +  7 )  = ; 1
3
4533, 11, 29, 30, 34, 35, 40, 16, 44decaddc 11026 . . . . . 6  |-  ( 6  + ; 4 7 )  = ; 5
3
463, 11deccl 10998 . . . . . 6  |- ; 2 6  e.  NN0
47 eqid 2443 . . . . . . 7  |- ; 1 2  = ; 1 2
485dec0h 11000 . . . . . . . 8  |-  5  = ; 0 5
49 eqid 2443 . . . . . . . 8  |- ; 2 6  = ; 2 6
5024addid2i 9771 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
5150oveq1i 6291 . . . . . . . . 9  |-  ( ( 0  +  2 )  +  1 )  =  ( 2  +  1 )
52 2p1e3 10665 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
5351, 52eqtri 2472 . . . . . . . 8  |-  ( ( 0  +  2 )  +  1 )  =  3
54 5cn 10621 . . . . . . . . 9  |-  5  e.  CC
55 6p5e11 11034 . . . . . . . . 9  |-  ( 6  +  5 )  = ; 1
1
5642, 54, 55addcomli 9775 . . . . . . . 8  |-  ( 5  +  6 )  = ; 1
1
5733, 5, 3, 11, 48, 49, 53, 2, 56decaddc 11026 . . . . . . 7  |-  ( 5  + ; 2 6 )  = ; 3
1
58 10nn0 10826 . . . . . . 7  |-  10  e.  NN0
59 eqid 2443 . . . . . . . 8  |- ; 5 2  = ; 5 2
6016dec0h 11000 . . . . . . . . 9  |-  3  = ; 0 3
61 dec10 11014 . . . . . . . . 9  |-  10  = ; 1 0
62 ax-1cn 9553 . . . . . . . . . 10  |-  1  e.  CC
6362addid2i 9771 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
64 3cn 10616 . . . . . . . . . 10  |-  3  e.  CC
6564addid1i 9770 . . . . . . . . 9  |-  ( 3  +  0 )  =  3
6633, 16, 2, 33, 60, 61, 63, 65decadd 11025 . . . . . . . 8  |-  ( 3  +  10 )  = ; 1
3
6754mulid1i 9601 . . . . . . . . . 10  |-  ( 5  x.  1 )  =  5
6862addid1i 9770 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
6967, 68oveq12i 6293 . . . . . . . . 9  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  ( 5  +  1 )
70 5p1e6 10669 . . . . . . . . 9  |-  ( 5  +  1 )  =  6
7169, 70eqtri 2472 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  6
7224mulid1i 9601 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
7372oveq1i 6291 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  3 )  =  ( 2  +  3 )
74 3p2e5 10674 . . . . . . . . . 10  |-  ( 3  +  2 )  =  5
7564, 24, 74addcomli 9775 . . . . . . . . 9  |-  ( 2  +  3 )  =  5
7673, 75, 483eqtri 2476 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  3 )  = ; 0
5
775, 3, 2, 16, 59, 66, 2, 5, 33, 71, 76decmac 11023 . . . . . . 7  |-  ( (; 5
2  x.  1 )  +  ( 3  +  10 ) )  = ; 6
5
782dec0h 11000 . . . . . . . 8  |-  1  = ; 0 1
79 5t2e10 10696 . . . . . . . . . 10  |-  ( 5  x.  2 )  =  10
80 00id 9758 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
8179, 80oveq12i 6293 . . . . . . . . 9  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  ( 10  +  0 )
82 10nn 10707 . . . . . . . . . . 11  |-  10  e.  NN
8382nncni 10552 . . . . . . . . . 10  |-  10  e.  CC
8483addid1i 9770 . . . . . . . . 9  |-  ( 10  +  0 )  =  10
8581, 84eqtri 2472 . . . . . . . 8  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  10
86 2t2e4 10691 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
8786oveq1i 6291 . . . . . . . . 9  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
8887, 39, 483eqtri 2476 . . . . . . . 8  |-  ( ( 2  x.  2 )  +  1 )  = ; 0
5
895, 3, 33, 2, 59, 78, 3, 5, 33, 85, 88decmac 11023 . . . . . . 7  |-  ( (; 5
2  x.  2 )  +  1 )  = ; 10 5
902, 3, 16, 2, 47, 57, 19, 5, 58, 77, 89decma2c 11024 . . . . . 6  |-  ( (; 5
2  x. ; 1 2 )  +  ( 5  + ; 2 6 ) )  = ;; 6 5 5
9163oveq2i 6292 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  =  ( ( 5  x.  5 )  +  1 )
92 5t5e25 11060 . . . . . . . . 9  |-  ( 5  x.  5 )  = ; 2
5
933, 5, 70, 92decsuc 11007 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  1 )  = ; 2
6
9491, 93eqtri 2472 . . . . . . 7  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  = ; 2
6
9554, 24, 79mulcomli 9606 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
9695, 61eqtri 2472 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
9764addid2i 9771 . . . . . . . 8  |-  ( 0  +  3 )  =  3
982, 33, 16, 96, 97decaddi 11028 . . . . . . 7  |-  ( ( 2  x.  5 )  +  3 )  = ; 1
3
995, 3, 33, 16, 59, 60, 5, 16, 2, 94, 98decmac 11023 . . . . . 6  |-  ( (; 5
2  x.  5 )  +  3 )  = ;; 2 6 3
1004, 5, 5, 16, 32, 45, 19, 16, 46, 90, 99decma2c 11024 . . . . 5  |-  ( (; 5
2  x. ;; 1 2 5 )  +  ( 6  + ; 4 7 ) )  = ;;; 6 5 5 3
10114dec0h 11000 . . . . . 6  |-  8  = ; 0 8
10250oveq2i 6292 . . . . . . 7  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  =  ( ( 5  x.  9 )  +  2 )
103 9cn 10629 . . . . . . . . 9  |-  9  e.  CC
104 9t5e45 11082 . . . . . . . . 9  |-  ( 9  x.  5 )  = ; 4
5
105103, 54, 104mulcomli 9606 . . . . . . . 8  |-  ( 5  x.  9 )  = ; 4
5
106 5p2e7 10679 . . . . . . . 8  |-  ( 5  +  2 )  =  7
10729, 5, 3, 105, 106decaddi 11028 . . . . . . 7  |-  ( ( 5  x.  9 )  +  2 )  = ; 4
7
108102, 107eqtri 2472 . . . . . 6  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  = ; 4
7
109 9t2e18 11079 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
110103, 24, 109mulcomli 9606 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
111 1p1e2 10655 . . . . . . 7  |-  ( 1  +  1 )  =  2
112 8p8e16 11045 . . . . . . 7  |-  ( 8  +  8 )  = ; 1
6
1132, 14, 14, 110, 111, 11, 112decaddci 11029 . . . . . 6  |-  ( ( 2  x.  9 )  +  8 )  = ; 2
6
1145, 3, 33, 14, 59, 101, 27, 11, 3, 108, 113decmac 11023 . . . . 5  |-  ( (; 5
2  x.  9 )  +  8 )  = ;; 4 7 6
1156, 27, 11, 14, 1, 28, 19, 11, 31, 100, 114decma2c 11024 . . . 4  |-  ( (; 5
2  x.  N )  + ; 6 8 )  = ;;;; 6 5 5 3 6
116 2exp16 14452 . . . 4  |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
117 eqid 2443 . . . . 5  |-  ( 2 ^ 8 )  =  ( 2 ^ 8 )
118 eqid 2443 . . . . 5  |-  ( ( 2 ^ 8 )  x.  ( 2 ^ 8 ) )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1193, 14, 26, 117, 118numexp2x 14442 . . . 4  |-  ( 2 ^; 1 6 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
120115, 116, 1193eqtr2i 2478 . . 3  |-  ( (; 5
2  x.  N )  + ; 6 8 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1219, 10, 14, 20, 21, 15, 22, 26, 120mod2xi 14432 . 2  |-  ( ( 2 ^; 1 6 )  mod 
N )  =  (; 6
8  mod  N )
122 6p1e7 10670 . . 3  |-  ( 6  +  1 )  =  7
123 eqid 2443 . . 3  |- ; 1 6  = ; 1 6
1242, 11, 122, 123decsuc 11007 . 2  |-  (; 1 6  +  1 )  = ; 1 7
12518nn0cni 10813 . . . 4  |- ;; 1 3 6  e.  CC
126125addid2i 9771 . . 3  |-  ( 0  + ;; 1 3 6 )  = ;; 1 3 6
1279nncni 10552 . . . . 5  |-  N  e.  CC
128127mul02i 9772 . . . 4  |-  ( 0  x.  N )  =  0
129128oveq1i 6291 . . 3  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  ( 0  + ;; 1 3 6 )
130 6t2e12 11061 . . . . 5  |-  ( 6  x.  2 )  = ; 1
2
1312, 3, 52, 130decsuc 11007 . . . 4  |-  ( ( 6  x.  2 )  +  1 )  = ; 1
3
1323, 11, 14, 28, 11, 2, 131, 25decmul1c 11031 . . 3  |-  (; 6 8  x.  2 )  = ;; 1 3 6
133126, 129, 1323eqtr4i 2482 . 2  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  (; 6
8  x.  2 )
1349, 10, 12, 13, 15, 18, 121, 124, 133modxp1i 14433 1  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383  (class class class)co 6281   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   NNcn 10542   2c2 10591   3c3 10592   4c4 10593   5c5 10594   6c6 10595   7c7 10596   8c8 10597   9c9 10598   10c10 10599  ;cdc 10984    mod cmo 11975   ^cexp 12145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-rp 11230  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146
This theorem is referenced by:  1259lem2  14491  1259lem4  14493
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