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Theorem 1259lem1 15089
Description: Lemma for 1259prm 15094. 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 10885 . . . . . 6  |-  1  e.  NN0
3 2nn0 10886 . . . . . 6  |-  2  e.  NN0
42, 3deccl 11065 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10889 . . . . 5  |-  5  e.  NN0
64, 5deccl 11065 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10774 . . . 4  |-  9  e.  NN
86, 7decnncl 11064 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2506 . 2  |-  N  e.  NN
10 2nn 10767 . 2  |-  2  e.  NN
11 6nn0 10890 . . 3  |-  6  e.  NN0
122, 11deccl 11065 . 2  |- ; 1 6  e.  NN0
13 0z 10948 . 2  |-  0  e.  ZZ
14 8nn0 10892 . . 3  |-  8  e.  NN0
1511, 14deccl 11065 . 2  |- ; 6 8  e.  NN0
16 3nn0 10887 . . . 4  |-  3  e.  NN0
172, 16deccl 11065 . . 3  |- ; 1 3  e.  NN0
1817, 11deccl 11065 . 2  |- ;; 1 3 6  e.  NN0
195, 3deccl 11065 . . . 4  |- ; 5 2  e.  NN0
2019nn0zi 10962 . . 3  |- ; 5 2  e.  ZZ
213, 14nn0expcli 12297 . . 3  |-  ( 2 ^ 8 )  e. 
NN0
22 eqid 2422 . . 3  |-  ( ( 2 ^ 8 )  mod  N )  =  ( ( 2 ^ 8 )  mod  N
)
2314nn0cni 10881 . . . 4  |-  8  e.  CC
24 2cn 10680 . . . 4  |-  2  e.  CC
25 8t2e16 11139 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
2623, 24, 25mulcomli 9650 . . 3  |-  ( 2  x.  8 )  = ; 1
6
27 9nn0 10893 . . . . 5  |-  9  e.  NN0
28 eqid 2422 . . . . 5  |- ; 6 8  = ; 6 8
29 4nn0 10888 . . . . . 6  |-  4  e.  NN0
30 7nn0 10891 . . . . . 6  |-  7  e.  NN0
3129, 30deccl 11065 . . . . 5  |- ; 4 7  e.  NN0
32 eqid 2422 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
33 0nn0 10884 . . . . . . 7  |-  0  e.  NN0
3411dec0h 11067 . . . . . . 7  |-  6  = ; 0 6
35 eqid 2422 . . . . . . 7  |- ; 4 7  = ; 4 7
36 4cn 10687 . . . . . . . . . 10  |-  4  e.  CC
3736addid2i 9821 . . . . . . . . 9  |-  ( 0  +  4 )  =  4
3837oveq1i 6311 . . . . . . . 8  |-  ( ( 0  +  4 )  +  1 )  =  ( 4  +  1 )
39 4p1e5 10736 . . . . . . . 8  |-  ( 4  +  1 )  =  5
4038, 39eqtri 2451 . . . . . . 7  |-  ( ( 0  +  4 )  +  1 )  =  5
41 7cn 10693 . . . . . . . 8  |-  7  e.  CC
42 6cn 10691 . . . . . . . 8  |-  6  e.  CC
43 7p6e13 11105 . . . . . . . 8  |-  ( 7  +  6 )  = ; 1
3
4441, 42, 43addcomli 9825 . . . . . . 7  |-  ( 6  +  7 )  = ; 1
3
4533, 11, 29, 30, 34, 35, 40, 16, 44decaddc 11093 . . . . . 6  |-  ( 6  + ; 4 7 )  = ; 5
3
463, 11deccl 11065 . . . . . 6  |- ; 2 6  e.  NN0
47 eqid 2422 . . . . . . 7  |- ; 1 2  = ; 1 2
485dec0h 11067 . . . . . . . 8  |-  5  = ; 0 5
49 eqid 2422 . . . . . . . 8  |- ; 2 6  = ; 2 6
5024addid2i 9821 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
5150oveq1i 6311 . . . . . . . . 9  |-  ( ( 0  +  2 )  +  1 )  =  ( 2  +  1 )
52 2p1e3 10733 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
5351, 52eqtri 2451 . . . . . . . 8  |-  ( ( 0  +  2 )  +  1 )  =  3
54 5cn 10689 . . . . . . . . 9  |-  5  e.  CC
55 6p5e11 11101 . . . . . . . . 9  |-  ( 6  +  5 )  = ; 1
1
5642, 54, 55addcomli 9825 . . . . . . . 8  |-  ( 5  +  6 )  = ; 1
1
5733, 5, 3, 11, 48, 49, 53, 2, 56decaddc 11093 . . . . . . 7  |-  ( 5  + ; 2 6 )  = ; 3
1
58 10nn0 10894 . . . . . . 7  |-  10  e.  NN0
59 eqid 2422 . . . . . . . 8  |- ; 5 2  = ; 5 2
6016dec0h 11067 . . . . . . . . 9  |-  3  = ; 0 3
61 dec10 11081 . . . . . . . . 9  |-  10  = ; 1 0
62 ax-1cn 9597 . . . . . . . . . 10  |-  1  e.  CC
6362addid2i 9821 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
64 3cn 10684 . . . . . . . . . 10  |-  3  e.  CC
6564addid1i 9820 . . . . . . . . 9  |-  ( 3  +  0 )  =  3
6633, 16, 2, 33, 60, 61, 63, 65decadd 11092 . . . . . . . 8  |-  ( 3  +  10 )  = ; 1
3
6754mulid1i 9645 . . . . . . . . . 10  |-  ( 5  x.  1 )  =  5
6862addid1i 9820 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
6967, 68oveq12i 6313 . . . . . . . . 9  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  ( 5  +  1 )
70 5p1e6 10737 . . . . . . . . 9  |-  ( 5  +  1 )  =  6
7169, 70eqtri 2451 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  6
7224mulid1i 9645 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
7372oveq1i 6311 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  3 )  =  ( 2  +  3 )
74 3p2e5 10742 . . . . . . . . . 10  |-  ( 3  +  2 )  =  5
7564, 24, 74addcomli 9825 . . . . . . . . 9  |-  ( 2  +  3 )  =  5
7673, 75, 483eqtri 2455 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  3 )  = ; 0
5
775, 3, 2, 16, 59, 66, 2, 5, 33, 71, 76decmac 11090 . . . . . . 7  |-  ( (; 5
2  x.  1 )  +  ( 3  +  10 ) )  = ; 6
5
782dec0h 11067 . . . . . . . 8  |-  1  = ; 0 1
79 5t2e10 10764 . . . . . . . . . 10  |-  ( 5  x.  2 )  =  10
80 00id 9808 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
8179, 80oveq12i 6313 . . . . . . . . 9  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  ( 10  +  0 )
82 10nn 10775 . . . . . . . . . . 11  |-  10  e.  NN
8382nncni 10619 . . . . . . . . . 10  |-  10  e.  CC
8483addid1i 9820 . . . . . . . . 9  |-  ( 10  +  0 )  =  10
8581, 84eqtri 2451 . . . . . . . 8  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  10
86 2t2e4 10759 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
8786oveq1i 6311 . . . . . . . . 9  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
8887, 39, 483eqtri 2455 . . . . . . . 8  |-  ( ( 2  x.  2 )  +  1 )  = ; 0
5
895, 3, 33, 2, 59, 78, 3, 5, 33, 85, 88decmac 11090 . . . . . . 7  |-  ( (; 5
2  x.  2 )  +  1 )  = ; 10 5
902, 3, 16, 2, 47, 57, 19, 5, 58, 77, 89decma2c 11091 . . . . . 6  |-  ( (; 5
2  x. ; 1 2 )  +  ( 5  + ; 2 6 ) )  = ;; 6 5 5
9163oveq2i 6312 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  =  ( ( 5  x.  5 )  +  1 )
92 5t5e25 11127 . . . . . . . . 9  |-  ( 5  x.  5 )  = ; 2
5
933, 5, 70, 92decsuc 11074 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  1 )  = ; 2
6
9491, 93eqtri 2451 . . . . . . 7  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  = ; 2
6
9554, 24, 79mulcomli 9650 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
9695, 61eqtri 2451 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
9764addid2i 9821 . . . . . . . 8  |-  ( 0  +  3 )  =  3
982, 33, 16, 96, 97decaddi 11095 . . . . . . 7  |-  ( ( 2  x.  5 )  +  3 )  = ; 1
3
995, 3, 33, 16, 59, 60, 5, 16, 2, 94, 98decmac 11090 . . . . . 6  |-  ( (; 5
2  x.  5 )  +  3 )  = ;; 2 6 3
1004, 5, 5, 16, 32, 45, 19, 16, 46, 90, 99decma2c 11091 . . . . 5  |-  ( (; 5
2  x. ;; 1 2 5 )  +  ( 6  + ; 4 7 ) )  = ;;; 6 5 5 3
10114dec0h 11067 . . . . . 6  |-  8  = ; 0 8
10250oveq2i 6312 . . . . . . 7  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  =  ( ( 5  x.  9 )  +  2 )
103 9cn 10697 . . . . . . . . 9  |-  9  e.  CC
104 9t5e45 11149 . . . . . . . . 9  |-  ( 9  x.  5 )  = ; 4
5
105103, 54, 104mulcomli 9650 . . . . . . . 8  |-  ( 5  x.  9 )  = ; 4
5
106 5p2e7 10747 . . . . . . . 8  |-  ( 5  +  2 )  =  7
10729, 5, 3, 105, 106decaddi 11095 . . . . . . 7  |-  ( ( 5  x.  9 )  +  2 )  = ; 4
7
108102, 107eqtri 2451 . . . . . 6  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  = ; 4
7
109 9t2e18 11146 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
110103, 24, 109mulcomli 9650 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
111 1p1e2 10723 . . . . . . 7  |-  ( 1  +  1 )  =  2
112 8p8e16 11112 . . . . . . 7  |-  ( 8  +  8 )  = ; 1
6
1132, 14, 14, 110, 111, 11, 112decaddci 11096 . . . . . 6  |-  ( ( 2  x.  9 )  +  8 )  = ; 2
6
1145, 3, 33, 14, 59, 101, 27, 11, 3, 108, 113decmac 11090 . . . . 5  |-  ( (; 5
2  x.  9 )  +  8 )  = ;; 4 7 6
1156, 27, 11, 14, 1, 28, 19, 11, 31, 100, 114decma2c 11091 . . . 4  |-  ( (; 5
2  x.  N )  + ; 6 8 )  = ;;;; 6 5 5 3 6
116 2exp16 15048 . . . 4  |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
117 eqid 2422 . . . . 5  |-  ( 2 ^ 8 )  =  ( 2 ^ 8 )
118 eqid 2422 . . . . 5  |-  ( ( 2 ^ 8 )  x.  ( 2 ^ 8 ) )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1193, 14, 26, 117, 118numexp2x 15038 . . . 4  |-  ( 2 ^; 1 6 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
120115, 116, 1193eqtr2i 2457 . . 3  |-  ( (; 5
2  x.  N )  + ; 6 8 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1219, 10, 14, 20, 21, 15, 22, 26, 120mod2xi 15028 . 2  |-  ( ( 2 ^; 1 6 )  mod 
N )  =  (; 6
8  mod  N )
122 6p1e7 10738 . . 3  |-  ( 6  +  1 )  =  7
123 eqid 2422 . . 3  |- ; 1 6  = ; 1 6
1242, 11, 122, 123decsuc 11074 . 2  |-  (; 1 6  +  1 )  = ; 1 7
12518nn0cni 10881 . . . 4  |- ;; 1 3 6  e.  CC
126125addid2i 9821 . . 3  |-  ( 0  + ;; 1 3 6 )  = ;; 1 3 6
1279nncni 10619 . . . . 5  |-  N  e.  CC
128127mul02i 9822 . . . 4  |-  ( 0  x.  N )  =  0
129128oveq1i 6311 . . 3  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  ( 0  + ;; 1 3 6 )
130 6t2e12 11128 . . . . 5  |-  ( 6  x.  2 )  = ; 1
2
1312, 3, 52, 130decsuc 11074 . . . 4  |-  ( ( 6  x.  2 )  +  1 )  = ; 1
3
1323, 11, 14, 28, 11, 2, 131, 25decmul1c 11098 . . 3  |-  (; 6 8  x.  2 )  = ;; 1 3 6
133126, 129, 1323eqtr4i 2461 . 2  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  (; 6
8  x.  2 )
1349, 10, 12, 13, 15, 18, 121, 124, 133modxp1i 15029 1  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437  (class class class)co 6301   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   NNcn 10609   2c2 10659   3c3 10660   4c4 10661   5c5 10662   6c6 10663   7c7 10664   8c8 10665   9c9 10666   10c10 10667  ;cdc 11051    mod cmo 12095   ^cexp 12271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-sup 7958  df-inf 7959  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272
This theorem is referenced by:  1259lem2  15090  1259lem4  15092
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