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Theorem 1259lem1 14151
Description: Lemma for 1259prm 14156. 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 10591 . . . . . 6  |-  1  e.  NN0
3 2nn0 10592 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10765 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10595 . . . . 5  |-  5  e.  NN0
64, 5deccl 10765 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10482 . . . 4  |-  9  e.  NN
86, 7decnncl 10764 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2511 . 2  |-  N  e.  NN
10 2nn 10475 . 2  |-  2  e.  NN
11 6nn0 10596 . . 3  |-  6  e.  NN0
122, 11deccl 10765 . 2  |- ; 1 6  e.  NN0
13 0z 10653 . 2  |-  0  e.  ZZ
14 8nn0 10598 . . 3  |-  8  e.  NN0
1511, 14deccl 10765 . 2  |- ; 6 8  e.  NN0
16 3nn0 10593 . . . 4  |-  3  e.  NN0
172, 16deccl 10765 . . 3  |- ; 1 3  e.  NN0
1817, 11deccl 10765 . 2  |- ;; 1 3 6  e.  NN0
195, 3deccl 10765 . . . 4  |- ; 5 2  e.  NN0
2019nn0zi 10667 . . 3  |- ; 5 2  e.  ZZ
213, 14nn0expcli 11887 . . 3  |-  ( 2 ^ 8 )  e. 
NN0
22 eqid 2441 . . 3  |-  ( ( 2 ^ 8 )  mod  N )  =  ( ( 2 ^ 8 )  mod  N
)
2314nn0cni 10587 . . . 4  |-  8  e.  CC
24 2cn 10388 . . . 4  |-  2  e.  CC
25 8t2e16 10839 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
2623, 24, 25mulcomli 9389 . . 3  |-  ( 2  x.  8 )  = ; 1
6
27 9nn0 10599 . . . . 5  |-  9  e.  NN0
28 eqid 2441 . . . . 5  |- ; 6 8  = ; 6 8
29 4nn0 10594 . . . . . 6  |-  4  e.  NN0
30 7nn0 10597 . . . . . 6  |-  7  e.  NN0
3129, 30deccl 10765 . . . . 5  |- ; 4 7  e.  NN0
32 eqid 2441 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
33 0nn0 10590 . . . . . . 7  |-  0  e.  NN0
3411dec0h 10767 . . . . . . 7  |-  6  = ; 0 6
35 eqid 2441 . . . . . . 7  |- ; 4 7  = ; 4 7
36 4cn 10395 . . . . . . . . . 10  |-  4  e.  CC
3736addid2i 9553 . . . . . . . . 9  |-  ( 0  +  4 )  =  4
3837oveq1i 6100 . . . . . . . 8  |-  ( ( 0  +  4 )  +  1 )  =  ( 4  +  1 )
39 4p1e5 10444 . . . . . . . 8  |-  ( 4  +  1 )  =  5
4038, 39eqtri 2461 . . . . . . 7  |-  ( ( 0  +  4 )  +  1 )  =  5
41 7cn 10401 . . . . . . . 8  |-  7  e.  CC
42 6cn 10399 . . . . . . . 8  |-  6  e.  CC
43 7p6e13 10805 . . . . . . . 8  |-  ( 7  +  6 )  = ; 1
3
4441, 42, 43addcomli 9557 . . . . . . 7  |-  ( 6  +  7 )  = ; 1
3
4533, 11, 29, 30, 34, 35, 40, 16, 44decaddc 10793 . . . . . 6  |-  ( 6  + ; 4 7 )  = ; 5
3
463, 11deccl 10765 . . . . . 6  |- ; 2 6  e.  NN0
47 eqid 2441 . . . . . . 7  |- ; 1 2  = ; 1 2
485dec0h 10767 . . . . . . . 8  |-  5  = ; 0 5
49 eqid 2441 . . . . . . . 8  |- ; 2 6  = ; 2 6
5024addid2i 9553 . . . . . . . . . 10  |-  ( 0  +  2 )  =  2
5150oveq1i 6100 . . . . . . . . 9  |-  ( ( 0  +  2 )  +  1 )  =  ( 2  +  1 )
52 2p1e3 10441 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
5351, 52eqtri 2461 . . . . . . . 8  |-  ( ( 0  +  2 )  +  1 )  =  3
54 5cn 10397 . . . . . . . . 9  |-  5  e.  CC
55 6p5e11 10801 . . . . . . . . 9  |-  ( 6  +  5 )  = ; 1
1
5642, 54, 55addcomli 9557 . . . . . . . 8  |-  ( 5  +  6 )  = ; 1
1
5733, 5, 3, 11, 48, 49, 53, 2, 56decaddc 10793 . . . . . . 7  |-  ( 5  + ; 2 6 )  = ; 3
1
58 10nn0 10600 . . . . . . 7  |-  10  e.  NN0
59 eqid 2441 . . . . . . . 8  |- ; 5 2  = ; 5 2
6016dec0h 10767 . . . . . . . . 9  |-  3  = ; 0 3
61 dec10 10781 . . . . . . . . 9  |-  10  = ; 1 0
62 ax-1cn 9336 . . . . . . . . . 10  |-  1  e.  CC
6362addid2i 9553 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
64 3cn 10392 . . . . . . . . . 10  |-  3  e.  CC
6564addid1i 9552 . . . . . . . . 9  |-  ( 3  +  0 )  =  3
6633, 16, 2, 33, 60, 61, 63, 65decadd 10792 . . . . . . . 8  |-  ( 3  +  10 )  = ; 1
3
6754mulid1i 9384 . . . . . . . . . 10  |-  ( 5  x.  1 )  =  5
6862addid1i 9552 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
6967, 68oveq12i 6102 . . . . . . . . 9  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  ( 5  +  1 )
70 5p1e6 10445 . . . . . . . . 9  |-  ( 5  +  1 )  =  6
7169, 70eqtri 2461 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 1  +  0 ) )  =  6
7224mulid1i 9384 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
7372oveq1i 6100 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  3 )  =  ( 2  +  3 )
74 3p2e5 10450 . . . . . . . . . 10  |-  ( 3  +  2 )  =  5
7564, 24, 74addcomli 9557 . . . . . . . . 9  |-  ( 2  +  3 )  =  5
7673, 75, 483eqtri 2465 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  3 )  = ; 0
5
775, 3, 2, 16, 59, 66, 2, 5, 33, 71, 76decmac 10790 . . . . . . 7  |-  ( (; 5
2  x.  1 )  +  ( 3  +  10 ) )  = ; 6
5
782dec0h 10767 . . . . . . . 8  |-  1  = ; 0 1
79 5t2e10 10472 . . . . . . . . . 10  |-  ( 5  x.  2 )  =  10
80 00id 9540 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
8179, 80oveq12i 6102 . . . . . . . . 9  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  ( 10  +  0 )
82 10nn 10483 . . . . . . . . . . 11  |-  10  e.  NN
8382nncni 10328 . . . . . . . . . 10  |-  10  e.  CC
8483addid1i 9552 . . . . . . . . 9  |-  ( 10  +  0 )  =  10
8581, 84eqtri 2461 . . . . . . . 8  |-  ( ( 5  x.  2 )  +  ( 0  +  0 ) )  =  10
86 2t2e4 10467 . . . . . . . . . 10  |-  ( 2  x.  2 )  =  4
8786oveq1i 6100 . . . . . . . . 9  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
8887, 39, 483eqtri 2465 . . . . . . . 8  |-  ( ( 2  x.  2 )  +  1 )  = ; 0
5
895, 3, 33, 2, 59, 78, 3, 5, 33, 85, 88decmac 10790 . . . . . . 7  |-  ( (; 5
2  x.  2 )  +  1 )  = ; 10 5
902, 3, 16, 2, 47, 57, 19, 5, 58, 77, 89decma2c 10791 . . . . . 6  |-  ( (; 5
2  x. ; 1 2 )  +  ( 5  + ; 2 6 ) )  = ;; 6 5 5
9163oveq2i 6101 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  =  ( ( 5  x.  5 )  +  1 )
92 5t5e25 10827 . . . . . . . . 9  |-  ( 5  x.  5 )  = ; 2
5
933, 5, 70, 92decsuc 10774 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  1 )  = ; 2
6
9491, 93eqtri 2461 . . . . . . 7  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  = ; 2
6
9554, 24, 79mulcomli 9389 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
9695, 61eqtri 2461 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
9764addid2i 9553 . . . . . . . 8  |-  ( 0  +  3 )  =  3
982, 33, 16, 96, 97decaddi 10795 . . . . . . 7  |-  ( ( 2  x.  5 )  +  3 )  = ; 1
3
995, 3, 33, 16, 59, 60, 5, 16, 2, 94, 98decmac 10790 . . . . . 6  |-  ( (; 5
2  x.  5 )  +  3 )  = ;; 2 6 3
1004, 5, 5, 16, 32, 45, 19, 16, 46, 90, 99decma2c 10791 . . . . 5  |-  ( (; 5
2  x. ;; 1 2 5 )  +  ( 6  + ; 4 7 ) )  = ;;; 6 5 5 3
10114dec0h 10767 . . . . . 6  |-  8  = ; 0 8
10250oveq2i 6101 . . . . . . 7  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  =  ( ( 5  x.  9 )  +  2 )
103 9cn 10405 . . . . . . . . 9  |-  9  e.  CC
104 9t5e45 10849 . . . . . . . . 9  |-  ( 9  x.  5 )  = ; 4
5
105103, 54, 104mulcomli 9389 . . . . . . . 8  |-  ( 5  x.  9 )  = ; 4
5
106 5p2e7 10455 . . . . . . . 8  |-  ( 5  +  2 )  =  7
10729, 5, 3, 105, 106decaddi 10795 . . . . . . 7  |-  ( ( 5  x.  9 )  +  2 )  = ; 4
7
108102, 107eqtri 2461 . . . . . 6  |-  ( ( 5  x.  9 )  +  ( 0  +  2 ) )  = ; 4
7
109 9t2e18 10846 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
110103, 24, 109mulcomli 9389 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
111 1p1e2 10431 . . . . . . 7  |-  ( 1  +  1 )  =  2
112 8p8e16 10812 . . . . . . 7  |-  ( 8  +  8 )  = ; 1
6
1132, 14, 14, 110, 111, 11, 112decaddci 10796 . . . . . 6  |-  ( ( 2  x.  9 )  +  8 )  = ; 2
6
1145, 3, 33, 14, 59, 101, 27, 11, 3, 108, 113decmac 10790 . . . . 5  |-  ( (; 5
2  x.  9 )  +  8 )  = ;; 4 7 6
1156, 27, 11, 14, 1, 28, 19, 11, 31, 100, 114decma2c 10791 . . . 4  |-  ( (; 5
2  x.  N )  + ; 6 8 )  = ;;;; 6 5 5 3 6
116 2exp16 14113 . . . 4  |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
117 eqid 2441 . . . . 5  |-  ( 2 ^ 8 )  =  ( 2 ^ 8 )
118 eqid 2441 . . . . 5  |-  ( ( 2 ^ 8 )  x.  ( 2 ^ 8 ) )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1193, 14, 26, 117, 118numexp2x 14104 . . . 4  |-  ( 2 ^; 1 6 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
120115, 116, 1193eqtr2i 2467 . . 3  |-  ( (; 5
2  x.  N )  + ; 6 8 )  =  ( ( 2 ^ 8 )  x.  (
2 ^ 8 ) )
1219, 10, 14, 20, 21, 15, 22, 26, 120mod2xi 14094 . 2  |-  ( ( 2 ^; 1 6 )  mod 
N )  =  (; 6
8  mod  N )
122 6p1e7 10446 . . 3  |-  ( 6  +  1 )  =  7
123 eqid 2441 . . 3  |- ; 1 6  = ; 1 6
1242, 11, 122, 123decsuc 10774 . 2  |-  (; 1 6  +  1 )  = ; 1 7
12518nn0cni 10587 . . . 4  |- ;; 1 3 6  e.  CC
126125addid2i 9553 . . 3  |-  ( 0  + ;; 1 3 6 )  = ;; 1 3 6
1279nncni 10328 . . . . 5  |-  N  e.  CC
128127mul02i 9554 . . . 4  |-  ( 0  x.  N )  =  0
129128oveq1i 6100 . . 3  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  ( 0  + ;; 1 3 6 )
130 6t2e12 10828 . . . . 5  |-  ( 6  x.  2 )  = ; 1
2
1312, 3, 52, 130decsuc 10774 . . . 4  |-  ( ( 6  x.  2 )  +  1 )  = ; 1
3
1323, 11, 14, 28, 11, 2, 131, 25decmul1c 10798 . . 3  |-  (; 6 8  x.  2 )  = ;; 1 3 6
133126, 129, 1323eqtr4i 2471 . 2  |-  ( ( 0  x.  N )  + ;; 1 3 6 )  =  (; 6
8  x.  2 )
1349, 10, 12, 13, 15, 18, 121, 124, 133modxp1i 14095 1  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364  (class class class)co 6090   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283   NNcn 10318   2c2 10367   3c3 10368   4c4 10369   5c5 10370   6c6 10371   7c7 10372   8c8 10373   9c9 10374   10c10 10375  ;cdc 10751    mod cmo 11704   ^cexp 11861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-rp 10988  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862
This theorem is referenced by:  1259lem2  14152  1259lem4  14154
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