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Theorem 1259lem2 15096
Description: Lemma for 1259prm 15100. Calculate a power mod. In decimal, we calculate  2 ^ 3 4  =  ( 2 ^ 1 7 ) ^ 2  ==  1
3 6 ^ 2  ==  1 4 N  +  8 7 0. (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
1259lem2  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )

Proof of Theorem 1259lem2
StepHypRef Expression
1 1259prm.1 . . 3  |-  N  = ;;; 1 2 5 9
2 1nn0 10887 . . . . . 6  |-  1  e.  NN0
3 2nn0 10888 . . . . . 6  |-  2  e.  NN0
42, 3deccl 11067 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10891 . . . . 5  |-  5  e.  NN0
64, 5deccl 11067 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10776 . . . 4  |-  9  e.  NN
86, 7decnncl 11066 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2507 . 2  |-  N  e.  NN
10 2nn 10769 . 2  |-  2  e.  NN
11 7nn0 10893 . . 3  |-  7  e.  NN0
122, 11deccl 11067 . 2  |- ; 1 7  e.  NN0
13 4nn0 10890 . . . 4  |-  4  e.  NN0
142, 13deccl 11067 . . 3  |- ; 1 4  e.  NN0
1514nn0zi 10964 . 2  |- ; 1 4  e.  ZZ
16 3nn0 10889 . . . 4  |-  3  e.  NN0
172, 16deccl 11067 . . 3  |- ; 1 3  e.  NN0
18 6nn0 10892 . . 3  |-  6  e.  NN0
1917, 18deccl 11067 . 2  |- ;; 1 3 6  e.  NN0
20 8nn0 10894 . . . 4  |-  8  e.  NN0
2120, 11deccl 11067 . . 3  |- ; 8 7  e.  NN0
22 0nn0 10886 . . 3  |-  0  e.  NN0
2321, 22deccl 11067 . 2  |- ;; 8 7 0  e.  NN0
2411259lem1 15095 . 2  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
25 eqid 2423 . . 3  |- ; 1 7  = ; 1 7
26 2cn 10682 . . . . . 6  |-  2  e.  CC
2726mulid1i 9647 . . . . 5  |-  ( 2  x.  1 )  =  2
2827oveq1i 6313 . . . 4  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
29 2p1e3 10735 . . . 4  |-  ( 2  +  1 )  =  3
3028, 29eqtri 2452 . . 3  |-  ( ( 2  x.  1 )  +  1 )  =  3
31 7cn 10695 . . . 4  |-  7  e.  CC
32 7t2e14 11135 . . . 4  |-  ( 7  x.  2 )  = ; 1
4
3331, 26, 32mulcomli 9652 . . 3  |-  ( 2  x.  7 )  = ; 1
4
343, 2, 11, 25, 13, 2, 30, 33decmul2c 11101 . 2  |-  ( 2  x. ; 1 7 )  = ; 3
4
35 9nn0 10895 . . . 4  |-  9  e.  NN0
36 eqid 2423 . . . 4  |- ;; 8 7 0  = ;; 8 7 0
37 eqid 2423 . . . . 5  |- ;; 1 2 5  = ;; 1 2 5
38 eqid 2423 . . . . . 6  |- ; 8 7  = ; 8 7
39 eqid 2423 . . . . . 6  |- ; 1 2  = ; 1 2
40 8p1e9 10742 . . . . . 6  |-  ( 8  +  1 )  =  9
41 7p2e9 10756 . . . . . 6  |-  ( 7  +  2 )  =  9
4220, 11, 2, 3, 38, 39, 40, 41decadd 11094 . . . . 5  |-  (; 8 7  + ; 1 2 )  = ; 9
9
43 9p7e16 11120 . . . . . 6  |-  ( 9  +  7 )  = ; 1
6
44 eqid 2423 . . . . . . 7  |- ; 1 4  = ; 1 4
45 3cn 10686 . . . . . . . . 9  |-  3  e.  CC
46 ax-1cn 9599 . . . . . . . . 9  |-  1  e.  CC
47 3p1e4 10737 . . . . . . . . 9  |-  ( 3  +  1 )  =  4
4845, 46, 47addcomli 9827 . . . . . . . 8  |-  ( 1  +  3 )  =  4
4913dec0h 11069 . . . . . . . 8  |-  4  = ; 0 4
5048, 49eqtri 2452 . . . . . . 7  |-  ( 1  +  3 )  = ; 0
4
5146mulid1i 9647 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
52 00id 9810 . . . . . . . . 9  |-  ( 0  +  0 )  =  0
5351, 52oveq12i 6315 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
5446addid1i 9822 . . . . . . . 8  |-  ( 1  +  0 )  =  1
5553, 54eqtri 2452 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
56 4cn 10689 . . . . . . . . . 10  |-  4  e.  CC
5756mulid1i 9647 . . . . . . . . 9  |-  ( 4  x.  1 )  =  4
5857oveq1i 6313 . . . . . . . 8  |-  ( ( 4  x.  1 )  +  4 )  =  ( 4  +  4 )
59 4p4e8 10748 . . . . . . . 8  |-  ( 4  +  4 )  =  8
6020dec0h 11069 . . . . . . . 8  |-  8  = ; 0 8
6158, 59, 603eqtri 2456 . . . . . . 7  |-  ( ( 4  x.  1 )  +  4 )  = ; 0
8
622, 13, 22, 13, 44, 50, 2, 20, 22, 55, 61decmac 11092 . . . . . 6  |-  ( (; 1
4  x.  1 )  +  ( 1  +  3 ) )  = ; 1
8
6318dec0h 11069 . . . . . . 7  |-  6  = ; 0 6
6426mulid2i 9648 . . . . . . . . 9  |-  ( 1  x.  2 )  =  2
6546addid2i 9823 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
6664, 65oveq12i 6315 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
6766, 29eqtri 2452 . . . . . . 7  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  3
68 4t2e8 10765 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
6968oveq1i 6313 . . . . . . . 8  |-  ( ( 4  x.  2 )  +  6 )  =  ( 8  +  6 )
70 8p6e14 11112 . . . . . . . 8  |-  ( 8  +  6 )  = ; 1
4
7169, 70eqtri 2452 . . . . . . 7  |-  ( ( 4  x.  2 )  +  6 )  = ; 1
4
722, 13, 22, 18, 44, 63, 3, 13, 2, 67, 71decmac 11092 . . . . . 6  |-  ( (; 1
4  x.  2 )  +  6 )  = ; 3
4
732, 3, 2, 18, 39, 43, 14, 13, 16, 62, 72decma2c 11093 . . . . 5  |-  ( (; 1
4  x. ; 1 2 )  +  ( 9  +  7 ) )  = ;; 1 8 4
7435dec0h 11069 . . . . . 6  |-  9  = ; 0 9
75 5cn 10691 . . . . . . . . 9  |-  5  e.  CC
7675mulid2i 9648 . . . . . . . 8  |-  ( 1  x.  5 )  =  5
7726addid2i 9823 . . . . . . . 8  |-  ( 0  +  2 )  =  2
7876, 77oveq12i 6315 . . . . . . 7  |-  ( ( 1  x.  5 )  +  ( 0  +  2 ) )  =  ( 5  +  2 )
79 5p2e7 10749 . . . . . . 7  |-  ( 5  +  2 )  =  7
8078, 79eqtri 2452 . . . . . 6  |-  ( ( 1  x.  5 )  +  ( 0  +  2 ) )  =  7
81 5t4e20 11128 . . . . . . . 8  |-  ( 5  x.  4 )  = ; 2
0
8275, 56, 81mulcomli 9652 . . . . . . 7  |-  ( 4  x.  5 )  = ; 2
0
83 9cn 10699 . . . . . . . 8  |-  9  e.  CC
8483addid2i 9823 . . . . . . 7  |-  ( 0  +  9 )  =  9
853, 22, 35, 82, 84decaddi 11097 . . . . . 6  |-  ( ( 4  x.  5 )  +  9 )  = ; 2
9
862, 13, 22, 35, 44, 74, 5, 35, 3, 80, 85decmac 11092 . . . . 5  |-  ( (; 1
4  x.  5 )  +  9 )  = ; 7
9
874, 5, 35, 35, 37, 42, 14, 35, 11, 73, 86decma2c 11093 . . . 4  |-  ( (; 1
4  x. ;; 1 2 5 )  +  (; 8 7  + ; 1 2 ) )  = ;;; 1 8 4 9
8883mulid2i 9648 . . . . . . . . 9  |-  ( 1  x.  9 )  =  9
8988oveq1i 6313 . . . . . . . 8  |-  ( ( 1  x.  9 )  +  3 )  =  ( 9  +  3 )
90 9p3e12 11116 . . . . . . . 8  |-  ( 9  +  3 )  = ; 1
2
9189, 90eqtri 2452 . . . . . . 7  |-  ( ( 1  x.  9 )  +  3 )  = ; 1
2
92 9t4e36 11150 . . . . . . . 8  |-  ( 9  x.  4 )  = ; 3
6
9383, 56, 92mulcomli 9652 . . . . . . 7  |-  ( 4  x.  9 )  = ; 3
6
9435, 2, 13, 44, 18, 16, 91, 93decmul1c 11100 . . . . . 6  |-  (; 1 4  x.  9 )  = ;; 1 2 6
9594oveq1i 6313 . . . . 5  |-  ( (; 1
4  x.  9 )  +  0 )  =  (;; 1 2 6  +  0 )
964, 18deccl 11067 . . . . . . 7  |- ;; 1 2 6  e.  NN0
9796nn0cni 10883 . . . . . 6  |- ;; 1 2 6  e.  CC
9897addid1i 9822 . . . . 5  |-  (;; 1 2 6  +  0 )  = ;; 1 2 6
9995, 98eqtri 2452 . . . 4  |-  ( (; 1
4  x.  9 )  +  0 )  = ;; 1 2 6
1006, 35, 21, 22, 1, 36, 14, 18, 4, 87, 99decma2c 11093 . . 3  |-  ( (; 1
4  x.  N )  + ;; 8 7 0 )  = ;;;; 1 8 4 9 6
101 eqid 2423 . . . 4  |- ;; 1 3 6  = ;; 1 3 6
10220, 2deccl 11067 . . . 4  |- ; 8 1  e.  NN0
103 eqid 2423 . . . . 5  |- ; 1 3  = ; 1 3
104 eqid 2423 . . . . 5  |- ; 8 1  = ; 8 1
10513, 22deccl 11067 . . . . 5  |- ; 4 0  e.  NN0
106 eqid 2423 . . . . . . 7  |- ; 4 0  = ; 4 0
10756addid2i 9823 . . . . . . 7  |-  ( 0  +  4 )  =  4
108 8cn 10697 . . . . . . . 8  |-  8  e.  CC
109108addid1i 9822 . . . . . . 7  |-  ( 8  +  0 )  =  8
11022, 20, 13, 22, 60, 106, 107, 109decadd 11094 . . . . . 6  |-  ( 8  + ; 4 0 )  = ; 4
8
111 4p1e5 10738 . . . . . . . 8  |-  ( 4  +  1 )  =  5
1125dec0h 11069 . . . . . . . 8  |-  5  = ; 0 5
113111, 112eqtri 2452 . . . . . . 7  |-  ( 4  +  1 )  = ; 0
5
11445mulid1i 9647 . . . . . . . . 9  |-  ( 3  x.  1 )  =  3
115114oveq1i 6313 . . . . . . . 8  |-  ( ( 3  x.  1 )  +  5 )  =  ( 3  +  5 )
116 5p3e8 10750 . . . . . . . . 9  |-  ( 5  +  3 )  =  8
11775, 45, 116addcomli 9827 . . . . . . . 8  |-  ( 3  +  5 )  =  8
118115, 117, 603eqtri 2456 . . . . . . 7  |-  ( ( 3  x.  1 )  +  5 )  = ; 0
8
1192, 16, 22, 5, 103, 113, 2, 20, 22, 55, 118decmac 11092 . . . . . 6  |-  ( (; 1
3  x.  1 )  +  ( 4  +  1 ) )  = ; 1
8
120 6cn 10693 . . . . . . . . 9  |-  6  e.  CC
121120mulid1i 9647 . . . . . . . 8  |-  ( 6  x.  1 )  =  6
122121oveq1i 6313 . . . . . . 7  |-  ( ( 6  x.  1 )  +  8 )  =  ( 6  +  8 )
123108, 120, 70addcomli 9827 . . . . . . 7  |-  ( 6  +  8 )  = ; 1
4
124122, 123eqtri 2452 . . . . . 6  |-  ( ( 6  x.  1 )  +  8 )  = ; 1
4
12517, 18, 13, 20, 101, 110, 2, 13, 2, 119, 124decmac 11092 . . . . 5  |-  ( (;; 1 3 6  x.  1 )  +  ( 8  + ; 4 0 ) )  = ;; 1 8 4
1262dec0h 11069 . . . . . 6  |-  1  = ; 0 1
12765, 126eqtri 2452 . . . . . . 7  |-  ( 0  +  1 )  = ; 0
1
12845mulid2i 9648 . . . . . . . . 9  |-  ( 1  x.  3 )  =  3
129128, 65oveq12i 6315 . . . . . . . 8  |-  ( ( 1  x.  3 )  +  ( 0  +  1 ) )  =  ( 3  +  1 )
130129, 47eqtri 2452 . . . . . . 7  |-  ( ( 1  x.  3 )  +  ( 0  +  1 ) )  =  4
131 3t3e9 10764 . . . . . . . . 9  |-  ( 3  x.  3 )  =  9
132131oveq1i 6313 . . . . . . . 8  |-  ( ( 3  x.  3 )  +  1 )  =  ( 9  +  1 )
133 9p1e10 10743 . . . . . . . 8  |-  ( 9  +  1 )  =  10
134 dec10 11083 . . . . . . . 8  |-  10  = ; 1 0
135132, 133, 1343eqtri 2456 . . . . . . 7  |-  ( ( 3  x.  3 )  +  1 )  = ; 1
0
1362, 16, 22, 2, 103, 127, 16, 22, 2, 130, 135decmac 11092 . . . . . 6  |-  ( (; 1
3  x.  3 )  +  ( 0  +  1 ) )  = ; 4
0
137 6t3e18 11131 . . . . . . 7  |-  ( 6  x.  3 )  = ; 1
8
1382, 20, 2, 137, 40decaddi 11097 . . . . . 6  |-  ( ( 6  x.  3 )  +  1 )  = ; 1
9
13917, 18, 22, 2, 101, 126, 16, 35, 2, 136, 138decmac 11092 . . . . 5  |-  ( (;; 1 3 6  x.  3 )  +  1 )  = ;; 4 0 9
1402, 16, 20, 2, 103, 104, 19, 35, 105, 125, 139decma2c 11093 . . . 4  |-  ( (;; 1 3 6  x. ; 1
3 )  + ; 8 1 )  = ;;; 1 8 4 9
14116dec0h 11069 . . . . . 6  |-  3  = ; 0 3
142120mulid2i 9648 . . . . . . . 8  |-  ( 1  x.  6 )  =  6
143142, 77oveq12i 6315 . . . . . . 7  |-  ( ( 1  x.  6 )  +  ( 0  +  2 ) )  =  ( 6  +  2 )
144 6p2e8 10753 . . . . . . 7  |-  ( 6  +  2 )  =  8
145143, 144eqtri 2452 . . . . . 6  |-  ( ( 1  x.  6 )  +  ( 0  +  2 ) )  =  8
146120, 45, 137mulcomli 9652 . . . . . . 7  |-  ( 3  x.  6 )  = ; 1
8
147 1p1e2 10725 . . . . . . 7  |-  ( 1  +  1 )  =  2
148 8p3e11 11109 . . . . . . 7  |-  ( 8  +  3 )  = ; 1
1
1492, 20, 16, 146, 147, 2, 148decaddci 11098 . . . . . 6  |-  ( ( 3  x.  6 )  +  3 )  = ; 2
1
1502, 16, 22, 16, 103, 141, 18, 2, 3, 145, 149decmac 11092 . . . . 5  |-  ( (; 1
3  x.  6 )  +  3 )  = ; 8
1
151 6t6e36 11134 . . . . 5  |-  ( 6  x.  6 )  = ; 3
6
15218, 17, 18, 101, 18, 16, 150, 151decmul1c 11100 . . . 4  |-  (;; 1 3 6  x.  6 )  = ;; 8 1 6
15319, 17, 18, 101, 18, 102, 140, 152decmul2c 11101 . . 3  |-  (;; 1 3 6  x. ;; 1 3 6 )  = ;;;; 1 8 4 9 6
154100, 153eqtr4i 2455 . 2  |-  ( (; 1
4  x.  N )  + ;; 8 7 0 )  =  (;; 1 3 6  x. ;; 1 3 6 )
1559, 10, 12, 15, 19, 23, 24, 34, 154mod2xi 15034 1  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1438  (class class class)co 6303   0cc0 9541   1c1 9542    + caddc 9544    x. cmul 9546   NNcn 10611   2c2 10661   3c3 10662   4c4 10663   5c5 10664   6c6 10665   7c7 10666   8c8 10667   9c9 10668   10c10 10669  ;cdc 11053    mod cmo 12097   ^cexp 12273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-sup 7960  df-inf 7961  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-rp 11305  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274
This theorem is referenced by:  1259lem3  15097  1259lem5  15099
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