MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1259lem2 Structured version   Unicode version

Theorem 1259lem2 14468
Description: Lemma for 1259prm 14472. 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 10807 . . . . . 6  |-  1  e.  NN0
3 2nn0 10808 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10986 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10811 . . . . 5  |-  5  e.  NN0
64, 5deccl 10986 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10696 . . . 4  |-  9  e.  NN
86, 7decnncl 10985 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2551 . 2  |-  N  e.  NN
10 2nn 10689 . 2  |-  2  e.  NN
11 7nn0 10813 . . 3  |-  7  e.  NN0
122, 11deccl 10986 . 2  |- ; 1 7  e.  NN0
13 4nn0 10810 . . . 4  |-  4  e.  NN0
142, 13deccl 10986 . . 3  |- ; 1 4  e.  NN0
1514nn0zi 10885 . 2  |- ; 1 4  e.  ZZ
16 3nn0 10809 . . . 4  |-  3  e.  NN0
172, 16deccl 10986 . . 3  |- ; 1 3  e.  NN0
18 6nn0 10812 . . 3  |-  6  e.  NN0
1917, 18deccl 10986 . 2  |- ;; 1 3 6  e.  NN0
20 8nn0 10814 . . . 4  |-  8  e.  NN0
2120, 11deccl 10986 . . 3  |- ; 8 7  e.  NN0
22 0nn0 10806 . . 3  |-  0  e.  NN0
2321, 22deccl 10986 . 2  |- ;; 8 7 0  e.  NN0
2411259lem1 14467 . 2  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
25 eqid 2467 . . 3  |- ; 1 7  = ; 1 7
26 2cn 10602 . . . . . 6  |-  2  e.  CC
2726mulid1i 9594 . . . . 5  |-  ( 2  x.  1 )  =  2
2827oveq1i 6292 . . . 4  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
29 2p1e3 10655 . . . 4  |-  ( 2  +  1 )  =  3
3028, 29eqtri 2496 . . 3  |-  ( ( 2  x.  1 )  +  1 )  =  3
31 7cn 10615 . . . 4  |-  7  e.  CC
32 7t2e14 11054 . . . 4  |-  ( 7  x.  2 )  = ; 1
4
3331, 26, 32mulcomli 9599 . . 3  |-  ( 2  x.  7 )  = ; 1
4
343, 2, 11, 25, 13, 2, 30, 33decmul2c 11020 . 2  |-  ( 2  x. ; 1 7 )  = ; 3
4
35 9nn0 10815 . . . 4  |-  9  e.  NN0
36 eqid 2467 . . . 4  |- ;; 8 7 0  = ;; 8 7 0
37 eqid 2467 . . . . 5  |- ;; 1 2 5  = ;; 1 2 5
38 eqid 2467 . . . . . 6  |- ; 8 7  = ; 8 7
39 eqid 2467 . . . . . 6  |- ; 1 2  = ; 1 2
40 8p1e9 10662 . . . . . 6  |-  ( 8  +  1 )  =  9
41 7p2e9 10676 . . . . . 6  |-  ( 7  +  2 )  =  9
4220, 11, 2, 3, 38, 39, 40, 41decadd 11013 . . . . 5  |-  (; 8 7  + ; 1 2 )  = ; 9
9
43 9p7e16 11039 . . . . . 6  |-  ( 9  +  7 )  = ; 1
6
44 eqid 2467 . . . . . . 7  |- ; 1 4  = ; 1 4
45 3cn 10606 . . . . . . . . 9  |-  3  e.  CC
46 ax-1cn 9546 . . . . . . . . 9  |-  1  e.  CC
47 3p1e4 10657 . . . . . . . . 9  |-  ( 3  +  1 )  =  4
4845, 46, 47addcomli 9767 . . . . . . . 8  |-  ( 1  +  3 )  =  4
4913dec0h 10988 . . . . . . . 8  |-  4  = ; 0 4
5048, 49eqtri 2496 . . . . . . 7  |-  ( 1  +  3 )  = ; 0
4
5146mulid1i 9594 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
52 00id 9750 . . . . . . . . 9  |-  ( 0  +  0 )  =  0
5351, 52oveq12i 6294 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
5446addid1i 9762 . . . . . . . 8  |-  ( 1  +  0 )  =  1
5553, 54eqtri 2496 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
56 4cn 10609 . . . . . . . . . 10  |-  4  e.  CC
5756mulid1i 9594 . . . . . . . . 9  |-  ( 4  x.  1 )  =  4
5857oveq1i 6292 . . . . . . . 8  |-  ( ( 4  x.  1 )  +  4 )  =  ( 4  +  4 )
59 4p4e8 10668 . . . . . . . 8  |-  ( 4  +  4 )  =  8
6020dec0h 10988 . . . . . . . 8  |-  8  = ; 0 8
6158, 59, 603eqtri 2500 . . . . . . 7  |-  ( ( 4  x.  1 )  +  4 )  = ; 0
8
622, 13, 22, 13, 44, 50, 2, 20, 22, 55, 61decmac 11011 . . . . . 6  |-  ( (; 1
4  x.  1 )  +  ( 1  +  3 ) )  = ; 1
8
6318dec0h 10988 . . . . . . 7  |-  6  = ; 0 6
6426mulid2i 9595 . . . . . . . . 9  |-  ( 1  x.  2 )  =  2
6546addid2i 9763 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
6664, 65oveq12i 6294 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
6766, 29eqtri 2496 . . . . . . 7  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  3
68 4t2e8 10685 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
6968oveq1i 6292 . . . . . . . 8  |-  ( ( 4  x.  2 )  +  6 )  =  ( 8  +  6 )
70 8p6e14 11031 . . . . . . . 8  |-  ( 8  +  6 )  = ; 1
4
7169, 70eqtri 2496 . . . . . . 7  |-  ( ( 4  x.  2 )  +  6 )  = ; 1
4
722, 13, 22, 18, 44, 63, 3, 13, 2, 67, 71decmac 11011 . . . . . 6  |-  ( (; 1
4  x.  2 )  +  6 )  = ; 3
4
732, 3, 2, 18, 39, 43, 14, 13, 16, 62, 72decma2c 11012 . . . . 5  |-  ( (; 1
4  x. ; 1 2 )  +  ( 9  +  7 ) )  = ;; 1 8 4
7435dec0h 10988 . . . . . 6  |-  9  = ; 0 9
75 5cn 10611 . . . . . . . . 9  |-  5  e.  CC
7675mulid2i 9595 . . . . . . . 8  |-  ( 1  x.  5 )  =  5
7726addid2i 9763 . . . . . . . 8  |-  ( 0  +  2 )  =  2
7876, 77oveq12i 6294 . . . . . . 7  |-  ( ( 1  x.  5 )  +  ( 0  +  2 ) )  =  ( 5  +  2 )
79 5p2e7 10669 . . . . . . 7  |-  ( 5  +  2 )  =  7
8078, 79eqtri 2496 . . . . . 6  |-  ( ( 1  x.  5 )  +  ( 0  +  2 ) )  =  7
81 5t4e20 11047 . . . . . . . 8  |-  ( 5  x.  4 )  = ; 2
0
8275, 56, 81mulcomli 9599 . . . . . . 7  |-  ( 4  x.  5 )  = ; 2
0
83 9cn 10619 . . . . . . . 8  |-  9  e.  CC
8483addid2i 9763 . . . . . . 7  |-  ( 0  +  9 )  =  9
853, 22, 35, 82, 84decaddi 11016 . . . . . 6  |-  ( ( 4  x.  5 )  +  9 )  = ; 2
9
862, 13, 22, 35, 44, 74, 5, 35, 3, 80, 85decmac 11011 . . . . 5  |-  ( (; 1
4  x.  5 )  +  9 )  = ; 7
9
874, 5, 35, 35, 37, 42, 14, 35, 11, 73, 86decma2c 11012 . . . 4  |-  ( (; 1
4  x. ;; 1 2 5 )  +  (; 8 7  + ; 1 2 ) )  = ;;; 1 8 4 9
8883mulid2i 9595 . . . . . . . . 9  |-  ( 1  x.  9 )  =  9
8988oveq1i 6292 . . . . . . . 8  |-  ( ( 1  x.  9 )  +  3 )  =  ( 9  +  3 )
90 9p3e12 11035 . . . . . . . 8  |-  ( 9  +  3 )  = ; 1
2
9189, 90eqtri 2496 . . . . . . 7  |-  ( ( 1  x.  9 )  +  3 )  = ; 1
2
92 9t4e36 11069 . . . . . . . 8  |-  ( 9  x.  4 )  = ; 3
6
9383, 56, 92mulcomli 9599 . . . . . . 7  |-  ( 4  x.  9 )  = ; 3
6
9435, 2, 13, 44, 18, 16, 91, 93decmul1c 11019 . . . . . 6  |-  (; 1 4  x.  9 )  = ;; 1 2 6
9594oveq1i 6292 . . . . 5  |-  ( (; 1
4  x.  9 )  +  0 )  =  (;; 1 2 6  +  0 )
964, 18deccl 10986 . . . . . . 7  |- ;; 1 2 6  e.  NN0
9796nn0cni 10803 . . . . . 6  |- ;; 1 2 6  e.  CC
9897addid1i 9762 . . . . 5  |-  (;; 1 2 6  +  0 )  = ;; 1 2 6
9995, 98eqtri 2496 . . . 4  |-  ( (; 1
4  x.  9 )  +  0 )  = ;; 1 2 6
1006, 35, 21, 22, 1, 36, 14, 18, 4, 87, 99decma2c 11012 . . 3  |-  ( (; 1
4  x.  N )  + ;; 8 7 0 )  = ;;;; 1 8 4 9 6
101 eqid 2467 . . . 4  |- ;; 1 3 6  = ;; 1 3 6
10220, 2deccl 10986 . . . 4  |- ; 8 1  e.  NN0
103 eqid 2467 . . . . 5  |- ; 1 3  = ; 1 3
104 eqid 2467 . . . . 5  |- ; 8 1  = ; 8 1
10513, 22deccl 10986 . . . . 5  |- ; 4 0  e.  NN0
106 eqid 2467 . . . . . . 7  |- ; 4 0  = ; 4 0
10756addid2i 9763 . . . . . . 7  |-  ( 0  +  4 )  =  4
108 8cn 10617 . . . . . . . 8  |-  8  e.  CC
109108addid1i 9762 . . . . . . 7  |-  ( 8  +  0 )  =  8
11022, 20, 13, 22, 60, 106, 107, 109decadd 11013 . . . . . 6  |-  ( 8  + ; 4 0 )  = ; 4
8
111 4p1e5 10658 . . . . . . . 8  |-  ( 4  +  1 )  =  5
1125dec0h 10988 . . . . . . . 8  |-  5  = ; 0 5
113111, 112eqtri 2496 . . . . . . 7  |-  ( 4  +  1 )  = ; 0
5
11445mulid1i 9594 . . . . . . . . 9  |-  ( 3  x.  1 )  =  3
115114oveq1i 6292 . . . . . . . 8  |-  ( ( 3  x.  1 )  +  5 )  =  ( 3  +  5 )
116 5p3e8 10670 . . . . . . . . 9  |-  ( 5  +  3 )  =  8
11775, 45, 116addcomli 9767 . . . . . . . 8  |-  ( 3  +  5 )  =  8
118115, 117, 603eqtri 2500 . . . . . . 7  |-  ( ( 3  x.  1 )  +  5 )  = ; 0
8
1192, 16, 22, 5, 103, 113, 2, 20, 22, 55, 118decmac 11011 . . . . . 6  |-  ( (; 1
3  x.  1 )  +  ( 4  +  1 ) )  = ; 1
8
120 6cn 10613 . . . . . . . . 9  |-  6  e.  CC
121120mulid1i 9594 . . . . . . . 8  |-  ( 6  x.  1 )  =  6
122121oveq1i 6292 . . . . . . 7  |-  ( ( 6  x.  1 )  +  8 )  =  ( 6  +  8 )
123108, 120, 70addcomli 9767 . . . . . . 7  |-  ( 6  +  8 )  = ; 1
4
124122, 123eqtri 2496 . . . . . 6  |-  ( ( 6  x.  1 )  +  8 )  = ; 1
4
12517, 18, 13, 20, 101, 110, 2, 13, 2, 119, 124decmac 11011 . . . . 5  |-  ( (;; 1 3 6  x.  1 )  +  ( 8  + ; 4 0 ) )  = ;; 1 8 4
1262dec0h 10988 . . . . . 6  |-  1  = ; 0 1
12765, 126eqtri 2496 . . . . . . 7  |-  ( 0  +  1 )  = ; 0
1
12845mulid2i 9595 . . . . . . . . 9  |-  ( 1  x.  3 )  =  3
129128, 65oveq12i 6294 . . . . . . . 8  |-  ( ( 1  x.  3 )  +  ( 0  +  1 ) )  =  ( 3  +  1 )
130129, 47eqtri 2496 . . . . . . 7  |-  ( ( 1  x.  3 )  +  ( 0  +  1 ) )  =  4
131 3t3e9 10684 . . . . . . . . 9  |-  ( 3  x.  3 )  =  9
132131oveq1i 6292 . . . . . . . 8  |-  ( ( 3  x.  3 )  +  1 )  =  ( 9  +  1 )
133 9p1e10 10663 . . . . . . . 8  |-  ( 9  +  1 )  =  10
134 dec10 11002 . . . . . . . 8  |-  10  = ; 1 0
135132, 133, 1343eqtri 2500 . . . . . . 7  |-  ( ( 3  x.  3 )  +  1 )  = ; 1
0
1362, 16, 22, 2, 103, 127, 16, 22, 2, 130, 135decmac 11011 . . . . . 6  |-  ( (; 1
3  x.  3 )  +  ( 0  +  1 ) )  = ; 4
0
137 6t3e18 11050 . . . . . . 7  |-  ( 6  x.  3 )  = ; 1
8
1382, 20, 2, 137, 40decaddi 11016 . . . . . 6  |-  ( ( 6  x.  3 )  +  1 )  = ; 1
9
13917, 18, 22, 2, 101, 126, 16, 35, 2, 136, 138decmac 11011 . . . . 5  |-  ( (;; 1 3 6  x.  3 )  +  1 )  = ;; 4 0 9
1402, 16, 20, 2, 103, 104, 19, 35, 105, 125, 139decma2c 11012 . . . 4  |-  ( (;; 1 3 6  x. ; 1
3 )  + ; 8 1 )  = ;;; 1 8 4 9
14116dec0h 10988 . . . . . 6  |-  3  = ; 0 3
142120mulid2i 9595 . . . . . . . 8  |-  ( 1  x.  6 )  =  6
143142, 77oveq12i 6294 . . . . . . 7  |-  ( ( 1  x.  6 )  +  ( 0  +  2 ) )  =  ( 6  +  2 )
144 6p2e8 10673 . . . . . . 7  |-  ( 6  +  2 )  =  8
145143, 144eqtri 2496 . . . . . 6  |-  ( ( 1  x.  6 )  +  ( 0  +  2 ) )  =  8
146120, 45, 137mulcomli 9599 . . . . . . 7  |-  ( 3  x.  6 )  = ; 1
8
147 1p1e2 10645 . . . . . . 7  |-  ( 1  +  1 )  =  2
148 8p3e11 11028 . . . . . . 7  |-  ( 8  +  3 )  = ; 1
1
1492, 20, 16, 146, 147, 2, 148decaddci 11017 . . . . . 6  |-  ( ( 3  x.  6 )  +  3 )  = ; 2
1
1502, 16, 22, 16, 103, 141, 18, 2, 3, 145, 149decmac 11011 . . . . 5  |-  ( (; 1
3  x.  6 )  +  3 )  = ; 8
1
151 6t6e36 11053 . . . . 5  |-  ( 6  x.  6 )  = ; 3
6
15218, 17, 18, 101, 18, 16, 150, 151decmul1c 11019 . . . 4  |-  (;; 1 3 6  x.  6 )  = ;; 8 1 6
15319, 17, 18, 101, 18, 102, 140, 152decmul2c 11020 . . 3  |-  (;; 1 3 6  x. ;; 1 3 6 )  = ;;;; 1 8 4 9 6
154100, 153eqtr4i 2499 . 2  |-  ( (; 1
4  x.  N )  + ;; 8 7 0 )  =  (;; 1 3 6  x. ;; 1 3 6 )
1559, 10, 12, 15, 19, 23, 24, 34, 154mod2xi 14410 1  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379  (class class class)co 6282   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493   NNcn 10532   2c2 10581   3c3 10582   4c4 10583   5c5 10584   6c6 10585   7c7 10586   8c8 10587   9c9 10588   10c10 10589  ;cdc 10972    mod cmo 11960   ^cexp 12130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-rp 11217  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131
This theorem is referenced by:  1259lem3  14469  1259lem5  14471
  Copyright terms: Public domain W3C validator