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Theorem 1259lem2 14635
Description: Lemma for 1259prm 14639. 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 10746 . . . . . 6  |-  1  e.  NN0
3 2nn0 10747 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10927 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10750 . . . . 5  |-  5  e.  NN0
64, 5deccl 10927 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10635 . . . 4  |-  9  e.  NN
86, 7decnncl 10926 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2476 . 2  |-  N  e.  NN
10 2nn 10628 . 2  |-  2  e.  NN
11 7nn0 10752 . . 3  |-  7  e.  NN0
122, 11deccl 10927 . 2  |- ; 1 7  e.  NN0
13 4nn0 10749 . . . 4  |-  4  e.  NN0
142, 13deccl 10927 . . 3  |- ; 1 4  e.  NN0
1514nn0zi 10824 . 2  |- ; 1 4  e.  ZZ
16 3nn0 10748 . . . 4  |-  3  e.  NN0
172, 16deccl 10927 . . 3  |- ; 1 3  e.  NN0
18 6nn0 10751 . . 3  |-  6  e.  NN0
1917, 18deccl 10927 . 2  |- ;; 1 3 6  e.  NN0
20 8nn0 10753 . . . 4  |-  8  e.  NN0
2120, 11deccl 10927 . . 3  |- ; 8 7  e.  NN0
22 0nn0 10745 . . 3  |-  0  e.  NN0
2321, 22deccl 10927 . 2  |- ;; 8 7 0  e.  NN0
2411259lem1 14634 . 2  |-  ( ( 2 ^; 1 7 )  mod 
N )  =  (;; 1 3 6  mod 
N )
25 eqid 2392 . . 3  |- ; 1 7  = ; 1 7
26 2cn 10541 . . . . . 6  |-  2  e.  CC
2726mulid1i 9527 . . . . 5  |-  ( 2  x.  1 )  =  2
2827oveq1i 6224 . . . 4  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
29 2p1e3 10594 . . . 4  |-  ( 2  +  1 )  =  3
3028, 29eqtri 2421 . . 3  |-  ( ( 2  x.  1 )  +  1 )  =  3
31 7cn 10554 . . . 4  |-  7  e.  CC
32 7t2e14 10995 . . . 4  |-  ( 7  x.  2 )  = ; 1
4
3331, 26, 32mulcomli 9532 . . 3  |-  ( 2  x.  7 )  = ; 1
4
343, 2, 11, 25, 13, 2, 30, 33decmul2c 10961 . 2  |-  ( 2  x. ; 1 7 )  = ; 3
4
35 9nn0 10754 . . . 4  |-  9  e.  NN0
36 eqid 2392 . . . 4  |- ;; 8 7 0  = ;; 8 7 0
37 eqid 2392 . . . . 5  |- ;; 1 2 5  = ;; 1 2 5
38 eqid 2392 . . . . . 6  |- ; 8 7  = ; 8 7
39 eqid 2392 . . . . . 6  |- ; 1 2  = ; 1 2
40 8p1e9 10601 . . . . . 6  |-  ( 8  +  1 )  =  9
41 7p2e9 10615 . . . . . 6  |-  ( 7  +  2 )  =  9
4220, 11, 2, 3, 38, 39, 40, 41decadd 10954 . . . . 5  |-  (; 8 7  + ; 1 2 )  = ; 9
9
43 9p7e16 10980 . . . . . 6  |-  ( 9  +  7 )  = ; 1
6
44 eqid 2392 . . . . . . 7  |- ; 1 4  = ; 1 4
45 3cn 10545 . . . . . . . . 9  |-  3  e.  CC
46 ax-1cn 9479 . . . . . . . . 9  |-  1  e.  CC
47 3p1e4 10596 . . . . . . . . 9  |-  ( 3  +  1 )  =  4
4845, 46, 47addcomli 9701 . . . . . . . 8  |-  ( 1  +  3 )  =  4
4913dec0h 10929 . . . . . . . 8  |-  4  = ; 0 4
5048, 49eqtri 2421 . . . . . . 7  |-  ( 1  +  3 )  = ; 0
4
5146mulid1i 9527 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
52 00id 9684 . . . . . . . . 9  |-  ( 0  +  0 )  =  0
5351, 52oveq12i 6226 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
5446addid1i 9696 . . . . . . . 8  |-  ( 1  +  0 )  =  1
5553, 54eqtri 2421 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
56 4cn 10548 . . . . . . . . . 10  |-  4  e.  CC
5756mulid1i 9527 . . . . . . . . 9  |-  ( 4  x.  1 )  =  4
5857oveq1i 6224 . . . . . . . 8  |-  ( ( 4  x.  1 )  +  4 )  =  ( 4  +  4 )
59 4p4e8 10607 . . . . . . . 8  |-  ( 4  +  4 )  =  8
6020dec0h 10929 . . . . . . . 8  |-  8  = ; 0 8
6158, 59, 603eqtri 2425 . . . . . . 7  |-  ( ( 4  x.  1 )  +  4 )  = ; 0
8
622, 13, 22, 13, 44, 50, 2, 20, 22, 55, 61decmac 10952 . . . . . 6  |-  ( (; 1
4  x.  1 )  +  ( 1  +  3 ) )  = ; 1
8
6318dec0h 10929 . . . . . . 7  |-  6  = ; 0 6
6426mulid2i 9528 . . . . . . . . 9  |-  ( 1  x.  2 )  =  2
6546addid2i 9697 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
6664, 65oveq12i 6226 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
6766, 29eqtri 2421 . . . . . . 7  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  3
68 4t2e8 10624 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
6968oveq1i 6224 . . . . . . . 8  |-  ( ( 4  x.  2 )  +  6 )  =  ( 8  +  6 )
70 8p6e14 10972 . . . . . . . 8  |-  ( 8  +  6 )  = ; 1
4
7169, 70eqtri 2421 . . . . . . 7  |-  ( ( 4  x.  2 )  +  6 )  = ; 1
4
722, 13, 22, 18, 44, 63, 3, 13, 2, 67, 71decmac 10952 . . . . . 6  |-  ( (; 1
4  x.  2 )  +  6 )  = ; 3
4
732, 3, 2, 18, 39, 43, 14, 13, 16, 62, 72decma2c 10953 . . . . 5  |-  ( (; 1
4  x. ; 1 2 )  +  ( 9  +  7 ) )  = ;; 1 8 4
7435dec0h 10929 . . . . . 6  |-  9  = ; 0 9
75 5cn 10550 . . . . . . . . 9  |-  5  e.  CC
7675mulid2i 9528 . . . . . . . 8  |-  ( 1  x.  5 )  =  5
7726addid2i 9697 . . . . . . . 8  |-  ( 0  +  2 )  =  2
7876, 77oveq12i 6226 . . . . . . 7  |-  ( ( 1  x.  5 )  +  ( 0  +  2 ) )  =  ( 5  +  2 )
79 5p2e7 10608 . . . . . . 7  |-  ( 5  +  2 )  =  7
8078, 79eqtri 2421 . . . . . 6  |-  ( ( 1  x.  5 )  +  ( 0  +  2 ) )  =  7
81 5t4e20 10988 . . . . . . . 8  |-  ( 5  x.  4 )  = ; 2
0
8275, 56, 81mulcomli 9532 . . . . . . 7  |-  ( 4  x.  5 )  = ; 2
0
83 9cn 10558 . . . . . . . 8  |-  9  e.  CC
8483addid2i 9697 . . . . . . 7  |-  ( 0  +  9 )  =  9
853, 22, 35, 82, 84decaddi 10957 . . . . . 6  |-  ( ( 4  x.  5 )  +  9 )  = ; 2
9
862, 13, 22, 35, 44, 74, 5, 35, 3, 80, 85decmac 10952 . . . . 5  |-  ( (; 1
4  x.  5 )  +  9 )  = ; 7
9
874, 5, 35, 35, 37, 42, 14, 35, 11, 73, 86decma2c 10953 . . . 4  |-  ( (; 1
4  x. ;; 1 2 5 )  +  (; 8 7  + ; 1 2 ) )  = ;;; 1 8 4 9
8883mulid2i 9528 . . . . . . . . 9  |-  ( 1  x.  9 )  =  9
8988oveq1i 6224 . . . . . . . 8  |-  ( ( 1  x.  9 )  +  3 )  =  ( 9  +  3 )
90 9p3e12 10976 . . . . . . . 8  |-  ( 9  +  3 )  = ; 1
2
9189, 90eqtri 2421 . . . . . . 7  |-  ( ( 1  x.  9 )  +  3 )  = ; 1
2
92 9t4e36 11010 . . . . . . . 8  |-  ( 9  x.  4 )  = ; 3
6
9383, 56, 92mulcomli 9532 . . . . . . 7  |-  ( 4  x.  9 )  = ; 3
6
9435, 2, 13, 44, 18, 16, 91, 93decmul1c 10960 . . . . . 6  |-  (; 1 4  x.  9 )  = ;; 1 2 6
9594oveq1i 6224 . . . . 5  |-  ( (; 1
4  x.  9 )  +  0 )  =  (;; 1 2 6  +  0 )
964, 18deccl 10927 . . . . . . 7  |- ;; 1 2 6  e.  NN0
9796nn0cni 10742 . . . . . 6  |- ;; 1 2 6  e.  CC
9897addid1i 9696 . . . . 5  |-  (;; 1 2 6  +  0 )  = ;; 1 2 6
9995, 98eqtri 2421 . . . 4  |-  ( (; 1
4  x.  9 )  +  0 )  = ;; 1 2 6
1006, 35, 21, 22, 1, 36, 14, 18, 4, 87, 99decma2c 10953 . . 3  |-  ( (; 1
4  x.  N )  + ;; 8 7 0 )  = ;;;; 1 8 4 9 6
101 eqid 2392 . . . 4  |- ;; 1 3 6  = ;; 1 3 6
10220, 2deccl 10927 . . . 4  |- ; 8 1  e.  NN0
103 eqid 2392 . . . . 5  |- ; 1 3  = ; 1 3
104 eqid 2392 . . . . 5  |- ; 8 1  = ; 8 1
10513, 22deccl 10927 . . . . 5  |- ; 4 0  e.  NN0
106 eqid 2392 . . . . . . 7  |- ; 4 0  = ; 4 0
10756addid2i 9697 . . . . . . 7  |-  ( 0  +  4 )  =  4
108 8cn 10556 . . . . . . . 8  |-  8  e.  CC
109108addid1i 9696 . . . . . . 7  |-  ( 8  +  0 )  =  8
11022, 20, 13, 22, 60, 106, 107, 109decadd 10954 . . . . . 6  |-  ( 8  + ; 4 0 )  = ; 4
8
111 4p1e5 10597 . . . . . . . 8  |-  ( 4  +  1 )  =  5
1125dec0h 10929 . . . . . . . 8  |-  5  = ; 0 5
113111, 112eqtri 2421 . . . . . . 7  |-  ( 4  +  1 )  = ; 0
5
11445mulid1i 9527 . . . . . . . . 9  |-  ( 3  x.  1 )  =  3
115114oveq1i 6224 . . . . . . . 8  |-  ( ( 3  x.  1 )  +  5 )  =  ( 3  +  5 )
116 5p3e8 10609 . . . . . . . . 9  |-  ( 5  +  3 )  =  8
11775, 45, 116addcomli 9701 . . . . . . . 8  |-  ( 3  +  5 )  =  8
118115, 117, 603eqtri 2425 . . . . . . 7  |-  ( ( 3  x.  1 )  +  5 )  = ; 0
8
1192, 16, 22, 5, 103, 113, 2, 20, 22, 55, 118decmac 10952 . . . . . 6  |-  ( (; 1
3  x.  1 )  +  ( 4  +  1 ) )  = ; 1
8
120 6cn 10552 . . . . . . . . 9  |-  6  e.  CC
121120mulid1i 9527 . . . . . . . 8  |-  ( 6  x.  1 )  =  6
122121oveq1i 6224 . . . . . . 7  |-  ( ( 6  x.  1 )  +  8 )  =  ( 6  +  8 )
123108, 120, 70addcomli 9701 . . . . . . 7  |-  ( 6  +  8 )  = ; 1
4
124122, 123eqtri 2421 . . . . . 6  |-  ( ( 6  x.  1 )  +  8 )  = ; 1
4
12517, 18, 13, 20, 101, 110, 2, 13, 2, 119, 124decmac 10952 . . . . 5  |-  ( (;; 1 3 6  x.  1 )  +  ( 8  + ; 4 0 ) )  = ;; 1 8 4
1262dec0h 10929 . . . . . 6  |-  1  = ; 0 1
12765, 126eqtri 2421 . . . . . . 7  |-  ( 0  +  1 )  = ; 0
1
12845mulid2i 9528 . . . . . . . . 9  |-  ( 1  x.  3 )  =  3
129128, 65oveq12i 6226 . . . . . . . 8  |-  ( ( 1  x.  3 )  +  ( 0  +  1 ) )  =  ( 3  +  1 )
130129, 47eqtri 2421 . . . . . . 7  |-  ( ( 1  x.  3 )  +  ( 0  +  1 ) )  =  4
131 3t3e9 10623 . . . . . . . . 9  |-  ( 3  x.  3 )  =  9
132131oveq1i 6224 . . . . . . . 8  |-  ( ( 3  x.  3 )  +  1 )  =  ( 9  +  1 )
133 9p1e10 10602 . . . . . . . 8  |-  ( 9  +  1 )  =  10
134 dec10 10943 . . . . . . . 8  |-  10  = ; 1 0
135132, 133, 1343eqtri 2425 . . . . . . 7  |-  ( ( 3  x.  3 )  +  1 )  = ; 1
0
1362, 16, 22, 2, 103, 127, 16, 22, 2, 130, 135decmac 10952 . . . . . 6  |-  ( (; 1
3  x.  3 )  +  ( 0  +  1 ) )  = ; 4
0
137 6t3e18 10991 . . . . . . 7  |-  ( 6  x.  3 )  = ; 1
8
1382, 20, 2, 137, 40decaddi 10957 . . . . . 6  |-  ( ( 6  x.  3 )  +  1 )  = ; 1
9
13917, 18, 22, 2, 101, 126, 16, 35, 2, 136, 138decmac 10952 . . . . 5  |-  ( (;; 1 3 6  x.  3 )  +  1 )  = ;; 4 0 9
1402, 16, 20, 2, 103, 104, 19, 35, 105, 125, 139decma2c 10953 . . . 4  |-  ( (;; 1 3 6  x. ; 1
3 )  + ; 8 1 )  = ;;; 1 8 4 9
14116dec0h 10929 . . . . . 6  |-  3  = ; 0 3
142120mulid2i 9528 . . . . . . . 8  |-  ( 1  x.  6 )  =  6
143142, 77oveq12i 6226 . . . . . . 7  |-  ( ( 1  x.  6 )  +  ( 0  +  2 ) )  =  ( 6  +  2 )
144 6p2e8 10612 . . . . . . 7  |-  ( 6  +  2 )  =  8
145143, 144eqtri 2421 . . . . . 6  |-  ( ( 1  x.  6 )  +  ( 0  +  2 ) )  =  8
146120, 45, 137mulcomli 9532 . . . . . . 7  |-  ( 3  x.  6 )  = ; 1
8
147 1p1e2 10584 . . . . . . 7  |-  ( 1  +  1 )  =  2
148 8p3e11 10969 . . . . . . 7  |-  ( 8  +  3 )  = ; 1
1
1492, 20, 16, 146, 147, 2, 148decaddci 10958 . . . . . 6  |-  ( ( 3  x.  6 )  +  3 )  = ; 2
1
1502, 16, 22, 16, 103, 141, 18, 2, 3, 145, 149decmac 10952 . . . . 5  |-  ( (; 1
3  x.  6 )  +  3 )  = ; 8
1
151 6t6e36 10994 . . . . 5  |-  ( 6  x.  6 )  = ; 3
6
15218, 17, 18, 101, 18, 16, 150, 151decmul1c 10960 . . . 4  |-  (;; 1 3 6  x.  6 )  = ;; 8 1 6
15319, 17, 18, 101, 18, 102, 140, 152decmul2c 10961 . . 3  |-  (;; 1 3 6  x. ;; 1 3 6 )  = ;;;; 1 8 4 9 6
154100, 153eqtr4i 2424 . 2  |-  ( (; 1
4  x.  N )  + ;; 8 7 0 )  =  (;; 1 3 6  x. ;; 1 3 6 )
1559, 10, 12, 15, 19, 23, 24, 34, 154mod2xi 14576 1  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399  (class class class)co 6214   0cc0 9421   1c1 9422    + caddc 9424    x. cmul 9426   NNcn 10470   2c2 10520   3c3 10521   4c4 10522   5c5 10523   6c6 10524   7c7 10525   8c8 10526   9c9 10527   10c10 10528  ;cdc 10913    mod cmo 11915   ^cexp 12088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-cnex 9477  ax-resscn 9478  ax-1cn 9479  ax-icn 9480  ax-addcl 9481  ax-addrcl 9482  ax-mulcl 9483  ax-mulrcl 9484  ax-mulcom 9485  ax-addass 9486  ax-mulass 9487  ax-distr 9488  ax-i2m1 9489  ax-1ne0 9490  ax-1rid 9491  ax-rnegex 9492  ax-rrecex 9493  ax-cnre 9494  ax-pre-lttri 9495  ax-pre-lttrn 9496  ax-pre-ltadd 9497  ax-pre-mulgt0 9498  ax-pre-sup 9499
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-nel 2590  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-riota 6176  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-2nd 6718  df-recs 6978  df-rdg 7012  df-er 7247  df-en 7454  df-dom 7455  df-sdom 7456  df-sup 7834  df-pnf 9559  df-mnf 9560  df-xr 9561  df-ltxr 9562  df-le 9563  df-sub 9738  df-neg 9739  df-div 10142  df-nn 10471  df-2 10529  df-3 10530  df-4 10531  df-5 10532  df-6 10533  df-7 10534  df-8 10535  df-9 10536  df-10 10537  df-n0 10731  df-z 10800  df-dec 10914  df-uz 11020  df-rp 11158  df-fl 11847  df-mod 11916  df-seq 12030  df-exp 12089
This theorem is referenced by:  1259lem3  14636  1259lem5  14638
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