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Theorem 1259lem5 14492
Description: Lemma for 1259prm 14493. Calculate the GCD of  2 ^ 3 4  -  1  ==  8 6 9 with  N  =  1 2 5 9. (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
1259lem5  |-  ( ( ( 2 ^; 3 4 )  - 
1 )  gcd  N
)  =  1

Proof of Theorem 1259lem5
StepHypRef Expression
1 2nn 10705 . . . 4  |-  2  e.  NN
2 3nn0 10825 . . . . 5  |-  3  e.  NN0
3 4nn0 10826 . . . . 5  |-  4  e.  NN0
42, 3deccl 11002 . . . 4  |- ; 3 4  e.  NN0
5 nnexpcl 12159 . . . 4  |-  ( ( 2  e.  NN  /\ ; 3 4  e.  NN0 )  -> 
( 2 ^; 3 4 )  e.  NN )
61, 4, 5mp2an 672 . . 3  |-  ( 2 ^; 3 4 )  e.  NN
7 nnm1nn0 10849 . . 3  |-  ( ( 2 ^; 3 4 )  e.  NN  ->  ( (
2 ^; 3 4 )  - 
1 )  e.  NN0 )
86, 7ax-mp 5 . 2  |-  ( ( 2 ^; 3 4 )  - 
1 )  e.  NN0
9 8nn0 10830 . . . 4  |-  8  e.  NN0
10 6nn0 10828 . . . 4  |-  6  e.  NN0
119, 10deccl 11002 . . 3  |- ; 8 6  e.  NN0
12 9nn0 10831 . . 3  |-  9  e.  NN0
1311, 12deccl 11002 . 2  |- ;; 8 6 9  e.  NN0
14 1259prm.1 . . 3  |-  N  = ;;; 1 2 5 9
15 1nn0 10823 . . . . . 6  |-  1  e.  NN0
16 2nn0 10824 . . . . . 6  |-  2  e.  NN0
1715, 16deccl 11002 . . . . 5  |- ; 1 2  e.  NN0
18 5nn0 10827 . . . . 5  |-  5  e.  NN0
1917, 18deccl 11002 . . . 4  |- ;; 1 2 5  e.  NN0
20 9nn 10712 . . . 4  |-  9  e.  NN
2119, 20decnncl 11001 . . 3  |- ;;; 1 2 5 9  e.  NN
2214, 21eqeltri 2551 . 2  |-  N  e.  NN
23141259lem2 14489 . . 3  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
24 6p1e7 10676 . . . . 5  |-  ( 6  +  1 )  =  7
25 eqid 2467 . . . . 5  |- ; 8 6  = ; 8 6
269, 10, 24, 25decsuc 11011 . . . 4  |-  (; 8 6  +  1 )  = ; 8 7
27 eqid 2467 . . . 4  |- ;; 8 6 9  = ;; 8 6 9
2811, 26, 27decsucc 11015 . . 3  |-  (;; 8 6 9  +  1 )  = ;; 8 7 0
2922, 6, 15, 13, 23, 28modsubi 14434 . 2  |-  ( ( ( 2 ^; 3 4 )  - 
1 )  mod  N
)  =  (;; 8 6 9  mod  N
)
302, 12deccl 11002 . . . 4  |- ; 3 9  e.  NN0
31 0nn0 10822 . . . 4  |-  0  e.  NN0
3230, 31deccl 11002 . . 3  |- ;; 3 9 0  e.  NN0
339, 12deccl 11002 . . . 4  |- ; 8 9  e.  NN0
3416, 15deccl 11002 . . . . . 6  |- ; 2 1  e.  NN0
3515, 2deccl 11002 . . . . . . 7  |- ; 1 3  e.  NN0
3634nn0zi 10901 . . . . . . . . 9  |- ; 2 1  e.  ZZ
3735nn0zi 10901 . . . . . . . . 9  |- ; 1 3  e.  ZZ
38 gcdcom 14034 . . . . . . . . 9  |-  ( (; 2
1  e.  ZZ  /\ ; 1 3  e.  ZZ )  -> 
(; 2 1  gcd ; 1 3 )  =  (; 1 3  gcd ; 2 1 ) )
3936, 37, 38mp2an 672 . . . . . . . 8  |-  (; 2 1  gcd ; 1 3 )  =  (; 1 3  gcd ; 2 1 )
40 3nn 10706 . . . . . . . . . . 11  |-  3  e.  NN
4115, 40decnncl 11001 . . . . . . . . . 10  |- ; 1 3  e.  NN
42 8nn 10711 . . . . . . . . . 10  |-  8  e.  NN
43 eqid 2467 . . . . . . . . . . 11  |- ; 1 3  = ; 1 3
449dec0h 11004 . . . . . . . . . . 11  |-  8  = ; 0 8
45 ax-1cn 9562 . . . . . . . . . . . . . 14  |-  1  e.  CC
4645mulid1i 9610 . . . . . . . . . . . . 13  |-  ( 1  x.  1 )  =  1
4745addid2i 9779 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
4846, 47oveq12i 6307 . . . . . . . . . . . 12  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  ( 1  +  1 )
49 1p1e2 10661 . . . . . . . . . . . 12  |-  ( 1  +  1 )  =  2
5048, 49eqtri 2496 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  2
51 3cn 10622 . . . . . . . . . . . . . 14  |-  3  e.  CC
5251mulid1i 9610 . . . . . . . . . . . . 13  |-  ( 3  x.  1 )  =  3
5352oveq1i 6305 . . . . . . . . . . . 12  |-  ( ( 3  x.  1 )  +  8 )  =  ( 3  +  8 )
54 8cn 10633 . . . . . . . . . . . . 13  |-  8  e.  CC
55 8p3e11 11044 . . . . . . . . . . . . 13  |-  ( 8  +  3 )  = ; 1
1
5654, 51, 55addcomli 9783 . . . . . . . . . . . 12  |-  ( 3  +  8 )  = ; 1
1
5753, 56eqtri 2496 . . . . . . . . . . 11  |-  ( ( 3  x.  1 )  +  8 )  = ; 1
1
5815, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57decmac 11027 . . . . . . . . . 10  |-  ( (; 1
3  x.  1 )  +  8 )  = ; 2
1
59 1nn 10559 . . . . . . . . . . 11  |-  1  e.  NN
60 8lt10 10751 . . . . . . . . . . 11  |-  8  <  10
6159, 2, 9, 60declti 11013 . . . . . . . . . 10  |-  8  < ; 1
3
6241, 15, 42, 58, 61ndvdsi 13944 . . . . . . . . 9  |-  -. ; 1 3  || ; 2 1
63 13prm 14476 . . . . . . . . . 10  |- ; 1 3  e.  Prime
64 coprm 14117 . . . . . . . . . 10  |-  ( (; 1
3  e.  Prime  /\ ; 2 1  e.  ZZ )  ->  ( -. ; 1 3  || ; 2 1  <->  (; 1 3  gcd ; 2 1 )  =  1 ) )
6563, 36, 64mp2an 672 . . . . . . . . 9  |-  ( -. ; 1
3  || ; 2 1  <->  (; 1 3  gcd ; 2 1 )  =  1 )
6662, 65mpbi 208 . . . . . . . 8  |-  (; 1 3  gcd ; 2 1 )  =  1
6739, 66eqtri 2496 . . . . . . 7  |-  (; 2 1  gcd ; 1 3 )  =  1
68 eqid 2467 . . . . . . . 8  |- ; 2 1  = ; 2 1
69 2cn 10618 . . . . . . . . . . 11  |-  2  e.  CC
7069mulid2i 9611 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
7145addid1i 9778 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
7270, 71oveq12i 6307 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 1  +  0 ) )  =  ( 2  +  1 )
73 2p1e3 10671 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
7472, 73eqtri 2496 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 1  +  0 ) )  =  3
7546oveq1i 6305 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  3 )  =  ( 1  +  3 )
76 3p1e4 10673 . . . . . . . . . 10  |-  ( 3  +  1 )  =  4
7751, 45, 76addcomli 9783 . . . . . . . . 9  |-  ( 1  +  3 )  =  4
783dec0h 11004 . . . . . . . . 9  |-  4  = ; 0 4
7975, 77, 783eqtri 2500 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  3 )  = ; 0
4
8016, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79decma2c 11028 . . . . . . 7  |-  ( ( 1  x. ; 2 1 )  + ; 1
3 )  = ; 3 4
8115, 35, 34, 67, 80gcdi 14435 . . . . . 6  |-  (; 3 4  gcd ; 2 1 )  =  1
82 eqid 2467 . . . . . . 7  |- ; 3 4  = ; 3 4
83 3t2e6 10699 . . . . . . . . . 10  |-  ( 3  x.  2 )  =  6
8451, 69, 83mulcomli 9615 . . . . . . . . 9  |-  ( 2  x.  3 )  =  6
8569addid1i 9778 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
8684, 85oveq12i 6307 . . . . . . . 8  |-  ( ( 2  x.  3 )  +  ( 2  +  0 ) )  =  ( 6  +  2 )
87 6p2e8 10689 . . . . . . . 8  |-  ( 6  +  2 )  =  8
8886, 87eqtri 2496 . . . . . . 7  |-  ( ( 2  x.  3 )  +  ( 2  +  0 ) )  =  8
89 4cn 10625 . . . . . . . . . 10  |-  4  e.  CC
90 4t2e8 10701 . . . . . . . . . 10  |-  ( 4  x.  2 )  =  8
9189, 69, 90mulcomli 9615 . . . . . . . . 9  |-  ( 2  x.  4 )  =  8
9291oveq1i 6305 . . . . . . . 8  |-  ( ( 2  x.  4 )  +  1 )  =  ( 8  +  1 )
93 8p1e9 10678 . . . . . . . 8  |-  ( 8  +  1 )  =  9
9412dec0h 11004 . . . . . . . 8  |-  9  = ; 0 9
9592, 93, 943eqtri 2500 . . . . . . 7  |-  ( ( 2  x.  4 )  +  1 )  = ; 0
9
962, 3, 16, 15, 82, 68, 16, 12, 31, 88, 95decma2c 11028 . . . . . 6  |-  ( ( 2  x. ; 3 4 )  + ; 2
1 )  = ; 8 9
9716, 34, 4, 81, 96gcdi 14435 . . . . 5  |-  (; 8 9  gcd ; 3 4 )  =  1
98 eqid 2467 . . . . . 6  |- ; 8 9  = ; 8 9
99 4p3e7 10683 . . . . . . . . 9  |-  ( 4  +  3 )  =  7
10089, 51, 99addcomli 9783 . . . . . . . 8  |-  ( 3  +  4 )  =  7
101100oveq2i 6306 . . . . . . 7  |-  ( ( 4  x.  8 )  +  ( 3  +  4 ) )  =  ( ( 4  x.  8 )  +  7 )
102 7nn0 10829 . . . . . . . 8  |-  7  e.  NN0
103 8t4e32 11078 . . . . . . . . 9  |-  ( 8  x.  4 )  = ; 3
2
10454, 89, 103mulcomli 9615 . . . . . . . 8  |-  ( 4  x.  8 )  = ; 3
2
105 7cn 10631 . . . . . . . . 9  |-  7  e.  CC
106 7p2e9 10692 . . . . . . . . 9  |-  ( 7  +  2 )  =  9
107105, 69, 106addcomli 9783 . . . . . . . 8  |-  ( 2  +  7 )  =  9
1082, 16, 102, 104, 107decaddi 11032 . . . . . . 7  |-  ( ( 4  x.  8 )  +  7 )  = ; 3
9
109101, 108eqtri 2496 . . . . . 6  |-  ( ( 4  x.  8 )  +  ( 3  +  4 ) )  = ; 3
9
110 9cn 10635 . . . . . . . 8  |-  9  e.  CC
111 9t4e36 11085 . . . . . . . 8  |-  ( 9  x.  4 )  = ; 3
6
112110, 89, 111mulcomli 9615 . . . . . . 7  |-  ( 4  x.  9 )  = ; 3
6
113 6p4e10 10691 . . . . . . 7  |-  ( 6  +  4 )  =  10
1142, 10, 3, 112, 76, 113decaddci2 11034 . . . . . 6  |-  ( ( 4  x.  9 )  +  4 )  = ; 4
0
1159, 12, 2, 3, 98, 82, 3, 31, 3, 109, 114decma2c 11028 . . . . 5  |-  ( ( 4  x. ; 8 9 )  + ; 3
4 )  = ;; 3 9 0
1163, 4, 33, 97, 115gcdi 14435 . . . 4  |-  (;; 3 9 0  gcd ; 8 9 )  =  1
117 eqid 2467 . . . . 5  |- ;; 3 9 0  = ;; 3 9 0
118 eqid 2467 . . . . . 6  |- ; 3 9  = ; 3 9
11954addid1i 9778 . . . . . . 7  |-  ( 8  +  0 )  =  8
120119, 44eqtri 2496 . . . . . 6  |-  ( 8  +  0 )  = ; 0
8
12169addid2i 9779 . . . . . . . 8  |-  ( 0  +  2 )  =  2
12284, 121oveq12i 6307 . . . . . . 7  |-  ( ( 2  x.  3 )  +  ( 0  +  2 ) )  =  ( 6  +  2 )
123122, 87eqtri 2496 . . . . . 6  |-  ( ( 2  x.  3 )  +  ( 0  +  2 ) )  =  8
124 9t2e18 11083 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
125110, 69, 124mulcomli 9615 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
126 8p8e16 11049 . . . . . . 7  |-  ( 8  +  8 )  = ; 1
6
12715, 9, 9, 125, 49, 10, 126decaddci 11033 . . . . . 6  |-  ( ( 2  x.  9 )  +  8 )  = ; 2
6
1282, 12, 31, 9, 118, 120, 16, 10, 16, 123, 127decma2c 11028 . . . . 5  |-  ( ( 2  x. ; 3 9 )  +  ( 8  +  0 ) )  = ; 8 6
129 2t0e0 10703 . . . . . . 7  |-  ( 2  x.  0 )  =  0
130129oveq1i 6305 . . . . . 6  |-  ( ( 2  x.  0 )  +  9 )  =  ( 0  +  9 )
131110addid2i 9779 . . . . . 6  |-  ( 0  +  9 )  =  9
132130, 131, 943eqtri 2500 . . . . 5  |-  ( ( 2  x.  0 )  +  9 )  = ; 0
9
13330, 31, 9, 12, 117, 98, 16, 12, 31, 128, 132decma2c 11028 . . . 4  |-  ( ( 2  x. ;; 3 9 0 )  + ; 8
9 )  = ;; 8 6 9
13416, 33, 32, 116, 133gcdi 14435 . . 3  |-  (;; 8 6 9  gcd ;; 3 9 0 )  =  1
13530nn0cni 10819 . . . . . . 7  |- ; 3 9  e.  CC
136135addid1i 9778 . . . . . 6  |-  (; 3 9  +  0 )  = ; 3 9
13754mulid2i 9611 . . . . . . . 8  |-  ( 1  x.  8 )  =  8
138137, 76oveq12i 6307 . . . . . . 7  |-  ( ( 1  x.  8 )  +  ( 3  +  1 ) )  =  ( 8  +  4 )
139 8p4e12 11045 . . . . . . 7  |-  ( 8  +  4 )  = ; 1
2
140138, 139eqtri 2496 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 3  +  1 ) )  = ; 1
2
141 6cn 10629 . . . . . . . . 9  |-  6  e.  CC
142141mulid2i 9611 . . . . . . . 8  |-  ( 1  x.  6 )  =  6
143142oveq1i 6305 . . . . . . 7  |-  ( ( 1  x.  6 )  +  9 )  =  ( 6  +  9 )
144 9p6e15 11054 . . . . . . . 8  |-  ( 9  +  6 )  = ; 1
5
145110, 141, 144addcomli 9783 . . . . . . 7  |-  ( 6  +  9 )  = ; 1
5
146143, 145eqtri 2496 . . . . . 6  |-  ( ( 1  x.  6 )  +  9 )  = ; 1
5
1479, 10, 2, 12, 25, 136, 15, 18, 15, 140, 146decma2c 11028 . . . . 5  |-  ( ( 1  x. ; 8 6 )  +  (; 3 9  +  0 ) )  = ;; 1 2 5
148110mulid2i 9611 . . . . . . 7  |-  ( 1  x.  9 )  =  9
149148oveq1i 6305 . . . . . 6  |-  ( ( 1  x.  9 )  +  0 )  =  ( 9  +  0 )
150110addid1i 9778 . . . . . 6  |-  ( 9  +  0 )  =  9
151149, 150, 943eqtri 2500 . . . . 5  |-  ( ( 1  x.  9 )  +  0 )  = ; 0
9
15211, 12, 30, 31, 27, 117, 15, 12, 31, 147, 151decma2c 11028 . . . 4  |-  ( ( 1  x. ;; 8 6 9 )  + ;; 3 9 0 )  = ;;; 1 2 5 9
153152, 14eqtr4i 2499 . . 3  |-  ( ( 1  x. ;; 8 6 9 )  + ;; 3 9 0 )  =  N
15415, 32, 13, 134, 153gcdi 14435 . 2  |-  ( N  gcd ;; 8 6 9 )  =  1
1558, 13, 22, 29, 154gcdmodi 14436 1  |-  ( ( ( 2 ^; 3 4 )  - 
1 )  gcd  N
)  =  1
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1379    e. wcel 1767   class class class wbr 4453  (class class class)co 6295   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    - cmin 9817   NNcn 10548   2c2 10597   3c3 10598   4c4 10599   5c5 10600   6c6 10601   7c7 10602   8c8 10603   9c9 10604   NN0cn0 10807   ZZcz 10876  ;cdc 10988   ^cexp 12146    || cdivides 13864    gcd cgcd 14020   Primecprime 14093
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-rp 11233  df-fz 11685  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-prm 14094
This theorem is referenced by:  1259prm  14493
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