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Theorem 1259lem5 13133
Description: Lemma for 1259prm 13134. 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 9877 . . . 4  |-  2  e.  NN
2 3nn0 9983 . . . . 5  |-  3  e.  NN0
3 4nn0 9984 . . . . 5  |-  4  e.  NN0
42, 3deccl 10138 . . . 4  |- ; 3 4  e.  NN0
5 nnexpcl 11116 . . . 4  |-  ( ( 2  e.  NN  /\ ; 3 4  e.  NN0 )  -> 
( 2 ^; 3 4 )  e.  NN )
61, 4, 5mp2an 653 . . 3  |-  ( 2 ^; 3 4 )  e.  NN
7 nnm1nn0 10005 . . 3  |-  ( ( 2 ^; 3 4 )  e.  NN  ->  ( (
2 ^; 3 4 )  - 
1 )  e.  NN0 )
86, 7ax-mp 8 . 2  |-  ( ( 2 ^; 3 4 )  - 
1 )  e.  NN0
9 8nn0 9988 . . . 4  |-  8  e.  NN0
10 6nn0 9986 . . . 4  |-  6  e.  NN0
119, 10deccl 10138 . . 3  |- ; 8 6  e.  NN0
12 9nn0 9989 . . 3  |-  9  e.  NN0
1311, 12deccl 10138 . 2  |- ;; 8 6 9  e.  NN0
14 1259prm.1 . . 3  |-  N  = ;;; 1 2 5 9
15 1nn0 9981 . . . . . 6  |-  1  e.  NN0
16 2nn0 9982 . . . . . 6  |-  2  e.  NN0
1715, 16deccl 10138 . . . . 5  |- ; 1 2  e.  NN0
18 5nn0 9985 . . . . 5  |-  5  e.  NN0
1917, 18deccl 10138 . . . 4  |- ;; 1 2 5  e.  NN0
20 9nn 9884 . . . 4  |-  9  e.  NN
2119, 20decnncl 10137 . . 3  |- ;;; 1 2 5 9  e.  NN
2214, 21eqeltri 2353 . 2  |-  N  e.  NN
23141259lem2 13130 . . 3  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
24 6p1e7 9851 . . . . 5  |-  ( 6  +  1 )  =  7
25 eqid 2283 . . . . 5  |- ; 8 6  = ; 8 6
269, 10, 24, 25decsuc 10147 . . . 4  |-  (; 8 6  +  1 )  = ; 8 7
27 eqid 2283 . . . 4  |- ;; 8 6 9  = ;; 8 6 9
2811, 26, 27decsucc 10151 . . 3  |-  (;; 8 6 9  +  1 )  = ;; 8 7 0
2922, 6, 15, 13, 23, 28modsubi 13087 . 2  |-  ( ( ( 2 ^; 3 4 )  - 
1 )  mod  N
)  =  (;; 8 6 9  mod  N
)
302, 12deccl 10138 . . . 4  |- ; 3 9  e.  NN0
31 0nn0 9980 . . . 4  |-  0  e.  NN0
3230, 31deccl 10138 . . 3  |- ;; 3 9 0  e.  NN0
339, 12deccl 10138 . . . 4  |- ; 8 9  e.  NN0
3416, 15deccl 10138 . . . . . 6  |- ; 2 1  e.  NN0
3515, 2deccl 10138 . . . . . . 7  |- ; 1 3  e.  NN0
3634nn0zi 10048 . . . . . . . . 9  |- ; 2 1  e.  ZZ
3735nn0zi 10048 . . . . . . . . 9  |- ; 1 3  e.  ZZ
38 gcdcom 12699 . . . . . . . . 9  |-  ( (; 2
1  e.  ZZ  /\ ; 1 3  e.  ZZ )  -> 
(; 2 1  gcd ; 1 3 )  =  (; 1 3  gcd ; 2 1 ) )
3936, 37, 38mp2an 653 . . . . . . . 8  |-  (; 2 1  gcd ; 1 3 )  =  (; 1 3  gcd ; 2 1 )
40 3nn 9878 . . . . . . . . . . 11  |-  3  e.  NN
4115, 40decnncl 10137 . . . . . . . . . 10  |- ; 1 3  e.  NN
42 8nn 9883 . . . . . . . . . 10  |-  8  e.  NN
43 eqid 2283 . . . . . . . . . . 11  |- ; 1 3  = ; 1 3
449dec0h 10140 . . . . . . . . . . 11  |-  8  = ; 0 8
45 ax-1cn 8795 . . . . . . . . . . . . . 14  |-  1  e.  CC
4645mulid1i 8839 . . . . . . . . . . . . 13  |-  ( 1  x.  1 )  =  1
4745addid2i 9000 . . . . . . . . . . . . 13  |-  ( 0  +  1 )  =  1
4846, 47oveq12i 5870 . . . . . . . . . . . 12  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  ( 1  +  1 )
49 1p1e2 9840 . . . . . . . . . . . 12  |-  ( 1  +  1 )  =  2
5048, 49eqtri 2303 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  2
51 3cn 9818 . . . . . . . . . . . . . 14  |-  3  e.  CC
5251mulid1i 8839 . . . . . . . . . . . . 13  |-  ( 3  x.  1 )  =  3
5352oveq1i 5868 . . . . . . . . . . . 12  |-  ( ( 3  x.  1 )  +  8 )  =  ( 3  +  8 )
5442nncni 9756 . . . . . . . . . . . . 13  |-  8  e.  CC
55 8p3e11 10180 . . . . . . . . . . . . 13  |-  ( 8  +  3 )  = ; 1
1
5654, 51, 55addcomli 9004 . . . . . . . . . . . 12  |-  ( 3  +  8 )  = ; 1
1
5753, 56eqtri 2303 . . . . . . . . . . 11  |-  ( ( 3  x.  1 )  +  8 )  = ; 1
1
5815, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57decmac 10163 . . . . . . . . . 10  |-  ( (; 1
3  x.  1 )  +  8 )  = ; 2
1
59 1nn 9757 . . . . . . . . . . 11  |-  1  e.  NN
60 8lt10 9923 . . . . . . . . . . 11  |-  8  <  10
6159, 2, 9, 60declti 10149 . . . . . . . . . 10  |-  8  < ; 1
3
6241, 15, 42, 58, 61ndvdsi 12609 . . . . . . . . 9  |-  -. ; 1 3  || ; 2 1
63 13prm 13117 . . . . . . . . . 10  |- ; 1 3  e.  Prime
64 coprm 12779 . . . . . . . . . 10  |-  ( (; 1
3  e.  Prime  /\ ; 2 1  e.  ZZ )  ->  ( -. ; 1 3  || ; 2 1  <->  (; 1 3  gcd ; 2 1 )  =  1 ) )
6563, 36, 64mp2an 653 . . . . . . . . 9  |-  ( -. ; 1
3  || ; 2 1  <->  (; 1 3  gcd ; 2 1 )  =  1 )
6662, 65mpbi 199 . . . . . . . 8  |-  (; 1 3  gcd ; 2 1 )  =  1
6739, 66eqtri 2303 . . . . . . 7  |-  (; 2 1  gcd ; 1 3 )  =  1
68 eqid 2283 . . . . . . . 8  |- ; 2 1  = ; 2 1
69 2cn 9816 . . . . . . . . . . 11  |-  2  e.  CC
7069mulid2i 8840 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
7145addid1i 8999 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
7270, 71oveq12i 5870 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 1  +  0 ) )  =  ( 2  +  1 )
73 2p1e3 9847 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
7472, 73eqtri 2303 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 1  +  0 ) )  =  3
7546oveq1i 5868 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  3 )  =  ( 1  +  3 )
76 3p1e4 9848 . . . . . . . . . 10  |-  ( 3  +  1 )  =  4
7751, 45, 76addcomli 9004 . . . . . . . . 9  |-  ( 1  +  3 )  =  4
783dec0h 10140 . . . . . . . . 9  |-  4  = ; 0 4
7975, 77, 783eqtri 2307 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  3 )  = ; 0
4
8016, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79decma2c 10164 . . . . . . 7  |-  ( ( 1  x. ; 2 1 )  + ; 1
3 )  = ; 3 4
8115, 35, 34, 67, 80gcdi 13088 . . . . . 6  |-  (; 3 4  gcd ; 2 1 )  =  1
82 eqid 2283 . . . . . . 7  |- ; 3 4  = ; 3 4
83 3t2e6 9872 . . . . . . . . . 10  |-  ( 3  x.  2 )  =  6
8451, 69, 83mulcomli 8844 . . . . . . . . 9  |-  ( 2  x.  3 )  =  6
8569addid1i 8999 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
8684, 85oveq12i 5870 . . . . . . . 8  |-  ( ( 2  x.  3 )  +  ( 2  +  0 ) )  =  ( 6  +  2 )
87 6p2e8 9864 . . . . . . . 8  |-  ( 6  +  2 )  =  8
8886, 87eqtri 2303 . . . . . . 7  |-  ( ( 2  x.  3 )  +  ( 2  +  0 ) )  =  8
89 4cn 9820 . . . . . . . . . 10  |-  4  e.  CC
90 4t2e8 9874 . . . . . . . . . 10  |-  ( 4  x.  2 )  =  8
9189, 69, 90mulcomli 8844 . . . . . . . . 9  |-  ( 2  x.  4 )  =  8
9291oveq1i 5868 . . . . . . . 8  |-  ( ( 2  x.  4 )  +  1 )  =  ( 8  +  1 )
93 8p1e9 9853 . . . . . . . 8  |-  ( 8  +  1 )  =  9
9412dec0h 10140 . . . . . . . 8  |-  9  = ; 0 9
9592, 93, 943eqtri 2307 . . . . . . 7  |-  ( ( 2  x.  4 )  +  1 )  = ; 0
9
962, 3, 16, 15, 82, 68, 16, 12, 31, 88, 95decma2c 10164 . . . . . 6  |-  ( ( 2  x. ; 3 4 )  + ; 2
1 )  = ; 8 9
9716, 34, 4, 81, 96gcdi 13088 . . . . 5  |-  (; 8 9  gcd ; 3 4 )  =  1
98 eqid 2283 . . . . . 6  |- ; 8 9  = ; 8 9
99 4p3e7 9858 . . . . . . . . 9  |-  ( 4  +  3 )  =  7
10089, 51, 99addcomli 9004 . . . . . . . 8  |-  ( 3  +  4 )  =  7
101100oveq2i 5869 . . . . . . 7  |-  ( ( 4  x.  8 )  +  ( 3  +  4 ) )  =  ( ( 4  x.  8 )  +  7 )
102 7nn0 9987 . . . . . . . 8  |-  7  e.  NN0
103 8t4e32 10214 . . . . . . . . 9  |-  ( 8  x.  4 )  = ; 3
2
10454, 89, 103mulcomli 8844 . . . . . . . 8  |-  ( 4  x.  8 )  = ; 3
2
105 7nn 9882 . . . . . . . . . 10  |-  7  e.  NN
106105nncni 9756 . . . . . . . . 9  |-  7  e.  CC
107 7p2e9 9867 . . . . . . . . 9  |-  ( 7  +  2 )  =  9
108106, 69, 107addcomli 9004 . . . . . . . 8  |-  ( 2  +  7 )  =  9
1092, 16, 102, 104, 108decaddi 10168 . . . . . . 7  |-  ( ( 4  x.  8 )  +  7 )  = ; 3
9
110101, 109eqtri 2303 . . . . . 6  |-  ( ( 4  x.  8 )  +  ( 3  +  4 ) )  = ; 3
9
11120nncni 9756 . . . . . . . 8  |-  9  e.  CC
112 9t4e36 10221 . . . . . . . 8  |-  ( 9  x.  4 )  = ; 3
6
113111, 89, 112mulcomli 8844 . . . . . . 7  |-  ( 4  x.  9 )  = ; 3
6
114 6p4e10 9866 . . . . . . 7  |-  ( 6  +  4 )  =  10
1152, 10, 3, 113, 76, 114decaddci2 10170 . . . . . 6  |-  ( ( 4  x.  9 )  +  4 )  = ; 4
0
1169, 12, 2, 3, 98, 82, 3, 31, 3, 110, 115decma2c 10164 . . . . 5  |-  ( ( 4  x. ; 8 9 )  + ; 3
4 )  = ;; 3 9 0
1173, 4, 33, 97, 116gcdi 13088 . . . 4  |-  (;; 3 9 0  gcd ; 8 9 )  =  1
118 eqid 2283 . . . . 5  |- ;; 3 9 0  = ;; 3 9 0
119 eqid 2283 . . . . . 6  |- ; 3 9  = ; 3 9
12054addid1i 8999 . . . . . . 7  |-  ( 8  +  0 )  =  8
121120, 44eqtri 2303 . . . . . 6  |-  ( 8  +  0 )  = ; 0
8
12269addid2i 9000 . . . . . . . 8  |-  ( 0  +  2 )  =  2
12384, 122oveq12i 5870 . . . . . . 7  |-  ( ( 2  x.  3 )  +  ( 0  +  2 ) )  =  ( 6  +  2 )
124123, 87eqtri 2303 . . . . . 6  |-  ( ( 2  x.  3 )  +  ( 0  +  2 ) )  =  8
125 9t2e18 10219 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
126111, 69, 125mulcomli 8844 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
127 8p8e16 10185 . . . . . . 7  |-  ( 8  +  8 )  = ; 1
6
12815, 9, 9, 126, 49, 10, 127decaddci 10169 . . . . . 6  |-  ( ( 2  x.  9 )  +  8 )  = ; 2
6
1292, 12, 31, 9, 119, 121, 16, 10, 16, 124, 128decma2c 10164 . . . . 5  |-  ( ( 2  x. ; 3 9 )  +  ( 8  +  0 ) )  = ; 8 6
13069mul01i 9002 . . . . . . 7  |-  ( 2  x.  0 )  =  0
131130oveq1i 5868 . . . . . 6  |-  ( ( 2  x.  0 )  +  9 )  =  ( 0  +  9 )
132111addid2i 9000 . . . . . 6  |-  ( 0  +  9 )  =  9
133131, 132, 943eqtri 2307 . . . . 5  |-  ( ( 2  x.  0 )  +  9 )  = ; 0
9
13430, 31, 9, 12, 118, 98, 16, 12, 31, 129, 133decma2c 10164 . . . 4  |-  ( ( 2  x. ;; 3 9 0 )  + ; 8
9 )  = ;; 8 6 9
13516, 33, 32, 117, 134gcdi 13088 . . 3  |-  (;; 8 6 9  gcd ;; 3 9 0 )  =  1
13630nn0cni 9977 . . . . . . 7  |- ; 3 9  e.  CC
137136addid1i 8999 . . . . . 6  |-  (; 3 9  +  0 )  = ; 3 9
13854mulid2i 8840 . . . . . . . 8  |-  ( 1  x.  8 )  =  8
139138, 76oveq12i 5870 . . . . . . 7  |-  ( ( 1  x.  8 )  +  ( 3  +  1 ) )  =  ( 8  +  4 )
140 8p4e12 10181 . . . . . . 7  |-  ( 8  +  4 )  = ; 1
2
141139, 140eqtri 2303 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 3  +  1 ) )  = ; 1
2
142 6nn 9881 . . . . . . . . . 10  |-  6  e.  NN
143142nncni 9756 . . . . . . . . 9  |-  6  e.  CC
144143mulid2i 8840 . . . . . . . 8  |-  ( 1  x.  6 )  =  6
145144oveq1i 5868 . . . . . . 7  |-  ( ( 1  x.  6 )  +  9 )  =  ( 6  +  9 )
146 9p6e15 10190 . . . . . . . 8  |-  ( 9  +  6 )  = ; 1
5
147111, 143, 146addcomli 9004 . . . . . . 7  |-  ( 6  +  9 )  = ; 1
5
148145, 147eqtri 2303 . . . . . 6  |-  ( ( 1  x.  6 )  +  9 )  = ; 1
5
1499, 10, 2, 12, 25, 137, 15, 18, 15, 141, 148decma2c 10164 . . . . 5  |-  ( ( 1  x. ; 8 6 )  +  (; 3 9  +  0 ) )  = ;; 1 2 5
150111mulid2i 8840 . . . . . . 7  |-  ( 1  x.  9 )  =  9
151150oveq1i 5868 . . . . . 6  |-  ( ( 1  x.  9 )  +  0 )  =  ( 9  +  0 )
152111addid1i 8999 . . . . . 6  |-  ( 9  +  0 )  =  9
153151, 152, 943eqtri 2307 . . . . 5  |-  ( ( 1  x.  9 )  +  0 )  = ; 0
9
15411, 12, 30, 31, 27, 118, 15, 12, 31, 149, 153decma2c 10164 . . . 4  |-  ( ( 1  x. ;; 8 6 9 )  + ;; 3 9 0 )  = ;;; 1 2 5 9
155154, 14eqtr4i 2306 . . 3  |-  ( ( 1  x. ;; 8 6 9 )  + ;; 3 9 0 )  =  N
15615, 32, 13, 135, 155gcdi 13088 . 2  |-  ( N  gcd ;; 8 6 9 )  =  1
1578, 13, 22, 29, 156gcdmodi 13089 1  |-  ( ( ( 2 ^; 3 4 )  - 
1 )  gcd  N
)  =  1
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   2c2 9795   3c3 9796   4c4 9797   5c5 9798   6c6 9799   7c7 9800   8c8 9801   9c9 9802   NN0cn0 9965   ZZcz 10024  ;cdc 10124   ^cexp 11104    || cdivides 12531    gcd cgcd 12685   Primecprime 12758
This theorem is referenced by:  1259prm  13134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759
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