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Theorem 4001lem4 14501
Description: Lemma for 4001prm 14502. Calculate the GCD of  2 ^ 8 0 0  -  1  ==  2 3 1 0 with  N  =  4 0 0 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
Hypothesis
Ref Expression
4001prm.1  |-  N  = ;;; 4 0 0 1
Assertion
Ref Expression
4001lem4  |-  ( ( ( 2 ^;; 8 0 0 )  - 
1 )  gcd  N
)  =  1

Proof of Theorem 4001lem4
StepHypRef Expression
1 2nn 10705 . . . 4  |-  2  e.  NN
2 8nn0 10830 . . . . . 6  |-  8  e.  NN0
3 0nn0 10822 . . . . . 6  |-  0  e.  NN0
42, 3deccl 11002 . . . . 5  |- ; 8 0  e.  NN0
54, 3deccl 11002 . . . 4  |- ;; 8 0 0  e.  NN0
6 nnexpcl 12159 . . . 4  |-  ( ( 2  e.  NN  /\ ;; 8 0 0  e. 
NN0 )  ->  (
2 ^;; 8 0 0 )  e.  NN )
71, 5, 6mp2an 672 . . 3  |-  ( 2 ^;; 8 0 0 )  e.  NN
8 nnm1nn0 10849 . . 3  |-  ( ( 2 ^;; 8 0 0 )  e.  NN  ->  ( (
2 ^;; 8 0 0 )  - 
1 )  e.  NN0 )
97, 8ax-mp 5 . 2  |-  ( ( 2 ^;; 8 0 0 )  - 
1 )  e.  NN0
10 2nn0 10824 . . . . 5  |-  2  e.  NN0
11 3nn0 10825 . . . . 5  |-  3  e.  NN0
1210, 11deccl 11002 . . . 4  |- ; 2 3  e.  NN0
13 1nn0 10823 . . . 4  |-  1  e.  NN0
1412, 13deccl 11002 . . 3  |- ;; 2 3 1  e.  NN0
1514, 3deccl 11002 . 2  |- ;;; 2 3 1 0  e.  NN0
16 4001prm.1 . . 3  |-  N  = ;;; 4 0 0 1
17 4nn0 10826 . . . . . 6  |-  4  e.  NN0
1817, 3deccl 11002 . . . . 5  |- ; 4 0  e.  NN0
1918, 3deccl 11002 . . . 4  |- ;; 4 0 0  e.  NN0
20 1nn 10559 . . . 4  |-  1  e.  NN
2119, 20decnncl 11001 . . 3  |- ;;; 4 0 0 1  e.  NN
2216, 21eqeltri 2551 . 2  |-  N  e.  NN
23164001lem2 14499 . . 3  |-  ( ( 2 ^;; 8 0 0 )  mod 
N )  =  (;;; 2 3 1 1  mod 
N )
24 0p1e1 10659 . . . 4  |-  ( 0  +  1 )  =  1
25 eqid 2467 . . . 4  |- ;;; 2 3 1 0  = ;;; 2 3 1 0
2614, 3, 24, 25decsuc 11011 . . 3  |-  (;;; 2 3 1 0  +  1 )  = ;;; 2 3 1 1
2722, 7, 13, 15, 23, 26modsubi 14434 . 2  |-  ( ( ( 2 ^;; 8 0 0 )  - 
1 )  mod  N
)  =  (;;; 2 3 1 0  mod  N )
28 6nn0 10828 . . . . . 6  |-  6  e.  NN0
2913, 28deccl 11002 . . . . 5  |- ; 1 6  e.  NN0
30 9nn0 10831 . . . . 5  |-  9  e.  NN0
3129, 30deccl 11002 . . . 4  |- ;; 1 6 9  e.  NN0
3231, 13deccl 11002 . . 3  |- ;;; 1 6 9 1  e.  NN0
3328, 13deccl 11002 . . . . 5  |- ; 6 1  e.  NN0
3433, 30deccl 11002 . . . 4  |- ;; 6 1 9  e.  NN0
35 5nn0 10827 . . . . . . 7  |-  5  e.  NN0
3617, 35deccl 11002 . . . . . 6  |- ; 4 5  e.  NN0
3736, 11deccl 11002 . . . . 5  |- ;; 4 5 3  e.  NN0
3829, 28deccl 11002 . . . . . 6  |- ;; 1 6 6  e.  NN0
3913, 10deccl 11002 . . . . . . . 8  |- ; 1 2  e.  NN0
4039, 13deccl 11002 . . . . . . 7  |- ;; 1 2 1  e.  NN0
4111, 13deccl 11002 . . . . . . . . 9  |- ; 3 1  e.  NN0
4213, 17deccl 11002 . . . . . . . . . 10  |- ; 1 4  e.  NN0
4342nn0zi 10901 . . . . . . . . . . . . 13  |- ; 1 4  e.  ZZ
4411nn0zi 10901 . . . . . . . . . . . . 13  |-  3  e.  ZZ
45 gcdcom 14034 . . . . . . . . . . . . 13  |-  ( (; 1
4  e.  ZZ  /\  3  e.  ZZ )  ->  (; 1 4  gcd  3
)  =  ( 3  gcd ; 1 4 ) )
4643, 44, 45mp2an 672 . . . . . . . . . . . 12  |-  (; 1 4  gcd  3
)  =  ( 3  gcd ; 1 4 )
47 3nn 10706 . . . . . . . . . . . . . 14  |-  3  e.  NN
48 4cn 10625 . . . . . . . . . . . . . . . 16  |-  4  e.  CC
49 3cn 10622 . . . . . . . . . . . . . . . 16  |-  3  e.  CC
50 4t3e12 11060 . . . . . . . . . . . . . . . 16  |-  ( 4  x.  3 )  = ; 1
2
5148, 49, 50mulcomli 9615 . . . . . . . . . . . . . . 15  |-  ( 3  x.  4 )  = ; 1
2
52 2p2e4 10665 . . . . . . . . . . . . . . 15  |-  ( 2  +  2 )  =  4
5313, 10, 10, 51, 52decaddi 11032 . . . . . . . . . . . . . 14  |-  ( ( 3  x.  4 )  +  2 )  = ; 1
4
54 2lt3 10715 . . . . . . . . . . . . . 14  |-  2  <  3
5547, 17, 1, 53, 54ndvdsi 13944 . . . . . . . . . . . . 13  |-  -.  3  || ; 1 4
56 3prm 14110 . . . . . . . . . . . . . 14  |-  3  e.  Prime
57 coprm 14117 . . . . . . . . . . . . . 14  |-  ( ( 3  e.  Prime  /\ ; 1 4  e.  ZZ )  ->  ( -.  3  || ; 1 4  <->  ( 3  gcd ; 1 4 )  =  1 ) )
5856, 43, 57mp2an 672 . . . . . . . . . . . . 13  |-  ( -.  3  || ; 1 4  <->  ( 3  gcd ; 1 4 )  =  1 )
5955, 58mpbi 208 . . . . . . . . . . . 12  |-  ( 3  gcd ; 1 4 )  =  1
6046, 59eqtri 2496 . . . . . . . . . . 11  |-  (; 1 4  gcd  3
)  =  1
61 eqid 2467 . . . . . . . . . . . 12  |- ; 1 4  = ; 1 4
6211dec0h 11004 . . . . . . . . . . . 12  |-  3  = ; 0 3
63 2t1e2 10696 . . . . . . . . . . . . . 14  |-  ( 2  x.  1 )  =  2
6463, 24oveq12i 6307 . . . . . . . . . . . . 13  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
65 2p1e3 10671 . . . . . . . . . . . . 13  |-  ( 2  +  1 )  =  3
6664, 65eqtri 2496 . . . . . . . . . . . 12  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  3
67 2cn 10618 . . . . . . . . . . . . . . 15  |-  2  e.  CC
68 4t2e8 10701 . . . . . . . . . . . . . . 15  |-  ( 4  x.  2 )  =  8
6948, 67, 68mulcomli 9615 . . . . . . . . . . . . . 14  |-  ( 2  x.  4 )  =  8
7069oveq1i 6305 . . . . . . . . . . . . 13  |-  ( ( 2  x.  4 )  +  3 )  =  ( 8  +  3 )
71 8p3e11 11044 . . . . . . . . . . . . 13  |-  ( 8  +  3 )  = ; 1
1
7270, 71eqtri 2496 . . . . . . . . . . . 12  |-  ( ( 2  x.  4 )  +  3 )  = ; 1
1
7313, 17, 3, 11, 61, 62, 10, 13, 13, 66, 72decma2c 11028 . . . . . . . . . . 11  |-  ( ( 2  x. ; 1 4 )  +  3 )  = ; 3 1
7410, 11, 42, 60, 73gcdi 14435 . . . . . . . . . 10  |-  (; 3 1  gcd ; 1 4 )  =  1
75 eqid 2467 . . . . . . . . . . 11  |- ; 3 1  = ; 3 1
7649mulid2i 9611 . . . . . . . . . . . . 13  |-  ( 1  x.  3 )  =  3
77 ax-1cn 9562 . . . . . . . . . . . . . 14  |-  1  e.  CC
7877addid1i 9778 . . . . . . . . . . . . 13  |-  ( 1  +  0 )  =  1
7976, 78oveq12i 6307 . . . . . . . . . . . 12  |-  ( ( 1  x.  3 )  +  ( 1  +  0 ) )  =  ( 3  +  1 )
80 3p1e4 10673 . . . . . . . . . . . 12  |-  ( 3  +  1 )  =  4
8179, 80eqtri 2496 . . . . . . . . . . 11  |-  ( ( 1  x.  3 )  +  ( 1  +  0 ) )  =  4
82 1t1e1 10695 . . . . . . . . . . . . 13  |-  ( 1  x.  1 )  =  1
8382oveq1i 6305 . . . . . . . . . . . 12  |-  ( ( 1  x.  1 )  +  4 )  =  ( 1  +  4 )
84 4p1e5 10674 . . . . . . . . . . . . 13  |-  ( 4  +  1 )  =  5
8548, 77, 84addcomli 9783 . . . . . . . . . . . 12  |-  ( 1  +  4 )  =  5
8635dec0h 11004 . . . . . . . . . . . 12  |-  5  = ; 0 5
8783, 85, 863eqtri 2500 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  +  4 )  = ; 0
5
8811, 13, 13, 17, 75, 61, 13, 35, 3, 81, 87decma2c 11028 . . . . . . . . . 10  |-  ( ( 1  x. ; 3 1 )  + ; 1
4 )  = ; 4 5
8913, 42, 41, 74, 88gcdi 14435 . . . . . . . . 9  |-  (; 4 5  gcd ; 3 1 )  =  1
90 eqid 2467 . . . . . . . . . 10  |- ; 4 5  = ; 4 5
9169, 80oveq12i 6307 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  ( 3  +  1 ) )  =  ( 8  +  4 )
92 8p4e12 11045 . . . . . . . . . . 11  |-  ( 8  +  4 )  = ; 1
2
9391, 92eqtri 2496 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  ( 3  +  1 ) )  = ; 1
2
94 5cn 10627 . . . . . . . . . . . . 13  |-  5  e.  CC
95 5t2e10 10702 . . . . . . . . . . . . 13  |-  ( 5  x.  2 )  =  10
9694, 67, 95mulcomli 9615 . . . . . . . . . . . 12  |-  ( 2  x.  5 )  =  10
97 dec10 11018 . . . . . . . . . . . 12  |-  10  = ; 1 0
9896, 97eqtri 2496 . . . . . . . . . . 11  |-  ( 2  x.  5 )  = ; 1
0
9913, 3, 24, 98decsuc 11011 . . . . . . . . . 10  |-  ( ( 2  x.  5 )  +  1 )  = ; 1
1
10017, 35, 11, 13, 90, 75, 10, 13, 13, 93, 99decma2c 11028 . . . . . . . . 9  |-  ( ( 2  x. ; 4 5 )  + ; 3
1 )  = ;; 1 2 1
10110, 41, 36, 89, 100gcdi 14435 . . . . . . . 8  |-  (;; 1 2 1  gcd ; 4 5 )  =  1
102 eqid 2467 . . . . . . . . 9  |- ;; 1 2 1  = ;; 1 2 1
103 eqid 2467 . . . . . . . . . 10  |- ; 1 2  = ; 1 2
10448addid1i 9778 . . . . . . . . . . 11  |-  ( 4  +  0 )  =  4
10517dec0h 11004 . . . . . . . . . . 11  |-  4  = ; 0 4
106104, 105eqtri 2496 . . . . . . . . . 10  |-  ( 4  +  0 )  = ; 0
4
107 00id 9766 . . . . . . . . . . . 12  |-  ( 0  +  0 )  =  0
10882, 107oveq12i 6307 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
109108, 78eqtri 2496 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
11067mulid2i 9611 . . . . . . . . . . . 12  |-  ( 1  x.  2 )  =  2
111110oveq1i 6305 . . . . . . . . . . 11  |-  ( ( 1  x.  2 )  +  4 )  =  ( 2  +  4 )
112 4p2e6 10682 . . . . . . . . . . . 12  |-  ( 4  +  2 )  =  6
11348, 67, 112addcomli 9783 . . . . . . . . . . 11  |-  ( 2  +  4 )  =  6
11428dec0h 11004 . . . . . . . . . . 11  |-  6  = ; 0 6
115111, 113, 1143eqtri 2500 . . . . . . . . . 10  |-  ( ( 1  x.  2 )  +  4 )  = ; 0
6
11613, 10, 3, 17, 103, 106, 13, 28, 3, 109, 115decma2c 11028 . . . . . . . . 9  |-  ( ( 1  x. ; 1 2 )  +  ( 4  +  0 ) )  = ; 1 6
11782oveq1i 6305 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  5 )  =  ( 1  +  5 )
118 5p1e6 10675 . . . . . . . . . . 11  |-  ( 5  +  1 )  =  6
11994, 77, 118addcomli 9783 . . . . . . . . . 10  |-  ( 1  +  5 )  =  6
120117, 119, 1143eqtri 2500 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  5 )  = ; 0
6
12139, 13, 17, 35, 102, 90, 13, 28, 3, 116, 120decma2c 11028 . . . . . . . 8  |-  ( ( 1  x. ;; 1 2 1 )  + ; 4
5 )  = ;; 1 6 6
12213, 36, 40, 101, 121gcdi 14435 . . . . . . 7  |-  (;; 1 6 6  gcd ;; 1 2 1 )  =  1
123 eqid 2467 . . . . . . . 8  |- ;; 1 6 6  = ;; 1 6 6
124 eqid 2467 . . . . . . . . 9  |- ; 1 6  = ; 1 6
12513, 10, 65, 103decsuc 11011 . . . . . . . . 9  |-  (; 1 2  +  1 )  = ; 1 3
126 1p1e2 10661 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
12763, 126oveq12i 6307 . . . . . . . . . 10  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
128127, 52eqtri 2496 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  4
129 6cn 10629 . . . . . . . . . . 11  |-  6  e.  CC
130 6t2e12 11065 . . . . . . . . . . 11  |-  ( 6  x.  2 )  = ; 1
2
131129, 67, 130mulcomli 9615 . . . . . . . . . 10  |-  ( 2  x.  6 )  = ; 1
2
132 3p2e5 10680 . . . . . . . . . . 11  |-  ( 3  +  2 )  =  5
13349, 67, 132addcomli 9783 . . . . . . . . . 10  |-  ( 2  +  3 )  =  5
13413, 10, 11, 131, 133decaddi 11032 . . . . . . . . 9  |-  ( ( 2  x.  6 )  +  3 )  = ; 1
5
13513, 28, 13, 11, 124, 125, 10, 35, 13, 128, 134decma2c 11028 . . . . . . . 8  |-  ( ( 2  x. ; 1 6 )  +  (; 1 2  +  1 ) )  = ; 4 5
13613, 10, 65, 131decsuc 11011 . . . . . . . 8  |-  ( ( 2  x.  6 )  +  1 )  = ; 1
3
13729, 28, 39, 13, 123, 102, 10, 11, 13, 135, 136decma2c 11028 . . . . . . 7  |-  ( ( 2  x. ;; 1 6 6 )  + ;; 1 2 1 )  = ;; 4 5 3
13810, 40, 38, 122, 137gcdi 14435 . . . . . 6  |-  (;; 4 5 3  gcd ;; 1 6 6 )  =  1
139 eqid 2467 . . . . . . 7  |- ;; 4 5 3  = ;; 4 5 3
14029nn0cni 10819 . . . . . . . . 9  |- ; 1 6  e.  CC
141140addid1i 9778 . . . . . . . 8  |-  (; 1 6  +  0 )  = ; 1 6
14248mulid2i 9611 . . . . . . . . . 10  |-  ( 1  x.  4 )  =  4
143142, 126oveq12i 6307 . . . . . . . . 9  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  ( 4  +  2 )
144143, 112eqtri 2496 . . . . . . . 8  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  6
14594mulid2i 9611 . . . . . . . . . 10  |-  ( 1  x.  5 )  =  5
146145oveq1i 6305 . . . . . . . . 9  |-  ( ( 1  x.  5 )  +  6 )  =  ( 5  +  6 )
147 6p5e11 11038 . . . . . . . . . 10  |-  ( 6  +  5 )  = ; 1
1
148129, 94, 147addcomli 9783 . . . . . . . . 9  |-  ( 5  +  6 )  = ; 1
1
149146, 148eqtri 2496 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  6 )  = ; 1
1
15017, 35, 13, 28, 90, 141, 13, 13, 13, 144, 149decma2c 11028 . . . . . . 7  |-  ( ( 1  x. ; 4 5 )  +  (; 1 6  +  0 ) )  = ; 6 1
15176oveq1i 6305 . . . . . . . 8  |-  ( ( 1  x.  3 )  +  6 )  =  ( 3  +  6 )
152 6p3e9 10690 . . . . . . . . 9  |-  ( 6  +  3 )  =  9
153129, 49, 152addcomli 9783 . . . . . . . 8  |-  ( 3  +  6 )  =  9
15430dec0h 11004 . . . . . . . 8  |-  9  = ; 0 9
155151, 153, 1543eqtri 2500 . . . . . . 7  |-  ( ( 1  x.  3 )  +  6 )  = ; 0
9
15636, 11, 29, 28, 139, 123, 13, 30, 3, 150, 155decma2c 11028 . . . . . 6  |-  ( ( 1  x. ;; 4 5 3 )  + ;; 1 6 6 )  = ;; 6 1 9
15713, 38, 37, 138, 156gcdi 14435 . . . . 5  |-  (;; 6 1 9  gcd ;; 4 5 3 )  =  1
158 eqid 2467 . . . . . 6  |- ;; 6 1 9  = ;; 6 1 9
159 7nn0 10829 . . . . . . 7  |-  7  e.  NN0
160 eqid 2467 . . . . . . 7  |- ; 6 1  = ; 6 1
161 5p2e7 10685 . . . . . . . 8  |-  ( 5  +  2 )  =  7
16217, 35, 10, 90, 161decaddi 11032 . . . . . . 7  |-  (; 4 5  +  2 )  = ; 4 7
163104oveq2i 6306 . . . . . . . 8  |-  ( ( 2  x.  6 )  +  ( 4  +  0 ) )  =  ( ( 2  x.  6 )  +  4 )
16413, 10, 17, 131, 113decaddi 11032 . . . . . . . 8  |-  ( ( 2  x.  6 )  +  4 )  = ; 1
6
165163, 164eqtri 2496 . . . . . . 7  |-  ( ( 2  x.  6 )  +  ( 4  +  0 ) )  = ; 1
6
16663oveq1i 6305 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  7 )  =  ( 2  +  7 )
167 7cn 10631 . . . . . . . . 9  |-  7  e.  CC
168 7p2e9 10692 . . . . . . . . 9  |-  ( 7  +  2 )  =  9
169167, 67, 168addcomli 9783 . . . . . . . 8  |-  ( 2  +  7 )  =  9
170166, 169, 1543eqtri 2500 . . . . . . 7  |-  ( ( 2  x.  1 )  +  7 )  = ; 0
9
17128, 13, 17, 159, 160, 162, 10, 30, 3, 165, 170decma2c 11028 . . . . . 6  |-  ( ( 2  x. ; 6 1 )  +  (; 4 5  +  2 ) )  = ;; 1 6 9
172 9cn 10635 . . . . . . . 8  |-  9  e.  CC
173 9t2e18 11083 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
174172, 67, 173mulcomli 9615 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
17513, 2, 11, 174, 126, 13, 71decaddci 11033 . . . . . 6  |-  ( ( 2  x.  9 )  +  3 )  = ; 2
1
17633, 30, 36, 11, 158, 139, 10, 13, 10, 171, 175decma2c 11028 . . . . 5  |-  ( ( 2  x. ;; 6 1 9 )  + ;; 4 5 3 )  = ;;; 1 6 9 1
17710, 37, 34, 157, 176gcdi 14435 . . . 4  |-  (;;; 1 6 9 1  gcd ;; 6 1 9 )  =  1
178 eqid 2467 . . . . 5  |- ;;; 1 6 9 1  = ;;; 1 6 9 1
179 eqid 2467 . . . . . 6  |- ;; 1 6 9  = ;; 1 6 9
18028, 13, 126, 160decsuc 11011 . . . . . 6  |-  (; 6 1  +  1 )  = ; 6 2
181 6p1e7 10676 . . . . . . . 8  |-  ( 6  +  1 )  =  7
182159dec0h 11004 . . . . . . . 8  |-  7  = ; 0 7
183181, 182eqtri 2496 . . . . . . 7  |-  ( 6  +  1 )  = ; 0
7
18482, 24oveq12i 6307 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  ( 1  +  1 )
185184, 126eqtri 2496 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  2
186129mulid2i 9611 . . . . . . . . 9  |-  ( 1  x.  6 )  =  6
187186oveq1i 6305 . . . . . . . 8  |-  ( ( 1  x.  6 )  +  7 )  =  ( 6  +  7 )
188 7p6e13 11042 . . . . . . . . 9  |-  ( 7  +  6 )  = ; 1
3
189167, 129, 188addcomli 9783 . . . . . . . 8  |-  ( 6  +  7 )  = ; 1
3
190187, 189eqtri 2496 . . . . . . 7  |-  ( ( 1  x.  6 )  +  7 )  = ; 1
3
19113, 28, 3, 159, 124, 183, 13, 11, 13, 185, 190decma2c 11028 . . . . . 6  |-  ( ( 1  x. ; 1 6 )  +  ( 6  +  1 ) )  = ; 2 3
192172mulid2i 9611 . . . . . . . 8  |-  ( 1  x.  9 )  =  9
193192oveq1i 6305 . . . . . . 7  |-  ( ( 1  x.  9 )  +  2 )  =  ( 9  +  2 )
194 9p2e11 11050 . . . . . . 7  |-  ( 9  +  2 )  = ; 1
1
195193, 194eqtri 2496 . . . . . 6  |-  ( ( 1  x.  9 )  +  2 )  = ; 1
1
19629, 30, 28, 10, 179, 180, 13, 13, 13, 191, 195decma2c 11028 . . . . 5  |-  ( ( 1  x. ;; 1 6 9 )  +  (; 6 1  +  1 ) )  = ;; 2 3 1
19782oveq1i 6305 . . . . . 6  |-  ( ( 1  x.  1 )  +  9 )  =  ( 1  +  9 )
198 9p1e10 10679 . . . . . . 7  |-  ( 9  +  1 )  =  10
199172, 77, 198addcomli 9783 . . . . . 6  |-  ( 1  +  9 )  =  10
200197, 199, 973eqtri 2500 . . . . 5  |-  ( ( 1  x.  1 )  +  9 )  = ; 1
0
20131, 13, 33, 30, 178, 158, 13, 3, 13, 196, 200decma2c 11028 . . . 4  |-  ( ( 1  x. ;;; 1 6 9 1 )  + ;; 6 1 9 )  = ;;; 2 3 1 0
20213, 34, 32, 177, 201gcdi 14435 . . 3  |-  (;;; 2 3 1 0  gcd ;;; 1 6 9 1 )  =  1
203 eqid 2467 . . . . . 6  |- ;; 2 3 1  = ;; 2 3 1
20431nn0cni 10819 . . . . . . 7  |- ;; 1 6 9  e.  CC
205204addid1i 9778 . . . . . 6  |-  (;; 1 6 9  +  0 )  = ;; 1 6 9
206 eqid 2467 . . . . . . 7  |- ; 2 3  = ; 2 3
20713, 28, 181, 124decsuc 11011 . . . . . . 7  |-  (; 1 6  +  1 )  = ; 1 7
208110, 126oveq12i 6307 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
209208, 52eqtri 2496 . . . . . . 7  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  4
21076oveq1i 6305 . . . . . . . 8  |-  ( ( 1  x.  3 )  +  7 )  =  ( 3  +  7 )
211 7p3e10 10693 . . . . . . . . 9  |-  ( 7  +  3 )  =  10
212167, 49, 211addcomli 9783 . . . . . . . 8  |-  ( 3  +  7 )  =  10
213210, 212, 973eqtri 2500 . . . . . . 7  |-  ( ( 1  x.  3 )  +  7 )  = ; 1
0
21410, 11, 13, 159, 206, 207, 13, 3, 13, 209, 213decma2c 11028 . . . . . 6  |-  ( ( 1  x. ; 2 3 )  +  (; 1 6  +  1 ) )  = ; 4 0
21512, 13, 29, 30, 203, 205, 13, 3, 13, 214, 200decma2c 11028 . . . . 5  |-  ( ( 1  x. ;; 2 3 1 )  +  (;; 1 6 9  +  0 ) )  = ;; 4 0 0
21677mul01i 9781 . . . . . . 7  |-  ( 1  x.  0 )  =  0
217216oveq1i 6305 . . . . . 6  |-  ( ( 1  x.  0 )  +  1 )  =  ( 0  +  1 )
21813dec0h 11004 . . . . . 6  |-  1  = ; 0 1
219217, 24, 2183eqtri 2500 . . . . 5  |-  ( ( 1  x.  0 )  +  1 )  = ; 0
1
22014, 3, 31, 13, 25, 178, 13, 13, 3, 215, 219decma2c 11028 . . . 4  |-  ( ( 1  x. ;;; 2 3 1 0 )  + ;;; 1 6 9 1 )  = ;;; 4 0 0 1
221220, 16eqtr4i 2499 . . 3  |-  ( ( 1  x. ;;; 2 3 1 0 )  + ;;; 1 6 9 1 )  =  N
22213, 32, 15, 202, 221gcdi 14435 . 2  |-  ( N  gcd ;;; 2 3 1 0 )  =  1
2239, 15, 22, 27, 222gcdmodi 14436 1  |-  ( ( ( 2 ^;; 8 0 0 )  - 
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   10c10 10605   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:  4001prm  14502
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