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Theorem 4001lem4 14290
Description: Lemma for 4001prm 14291. 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 10594 . . . 4  |-  2  e.  NN
2 8nn0 10717 . . . . . 6  |-  8  e.  NN0
3 0nn0 10709 . . . . . 6  |-  0  e.  NN0
42, 3deccl 10884 . . . . 5  |- ; 8 0  e.  NN0
54, 3deccl 10884 . . . 4  |- ;; 8 0 0  e.  NN0
6 nnexpcl 11999 . . . 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 10736 . . 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 10711 . . . . 5  |-  2  e.  NN0
11 3nn0 10712 . . . . 5  |-  3  e.  NN0
1210, 11deccl 10884 . . . 4  |- ; 2 3  e.  NN0
13 1nn0 10710 . . . 4  |-  1  e.  NN0
1412, 13deccl 10884 . . 3  |- ;; 2 3 1  e.  NN0
1514, 3deccl 10884 . 2  |- ;;; 2 3 1 0  e.  NN0
16 4001prm.1 . . 3  |-  N  = ;;; 4 0 0 1
17 4nn0 10713 . . . . . 6  |-  4  e.  NN0
1817, 3deccl 10884 . . . . 5  |- ; 4 0  e.  NN0
1918, 3deccl 10884 . . . 4  |- ;; 4 0 0  e.  NN0
20 1nn 10448 . . . 4  |-  1  e.  NN
2119, 20decnncl 10883 . . 3  |- ;;; 4 0 0 1  e.  NN
2216, 21eqeltri 2538 . 2  |-  N  e.  NN
23164001lem2 14288 . . 3  |-  ( ( 2 ^;; 8 0 0 )  mod 
N )  =  (;;; 2 3 1 1  mod 
N )
24 0p1e1 10548 . . . 4  |-  ( 0  +  1 )  =  1
25 eqid 2454 . . . 4  |- ;;; 2 3 1 0  = ;;; 2 3 1 0
2614, 3, 24, 25decsuc 10893 . . 3  |-  (;;; 2 3 1 0  +  1 )  = ;;; 2 3 1 1
2722, 7, 13, 15, 23, 26modsubi 14223 . 2  |-  ( ( ( 2 ^;; 8 0 0 )  - 
1 )  mod  N
)  =  (;;; 2 3 1 0  mod  N )
28 6nn0 10715 . . . . . 6  |-  6  e.  NN0
2913, 28deccl 10884 . . . . 5  |- ; 1 6  e.  NN0
30 9nn0 10718 . . . . 5  |-  9  e.  NN0
3129, 30deccl 10884 . . . 4  |- ;; 1 6 9  e.  NN0
3231, 13deccl 10884 . . 3  |- ;;; 1 6 9 1  e.  NN0
3328, 13deccl 10884 . . . . 5  |- ; 6 1  e.  NN0
3433, 30deccl 10884 . . . 4  |- ;; 6 1 9  e.  NN0
35 5nn0 10714 . . . . . . 7  |-  5  e.  NN0
3617, 35deccl 10884 . . . . . 6  |- ; 4 5  e.  NN0
3736, 11deccl 10884 . . . . 5  |- ;; 4 5 3  e.  NN0
3829, 28deccl 10884 . . . . . 6  |- ;; 1 6 6  e.  NN0
3913, 10deccl 10884 . . . . . . . 8  |- ; 1 2  e.  NN0
4039, 13deccl 10884 . . . . . . 7  |- ;; 1 2 1  e.  NN0
4111, 13deccl 10884 . . . . . . . . 9  |- ; 3 1  e.  NN0
4213, 17deccl 10884 . . . . . . . . . 10  |- ; 1 4  e.  NN0
4342nn0zi 10786 . . . . . . . . . . . . 13  |- ; 1 4  e.  ZZ
4411nn0zi 10786 . . . . . . . . . . . . 13  |-  3  e.  ZZ
45 gcdcom 13826 . . . . . . . . . . . . 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 10595 . . . . . . . . . . . . . 14  |-  3  e.  NN
48 4cn 10514 . . . . . . . . . . . . . . . 16  |-  4  e.  CC
49 3cn 10511 . . . . . . . . . . . . . . . 16  |-  3  e.  CC
50 4t3e12 10942 . . . . . . . . . . . . . . . 16  |-  ( 4  x.  3 )  = ; 1
2
5148, 49, 50mulcomli 9508 . . . . . . . . . . . . . . 15  |-  ( 3  x.  4 )  = ; 1
2
52 2p2e4 10554 . . . . . . . . . . . . . . 15  |-  ( 2  +  2 )  =  4
5313, 10, 10, 51, 52decaddi 10914 . . . . . . . . . . . . . 14  |-  ( ( 3  x.  4 )  +  2 )  = ; 1
4
54 2lt3 10604 . . . . . . . . . . . . . 14  |-  2  <  3
5547, 17, 1, 53, 54ndvdsi 13736 . . . . . . . . . . . . 13  |-  -.  3  || ; 1 4
56 3prm 13902 . . . . . . . . . . . . . 14  |-  3  e.  Prime
57 coprm 13908 . . . . . . . . . . . . . 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 2483 . . . . . . . . . . 11  |-  (; 1 4  gcd  3
)  =  1
61 eqid 2454 . . . . . . . . . . . 12  |- ; 1 4  = ; 1 4
6211dec0h 10886 . . . . . . . . . . . 12  |-  3  = ; 0 3
63 2t1e2 10585 . . . . . . . . . . . . . 14  |-  ( 2  x.  1 )  =  2
6463, 24oveq12i 6215 . . . . . . . . . . . . 13  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
65 2p1e3 10560 . . . . . . . . . . . . 13  |-  ( 2  +  1 )  =  3
6664, 65eqtri 2483 . . . . . . . . . . . 12  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  3
67 2cn 10507 . . . . . . . . . . . . . . 15  |-  2  e.  CC
68 4t2e8 10590 . . . . . . . . . . . . . . 15  |-  ( 4  x.  2 )  =  8
6948, 67, 68mulcomli 9508 . . . . . . . . . . . . . 14  |-  ( 2  x.  4 )  =  8
7069oveq1i 6213 . . . . . . . . . . . . 13  |-  ( ( 2  x.  4 )  +  3 )  =  ( 8  +  3 )
71 8p3e11 10926 . . . . . . . . . . . . 13  |-  ( 8  +  3 )  = ; 1
1
7270, 71eqtri 2483 . . . . . . . . . . . 12  |-  ( ( 2  x.  4 )  +  3 )  = ; 1
1
7313, 17, 3, 11, 61, 62, 10, 13, 13, 66, 72decma2c 10910 . . . . . . . . . . 11  |-  ( ( 2  x. ; 1 4 )  +  3 )  = ; 3 1
7410, 11, 42, 60, 73gcdi 14224 . . . . . . . . . 10  |-  (; 3 1  gcd ; 1 4 )  =  1
75 eqid 2454 . . . . . . . . . . 11  |- ; 3 1  = ; 3 1
7649mulid2i 9504 . . . . . . . . . . . . 13  |-  ( 1  x.  3 )  =  3
77 ax-1cn 9455 . . . . . . . . . . . . . 14  |-  1  e.  CC
7877addid1i 9671 . . . . . . . . . . . . 13  |-  ( 1  +  0 )  =  1
7976, 78oveq12i 6215 . . . . . . . . . . . 12  |-  ( ( 1  x.  3 )  +  ( 1  +  0 ) )  =  ( 3  +  1 )
80 3p1e4 10562 . . . . . . . . . . . 12  |-  ( 3  +  1 )  =  4
8179, 80eqtri 2483 . . . . . . . . . . 11  |-  ( ( 1  x.  3 )  +  ( 1  +  0 ) )  =  4
82 1t1e1 10584 . . . . . . . . . . . . 13  |-  ( 1  x.  1 )  =  1
8382oveq1i 6213 . . . . . . . . . . . 12  |-  ( ( 1  x.  1 )  +  4 )  =  ( 1  +  4 )
84 4p1e5 10563 . . . . . . . . . . . . 13  |-  ( 4  +  1 )  =  5
8548, 77, 84addcomli 9676 . . . . . . . . . . . 12  |-  ( 1  +  4 )  =  5
8635dec0h 10886 . . . . . . . . . . . 12  |-  5  = ; 0 5
8783, 85, 863eqtri 2487 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  +  4 )  = ; 0
5
8811, 13, 13, 17, 75, 61, 13, 35, 3, 81, 87decma2c 10910 . . . . . . . . . 10  |-  ( ( 1  x. ; 3 1 )  + ; 1
4 )  = ; 4 5
8913, 42, 41, 74, 88gcdi 14224 . . . . . . . . 9  |-  (; 4 5  gcd ; 3 1 )  =  1
90 eqid 2454 . . . . . . . . . 10  |- ; 4 5  = ; 4 5
9169, 80oveq12i 6215 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  ( 3  +  1 ) )  =  ( 8  +  4 )
92 8p4e12 10927 . . . . . . . . . . 11  |-  ( 8  +  4 )  = ; 1
2
9391, 92eqtri 2483 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  ( 3  +  1 ) )  = ; 1
2
94 5cn 10516 . . . . . . . . . . . . 13  |-  5  e.  CC
95 5t2e10 10591 . . . . . . . . . . . . 13  |-  ( 5  x.  2 )  =  10
9694, 67, 95mulcomli 9508 . . . . . . . . . . . 12  |-  ( 2  x.  5 )  =  10
97 dec10 10900 . . . . . . . . . . . 12  |-  10  = ; 1 0
9896, 97eqtri 2483 . . . . . . . . . . 11  |-  ( 2  x.  5 )  = ; 1
0
9913, 3, 24, 98decsuc 10893 . . . . . . . . . 10  |-  ( ( 2  x.  5 )  +  1 )  = ; 1
1
10017, 35, 11, 13, 90, 75, 10, 13, 13, 93, 99decma2c 10910 . . . . . . . . 9  |-  ( ( 2  x. ; 4 5 )  + ; 3
1 )  = ;; 1 2 1
10110, 41, 36, 89, 100gcdi 14224 . . . . . . . 8  |-  (;; 1 2 1  gcd ; 4 5 )  =  1
102 eqid 2454 . . . . . . . . 9  |- ;; 1 2 1  = ;; 1 2 1
103 eqid 2454 . . . . . . . . . 10  |- ; 1 2  = ; 1 2
10448addid1i 9671 . . . . . . . . . . 11  |-  ( 4  +  0 )  =  4
10517dec0h 10886 . . . . . . . . . . 11  |-  4  = ; 0 4
106104, 105eqtri 2483 . . . . . . . . . 10  |-  ( 4  +  0 )  = ; 0
4
107 00id 9659 . . . . . . . . . . . 12  |-  ( 0  +  0 )  =  0
10882, 107oveq12i 6215 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
109108, 78eqtri 2483 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
11067mulid2i 9504 . . . . . . . . . . . 12  |-  ( 1  x.  2 )  =  2
111110oveq1i 6213 . . . . . . . . . . 11  |-  ( ( 1  x.  2 )  +  4 )  =  ( 2  +  4 )
112 4p2e6 10571 . . . . . . . . . . . 12  |-  ( 4  +  2 )  =  6
11348, 67, 112addcomli 9676 . . . . . . . . . . 11  |-  ( 2  +  4 )  =  6
11428dec0h 10886 . . . . . . . . . . 11  |-  6  = ; 0 6
115111, 113, 1143eqtri 2487 . . . . . . . . . 10  |-  ( ( 1  x.  2 )  +  4 )  = ; 0
6
11613, 10, 3, 17, 103, 106, 13, 28, 3, 109, 115decma2c 10910 . . . . . . . . 9  |-  ( ( 1  x. ; 1 2 )  +  ( 4  +  0 ) )  = ; 1 6
11782oveq1i 6213 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  5 )  =  ( 1  +  5 )
118 5p1e6 10564 . . . . . . . . . . 11  |-  ( 5  +  1 )  =  6
11994, 77, 118addcomli 9676 . . . . . . . . . 10  |-  ( 1  +  5 )  =  6
120117, 119, 1143eqtri 2487 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  5 )  = ; 0
6
12139, 13, 17, 35, 102, 90, 13, 28, 3, 116, 120decma2c 10910 . . . . . . . 8  |-  ( ( 1  x. ;; 1 2 1 )  + ; 4
5 )  = ;; 1 6 6
12213, 36, 40, 101, 121gcdi 14224 . . . . . . 7  |-  (;; 1 6 6  gcd ;; 1 2 1 )  =  1
123 eqid 2454 . . . . . . . 8  |- ;; 1 6 6  = ;; 1 6 6
124 eqid 2454 . . . . . . . . 9  |- ; 1 6  = ; 1 6
12513, 10, 65, 103decsuc 10893 . . . . . . . . 9  |-  (; 1 2  +  1 )  = ; 1 3
126 1p1e2 10550 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
12763, 126oveq12i 6215 . . . . . . . . . 10  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
128127, 52eqtri 2483 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  4
129 6cn 10518 . . . . . . . . . . 11  |-  6  e.  CC
130 6t2e12 10947 . . . . . . . . . . 11  |-  ( 6  x.  2 )  = ; 1
2
131129, 67, 130mulcomli 9508 . . . . . . . . . 10  |-  ( 2  x.  6 )  = ; 1
2
132 3p2e5 10569 . . . . . . . . . . 11  |-  ( 3  +  2 )  =  5
13349, 67, 132addcomli 9676 . . . . . . . . . 10  |-  ( 2  +  3 )  =  5
13413, 10, 11, 131, 133decaddi 10914 . . . . . . . . 9  |-  ( ( 2  x.  6 )  +  3 )  = ; 1
5
13513, 28, 13, 11, 124, 125, 10, 35, 13, 128, 134decma2c 10910 . . . . . . . 8  |-  ( ( 2  x. ; 1 6 )  +  (; 1 2  +  1 ) )  = ; 4 5
13613, 10, 65, 131decsuc 10893 . . . . . . . 8  |-  ( ( 2  x.  6 )  +  1 )  = ; 1
3
13729, 28, 39, 13, 123, 102, 10, 11, 13, 135, 136decma2c 10910 . . . . . . 7  |-  ( ( 2  x. ;; 1 6 6 )  + ;; 1 2 1 )  = ;; 4 5 3
13810, 40, 38, 122, 137gcdi 14224 . . . . . 6  |-  (;; 4 5 3  gcd ;; 1 6 6 )  =  1
139 eqid 2454 . . . . . . 7  |- ;; 4 5 3  = ;; 4 5 3
14029nn0cni 10706 . . . . . . . . 9  |- ; 1 6  e.  CC
141140addid1i 9671 . . . . . . . 8  |-  (; 1 6  +  0 )  = ; 1 6
14248mulid2i 9504 . . . . . . . . . 10  |-  ( 1  x.  4 )  =  4
143142, 126oveq12i 6215 . . . . . . . . 9  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  ( 4  +  2 )
144143, 112eqtri 2483 . . . . . . . 8  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  6
14594mulid2i 9504 . . . . . . . . . 10  |-  ( 1  x.  5 )  =  5
146145oveq1i 6213 . . . . . . . . 9  |-  ( ( 1  x.  5 )  +  6 )  =  ( 5  +  6 )
147 6p5e11 10920 . . . . . . . . . 10  |-  ( 6  +  5 )  = ; 1
1
148129, 94, 147addcomli 9676 . . . . . . . . 9  |-  ( 5  +  6 )  = ; 1
1
149146, 148eqtri 2483 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  6 )  = ; 1
1
15017, 35, 13, 28, 90, 141, 13, 13, 13, 144, 149decma2c 10910 . . . . . . 7  |-  ( ( 1  x. ; 4 5 )  +  (; 1 6  +  0 ) )  = ; 6 1
15176oveq1i 6213 . . . . . . . 8  |-  ( ( 1  x.  3 )  +  6 )  =  ( 3  +  6 )
152 6p3e9 10579 . . . . . . . . 9  |-  ( 6  +  3 )  =  9
153129, 49, 152addcomli 9676 . . . . . . . 8  |-  ( 3  +  6 )  =  9
15430dec0h 10886 . . . . . . . 8  |-  9  = ; 0 9
155151, 153, 1543eqtri 2487 . . . . . . 7  |-  ( ( 1  x.  3 )  +  6 )  = ; 0
9
15636, 11, 29, 28, 139, 123, 13, 30, 3, 150, 155decma2c 10910 . . . . . 6  |-  ( ( 1  x. ;; 4 5 3 )  + ;; 1 6 6 )  = ;; 6 1 9
15713, 38, 37, 138, 156gcdi 14224 . . . . 5  |-  (;; 6 1 9  gcd ;; 4 5 3 )  =  1
158 eqid 2454 . . . . . 6  |- ;; 6 1 9  = ;; 6 1 9
159 7nn0 10716 . . . . . . 7  |-  7  e.  NN0
160 eqid 2454 . . . . . . 7  |- ; 6 1  = ; 6 1
161 5p2e7 10574 . . . . . . . 8  |-  ( 5  +  2 )  =  7
16217, 35, 10, 90, 161decaddi 10914 . . . . . . 7  |-  (; 4 5  +  2 )  = ; 4 7
163104oveq2i 6214 . . . . . . . 8  |-  ( ( 2  x.  6 )  +  ( 4  +  0 ) )  =  ( ( 2  x.  6 )  +  4 )
16413, 10, 17, 131, 113decaddi 10914 . . . . . . . 8  |-  ( ( 2  x.  6 )  +  4 )  = ; 1
6
165163, 164eqtri 2483 . . . . . . 7  |-  ( ( 2  x.  6 )  +  ( 4  +  0 ) )  = ; 1
6
16663oveq1i 6213 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  7 )  =  ( 2  +  7 )
167 7cn 10520 . . . . . . . . 9  |-  7  e.  CC
168 7p2e9 10581 . . . . . . . . 9  |-  ( 7  +  2 )  =  9
169167, 67, 168addcomli 9676 . . . . . . . 8  |-  ( 2  +  7 )  =  9
170166, 169, 1543eqtri 2487 . . . . . . 7  |-  ( ( 2  x.  1 )  +  7 )  = ; 0
9
17128, 13, 17, 159, 160, 162, 10, 30, 3, 165, 170decma2c 10910 . . . . . 6  |-  ( ( 2  x. ; 6 1 )  +  (; 4 5  +  2 ) )  = ;; 1 6 9
172 9cn 10524 . . . . . . . 8  |-  9  e.  CC
173 9t2e18 10965 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
174172, 67, 173mulcomli 9508 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
17513, 2, 11, 174, 126, 13, 71decaddci 10915 . . . . . 6  |-  ( ( 2  x.  9 )  +  3 )  = ; 2
1
17633, 30, 36, 11, 158, 139, 10, 13, 10, 171, 175decma2c 10910 . . . . 5  |-  ( ( 2  x. ;; 6 1 9 )  + ;; 4 5 3 )  = ;;; 1 6 9 1
17710, 37, 34, 157, 176gcdi 14224 . . . 4  |-  (;;; 1 6 9 1  gcd ;; 6 1 9 )  =  1
178 eqid 2454 . . . . 5  |- ;;; 1 6 9 1  = ;;; 1 6 9 1
179 eqid 2454 . . . . . 6  |- ;; 1 6 9  = ;; 1 6 9
18028, 13, 126, 160decsuc 10893 . . . . . 6  |-  (; 6 1  +  1 )  = ; 6 2
181 6p1e7 10565 . . . . . . . 8  |-  ( 6  +  1 )  =  7
182159dec0h 10886 . . . . . . . 8  |-  7  = ; 0 7
183181, 182eqtri 2483 . . . . . . 7  |-  ( 6  +  1 )  = ; 0
7
18482, 24oveq12i 6215 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  ( 1  +  1 )
185184, 126eqtri 2483 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  2
186129mulid2i 9504 . . . . . . . . 9  |-  ( 1  x.  6 )  =  6
187186oveq1i 6213 . . . . . . . 8  |-  ( ( 1  x.  6 )  +  7 )  =  ( 6  +  7 )
188 7p6e13 10924 . . . . . . . . 9  |-  ( 7  +  6 )  = ; 1
3
189167, 129, 188addcomli 9676 . . . . . . . 8  |-  ( 6  +  7 )  = ; 1
3
190187, 189eqtri 2483 . . . . . . 7  |-  ( ( 1  x.  6 )  +  7 )  = ; 1
3
19113, 28, 3, 159, 124, 183, 13, 11, 13, 185, 190decma2c 10910 . . . . . 6  |-  ( ( 1  x. ; 1 6 )  +  ( 6  +  1 ) )  = ; 2 3
192172mulid2i 9504 . . . . . . . 8  |-  ( 1  x.  9 )  =  9
193192oveq1i 6213 . . . . . . 7  |-  ( ( 1  x.  9 )  +  2 )  =  ( 9  +  2 )
194 9p2e11 10932 . . . . . . 7  |-  ( 9  +  2 )  = ; 1
1
195193, 194eqtri 2483 . . . . . 6  |-  ( ( 1  x.  9 )  +  2 )  = ; 1
1
19629, 30, 28, 10, 179, 180, 13, 13, 13, 191, 195decma2c 10910 . . . . 5  |-  ( ( 1  x. ;; 1 6 9 )  +  (; 6 1  +  1 ) )  = ;; 2 3 1
19782oveq1i 6213 . . . . . 6  |-  ( ( 1  x.  1 )  +  9 )  =  ( 1  +  9 )
198 9p1e10 10568 . . . . . . 7  |-  ( 9  +  1 )  =  10
199172, 77, 198addcomli 9676 . . . . . 6  |-  ( 1  +  9 )  =  10
200197, 199, 973eqtri 2487 . . . . 5  |-  ( ( 1  x.  1 )  +  9 )  = ; 1
0
20131, 13, 33, 30, 178, 158, 13, 3, 13, 196, 200decma2c 10910 . . . 4  |-  ( ( 1  x. ;;; 1 6 9 1 )  + ;; 6 1 9 )  = ;;; 2 3 1 0
20213, 34, 32, 177, 201gcdi 14224 . . 3  |-  (;;; 2 3 1 0  gcd ;;; 1 6 9 1 )  =  1
203 eqid 2454 . . . . . 6  |- ;; 2 3 1  = ;; 2 3 1
20431nn0cni 10706 . . . . . . 7  |- ;; 1 6 9  e.  CC
205204addid1i 9671 . . . . . 6  |-  (;; 1 6 9  +  0 )  = ;; 1 6 9
206 eqid 2454 . . . . . . 7  |- ; 2 3  = ; 2 3
20713, 28, 181, 124decsuc 10893 . . . . . . 7  |-  (; 1 6  +  1 )  = ; 1 7
208110, 126oveq12i 6215 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
209208, 52eqtri 2483 . . . . . . 7  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  4
21076oveq1i 6213 . . . . . . . 8  |-  ( ( 1  x.  3 )  +  7 )  =  ( 3  +  7 )
211 7p3e10 10582 . . . . . . . . 9  |-  ( 7  +  3 )  =  10
212167, 49, 211addcomli 9676 . . . . . . . 8  |-  ( 3  +  7 )  =  10
213210, 212, 973eqtri 2487 . . . . . . 7  |-  ( ( 1  x.  3 )  +  7 )  = ; 1
0
21410, 11, 13, 159, 206, 207, 13, 3, 13, 209, 213decma2c 10910 . . . . . 6  |-  ( ( 1  x. ; 2 3 )  +  (; 1 6  +  1 ) )  = ; 4 0
21512, 13, 29, 30, 203, 205, 13, 3, 13, 214, 200decma2c 10910 . . . . 5  |-  ( ( 1  x. ;; 2 3 1 )  +  (;; 1 6 9  +  0 ) )  = ;; 4 0 0
21677mul01i 9674 . . . . . . 7  |-  ( 1  x.  0 )  =  0
217216oveq1i 6213 . . . . . 6  |-  ( ( 1  x.  0 )  +  1 )  =  ( 0  +  1 )
21813dec0h 10886 . . . . . 6  |-  1  = ; 0 1
219217, 24, 2183eqtri 2487 . . . . 5  |-  ( ( 1  x.  0 )  +  1 )  = ; 0
1
22014, 3, 31, 13, 25, 178, 13, 13, 3, 215, 219decma2c 10910 . . . 4  |-  ( ( 1  x. ;;; 2 3 1 0 )  + ;;; 1 6 9 1 )  = ;;; 4 0 0 1
221220, 16eqtr4i 2486 . . 3  |-  ( ( 1  x. ;;; 2 3 1 0 )  + ;;; 1 6 9 1 )  =  N
22213, 32, 15, 202, 221gcdi 14224 . 2  |-  ( N  gcd ;;; 2 3 1 0 )  =  1
2239, 15, 22, 27, 222gcdmodi 14225 1  |-  ( ( ( 2 ^;; 8 0 0 )  - 
1 )  gcd  N
)  =  1
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1370    e. wcel 1758   class class class wbr 4403  (class class class)co 6203   0cc0 9397   1c1 9398    + caddc 9400    x. cmul 9402    - cmin 9710   NNcn 10437   2c2 10486   3c3 10487   4c4 10488   5c5 10489   6c6 10490   7c7 10491   8c8 10492   9c9 10493   10c10 10494   NN0cn0 10694   ZZcz 10761  ;cdc 10870   ^cexp 11986    || cdivides 13657    gcd cgcd 13812   Primecprime 13885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-rp 11107  df-fz 11559  df-fl 11763  df-mod 11830  df-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-dvds 13658  df-gcd 13813  df-prm 13886
This theorem is referenced by:  4001prm  14291
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