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Theorem 4001lem4 13463
Description: Lemma for 4001prm 13464. 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 10133 . . . 4  |-  2  e.  NN
2 8nn0 10244 . . . . . 6  |-  8  e.  NN0
3 0nn0 10236 . . . . . 6  |-  0  e.  NN0
42, 3deccl 10396 . . . . 5  |- ; 8 0  e.  NN0
54, 3deccl 10396 . . . 4  |- ;; 8 0 0  e.  NN0
6 nnexpcl 11394 . . . 4  |-  ( ( 2  e.  NN  /\ ;; 8 0 0  e. 
NN0 )  ->  (
2 ^;; 8 0 0 )  e.  NN )
71, 5, 6mp2an 654 . . 3  |-  ( 2 ^;; 8 0 0 )  e.  NN
8 nnm1nn0 10261 . . 3  |-  ( ( 2 ^;; 8 0 0 )  e.  NN  ->  ( (
2 ^;; 8 0 0 )  - 
1 )  e.  NN0 )
97, 8ax-mp 8 . 2  |-  ( ( 2 ^;; 8 0 0 )  - 
1 )  e.  NN0
10 2nn0 10238 . . . . 5  |-  2  e.  NN0
11 3nn0 10239 . . . . 5  |-  3  e.  NN0
1210, 11deccl 10396 . . . 4  |- ; 2 3  e.  NN0
13 1nn0 10237 . . . 4  |-  1  e.  NN0
1412, 13deccl 10396 . . 3  |- ;; 2 3 1  e.  NN0
1514, 3deccl 10396 . 2  |- ;;; 2 3 1 0  e.  NN0
16 4001prm.1 . . 3  |-  N  = ;;; 4 0 0 1
17 4nn0 10240 . . . . . 6  |-  4  e.  NN0
1817, 3deccl 10396 . . . . 5  |- ; 4 0  e.  NN0
1918, 3deccl 10396 . . . 4  |- ;; 4 0 0  e.  NN0
20 1nn 10011 . . . 4  |-  1  e.  NN
2119, 20decnncl 10395 . . 3  |- ;;; 4 0 0 1  e.  NN
2216, 21eqeltri 2506 . 2  |-  N  e.  NN
23164001lem2 13461 . . 3  |-  ( ( 2 ^;; 8 0 0 )  mod 
N )  =  (;;; 2 3 1 1  mod 
N )
24 0p1e1 10093 . . . 4  |-  ( 0  +  1 )  =  1
25 eqid 2436 . . . 4  |- ;;; 2 3 1 0  = ;;; 2 3 1 0
2614, 3, 24, 25decsuc 10405 . . 3  |-  (;;; 2 3 1 0  +  1 )  = ;;; 2 3 1 1
2722, 7, 13, 15, 23, 26modsubi 13408 . 2  |-  ( ( ( 2 ^;; 8 0 0 )  - 
1 )  mod  N
)  =  (;;; 2 3 1 0  mod  N )
28 6nn0 10242 . . . . . 6  |-  6  e.  NN0
2913, 28deccl 10396 . . . . 5  |- ; 1 6  e.  NN0
30 9nn0 10245 . . . . 5  |-  9  e.  NN0
3129, 30deccl 10396 . . . 4  |- ;; 1 6 9  e.  NN0
3231, 13deccl 10396 . . 3  |- ;;; 1 6 9 1  e.  NN0
3328, 13deccl 10396 . . . . 5  |- ; 6 1  e.  NN0
3433, 30deccl 10396 . . . 4  |- ;; 6 1 9  e.  NN0
35 5nn0 10241 . . . . . . 7  |-  5  e.  NN0
3617, 35deccl 10396 . . . . . 6  |- ; 4 5  e.  NN0
3736, 11deccl 10396 . . . . 5  |- ;; 4 5 3  e.  NN0
3829, 28deccl 10396 . . . . . 6  |- ;; 1 6 6  e.  NN0
3913, 10deccl 10396 . . . . . . . 8  |- ; 1 2  e.  NN0
4039, 13deccl 10396 . . . . . . 7  |- ;; 1 2 1  e.  NN0
4111, 13deccl 10396 . . . . . . . . 9  |- ; 3 1  e.  NN0
4213, 17deccl 10396 . . . . . . . . . 10  |- ; 1 4  e.  NN0
4342nn0zi 10306 . . . . . . . . . . . . 13  |- ; 1 4  e.  ZZ
4411nn0zi 10306 . . . . . . . . . . . . 13  |-  3  e.  ZZ
45 gcdcom 13020 . . . . . . . . . . . . 13  |-  ( (; 1
4  e.  ZZ  /\  3  e.  ZZ )  ->  (; 1 4  gcd  3
)  =  ( 3  gcd ; 1 4 ) )
4643, 44, 45mp2an 654 . . . . . . . . . . . 12  |-  (; 1 4  gcd  3
)  =  ( 3  gcd ; 1 4 )
47 3nn 10134 . . . . . . . . . . . . . 14  |-  3  e.  NN
48 4cn 10074 . . . . . . . . . . . . . . . 16  |-  4  e.  CC
49 3cn 10072 . . . . . . . . . . . . . . . 16  |-  3  e.  CC
50 4t3e12 10454 . . . . . . . . . . . . . . . 16  |-  ( 4  x.  3 )  = ; 1
2
5148, 49, 50mulcomli 9097 . . . . . . . . . . . . . . 15  |-  ( 3  x.  4 )  = ; 1
2
52 2p2e4 10098 . . . . . . . . . . . . . . 15  |-  ( 2  +  2 )  =  4
5313, 10, 10, 51, 52decaddi 10426 . . . . . . . . . . . . . 14  |-  ( ( 3  x.  4 )  +  2 )  = ; 1
4
54 2lt3 10143 . . . . . . . . . . . . . 14  |-  2  <  3
5547, 17, 1, 53, 54ndvdsi 12930 . . . . . . . . . . . . 13  |-  -.  3  || ; 1 4
56 3prm 13096 . . . . . . . . . . . . . 14  |-  3  e.  Prime
57 coprm 13100 . . . . . . . . . . . . . 14  |-  ( ( 3  e.  Prime  /\ ; 1 4  e.  ZZ )  ->  ( -.  3  || ; 1 4  <->  ( 3  gcd ; 1 4 )  =  1 ) )
5856, 43, 57mp2an 654 . . . . . . . . . . . . 13  |-  ( -.  3  || ; 1 4  <->  ( 3  gcd ; 1 4 )  =  1 )
5955, 58mpbi 200 . . . . . . . . . . . 12  |-  ( 3  gcd ; 1 4 )  =  1
6046, 59eqtri 2456 . . . . . . . . . . 11  |-  (; 1 4  gcd  3
)  =  1
61 eqid 2436 . . . . . . . . . . . 12  |- ; 1 4  = ; 1 4
6211dec0h 10398 . . . . . . . . . . . 12  |-  3  = ; 0 3
63 2cn 10070 . . . . . . . . . . . . . . 15  |-  2  e.  CC
6463mulid1i 9092 . . . . . . . . . . . . . 14  |-  ( 2  x.  1 )  =  2
6564, 24oveq12i 6093 . . . . . . . . . . . . 13  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
66 2p1e3 10103 . . . . . . . . . . . . 13  |-  ( 2  +  1 )  =  3
6765, 66eqtri 2456 . . . . . . . . . . . 12  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  3
68 4t2e8 10130 . . . . . . . . . . . . . . 15  |-  ( 4  x.  2 )  =  8
6948, 63, 68mulcomli 9097 . . . . . . . . . . . . . 14  |-  ( 2  x.  4 )  =  8
7069oveq1i 6091 . . . . . . . . . . . . 13  |-  ( ( 2  x.  4 )  +  3 )  =  ( 8  +  3 )
71 8p3e11 10438 . . . . . . . . . . . . 13  |-  ( 8  +  3 )  = ; 1
1
7270, 71eqtri 2456 . . . . . . . . . . . 12  |-  ( ( 2  x.  4 )  +  3 )  = ; 1
1
7313, 17, 3, 11, 61, 62, 10, 13, 13, 67, 72decma2c 10422 . . . . . . . . . . 11  |-  ( ( 2  x. ; 1 4 )  +  3 )  = ; 3 1
7410, 11, 42, 60, 73gcdi 13409 . . . . . . . . . 10  |-  (; 3 1  gcd ; 1 4 )  =  1
75 eqid 2436 . . . . . . . . . . 11  |- ; 3 1  = ; 3 1
7649mulid2i 9093 . . . . . . . . . . . . 13  |-  ( 1  x.  3 )  =  3
77 ax-1cn 9048 . . . . . . . . . . . . . 14  |-  1  e.  CC
7877addid1i 9253 . . . . . . . . . . . . 13  |-  ( 1  +  0 )  =  1
7976, 78oveq12i 6093 . . . . . . . . . . . 12  |-  ( ( 1  x.  3 )  +  ( 1  +  0 ) )  =  ( 3  +  1 )
80 3p1e4 10104 . . . . . . . . . . . 12  |-  ( 3  +  1 )  =  4
8179, 80eqtri 2456 . . . . . . . . . . 11  |-  ( ( 1  x.  3 )  +  ( 1  +  0 ) )  =  4
82 1t1e1 10126 . . . . . . . . . . . . 13  |-  ( 1  x.  1 )  =  1
8382oveq1i 6091 . . . . . . . . . . . 12  |-  ( ( 1  x.  1 )  +  4 )  =  ( 1  +  4 )
84 4p1e5 10105 . . . . . . . . . . . . 13  |-  ( 4  +  1 )  =  5
8548, 77, 84addcomli 9258 . . . . . . . . . . . 12  |-  ( 1  +  4 )  =  5
8635dec0h 10398 . . . . . . . . . . . 12  |-  5  = ; 0 5
8783, 85, 863eqtri 2460 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  +  4 )  = ; 0
5
8811, 13, 13, 17, 75, 61, 13, 35, 3, 81, 87decma2c 10422 . . . . . . . . . 10  |-  ( ( 1  x. ; 3 1 )  + ; 1
4 )  = ; 4 5
8913, 42, 41, 74, 88gcdi 13409 . . . . . . . . 9  |-  (; 4 5  gcd ; 3 1 )  =  1
90 eqid 2436 . . . . . . . . . 10  |- ; 4 5  = ; 4 5
9169, 80oveq12i 6093 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  ( 3  +  1 ) )  =  ( 8  +  4 )
92 8p4e12 10439 . . . . . . . . . . 11  |-  ( 8  +  4 )  = ; 1
2
9391, 92eqtri 2456 . . . . . . . . . 10  |-  ( ( 2  x.  4 )  +  ( 3  +  1 ) )  = ; 1
2
94 5nn 10136 . . . . . . . . . . . . . 14  |-  5  e.  NN
9594nncni 10010 . . . . . . . . . . . . 13  |-  5  e.  CC
96 5t2e10 10131 . . . . . . . . . . . . 13  |-  ( 5  x.  2 )  =  10
9795, 63, 96mulcomli 9097 . . . . . . . . . . . 12  |-  ( 2  x.  5 )  =  10
98 dec10 10412 . . . . . . . . . . . 12  |-  10  = ; 1 0
9997, 98eqtri 2456 . . . . . . . . . . 11  |-  ( 2  x.  5 )  = ; 1
0
10013, 3, 24, 99decsuc 10405 . . . . . . . . . 10  |-  ( ( 2  x.  5 )  +  1 )  = ; 1
1
10117, 35, 11, 13, 90, 75, 10, 13, 13, 93, 100decma2c 10422 . . . . . . . . 9  |-  ( ( 2  x. ; 4 5 )  + ; 3
1 )  = ;; 1 2 1
10210, 41, 36, 89, 101gcdi 13409 . . . . . . . 8  |-  (;; 1 2 1  gcd ; 4 5 )  =  1
103 eqid 2436 . . . . . . . . 9  |- ;; 1 2 1  = ;; 1 2 1
104 eqid 2436 . . . . . . . . . 10  |- ; 1 2  = ; 1 2
10548addid1i 9253 . . . . . . . . . . 11  |-  ( 4  +  0 )  =  4
10617dec0h 10398 . . . . . . . . . . 11  |-  4  = ; 0 4
107105, 106eqtri 2456 . . . . . . . . . 10  |-  ( 4  +  0 )  = ; 0
4
108 00id 9241 . . . . . . . . . . . 12  |-  ( 0  +  0 )  =  0
10982, 108oveq12i 6093 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
110109, 78eqtri 2456 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
11163mulid2i 9093 . . . . . . . . . . . 12  |-  ( 1  x.  2 )  =  2
112111oveq1i 6091 . . . . . . . . . . 11  |-  ( ( 1  x.  2 )  +  4 )  =  ( 2  +  4 )
113 4p2e6 10113 . . . . . . . . . . . 12  |-  ( 4  +  2 )  =  6
11448, 63, 113addcomli 9258 . . . . . . . . . . 11  |-  ( 2  +  4 )  =  6
11528dec0h 10398 . . . . . . . . . . 11  |-  6  = ; 0 6
116112, 114, 1153eqtri 2460 . . . . . . . . . 10  |-  ( ( 1  x.  2 )  +  4 )  = ; 0
6
11713, 10, 3, 17, 104, 107, 13, 28, 3, 110, 116decma2c 10422 . . . . . . . . 9  |-  ( ( 1  x. ; 1 2 )  +  ( 4  +  0 ) )  = ; 1 6
11882oveq1i 6091 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  5 )  =  ( 1  +  5 )
119 5p1e6 10106 . . . . . . . . . . 11  |-  ( 5  +  1 )  =  6
12095, 77, 119addcomli 9258 . . . . . . . . . 10  |-  ( 1  +  5 )  =  6
121118, 120, 1153eqtri 2460 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  5 )  = ; 0
6
12239, 13, 17, 35, 103, 90, 13, 28, 3, 117, 121decma2c 10422 . . . . . . . 8  |-  ( ( 1  x. ;; 1 2 1 )  + ; 4
5 )  = ;; 1 6 6
12313, 36, 40, 102, 122gcdi 13409 . . . . . . 7  |-  (;; 1 6 6  gcd ;; 1 2 1 )  =  1
124 eqid 2436 . . . . . . . 8  |- ;; 1 6 6  = ;; 1 6 6
125 eqid 2436 . . . . . . . . 9  |- ; 1 6  = ; 1 6
12613, 10, 66, 104decsuc 10405 . . . . . . . . 9  |-  (; 1 2  +  1 )  = ; 1 3
127 1p1e2 10094 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
12864, 127oveq12i 6093 . . . . . . . . . 10  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
129128, 52eqtri 2456 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  4
130 6nn 10137 . . . . . . . . . . . 12  |-  6  e.  NN
131130nncni 10010 . . . . . . . . . . 11  |-  6  e.  CC
132 6t2e12 10459 . . . . . . . . . . 11  |-  ( 6  x.  2 )  = ; 1
2
133131, 63, 132mulcomli 9097 . . . . . . . . . 10  |-  ( 2  x.  6 )  = ; 1
2
134 3p2e5 10111 . . . . . . . . . . 11  |-  ( 3  +  2 )  =  5
13549, 63, 134addcomli 9258 . . . . . . . . . 10  |-  ( 2  +  3 )  =  5
13613, 10, 11, 133, 135decaddi 10426 . . . . . . . . 9  |-  ( ( 2  x.  6 )  +  3 )  = ; 1
5
13713, 28, 13, 11, 125, 126, 10, 35, 13, 129, 136decma2c 10422 . . . . . . . 8  |-  ( ( 2  x. ; 1 6 )  +  (; 1 2  +  1 ) )  = ; 4 5
13813, 10, 66, 133decsuc 10405 . . . . . . . 8  |-  ( ( 2  x.  6 )  +  1 )  = ; 1
3
13929, 28, 39, 13, 124, 103, 10, 11, 13, 137, 138decma2c 10422 . . . . . . 7  |-  ( ( 2  x. ;; 1 6 6 )  + ;; 1 2 1 )  = ;; 4 5 3
14010, 40, 38, 123, 139gcdi 13409 . . . . . 6  |-  (;; 4 5 3  gcd ;; 1 6 6 )  =  1
141 eqid 2436 . . . . . . 7  |- ;; 4 5 3  = ;; 4 5 3
14229nn0cni 10233 . . . . . . . . 9  |- ; 1 6  e.  CC
143142addid1i 9253 . . . . . . . 8  |-  (; 1 6  +  0 )  = ; 1 6
14448mulid2i 9093 . . . . . . . . . 10  |-  ( 1  x.  4 )  =  4
145144, 127oveq12i 6093 . . . . . . . . 9  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  ( 4  +  2 )
146145, 113eqtri 2456 . . . . . . . 8  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  6
14795mulid2i 9093 . . . . . . . . . 10  |-  ( 1  x.  5 )  =  5
148147oveq1i 6091 . . . . . . . . 9  |-  ( ( 1  x.  5 )  +  6 )  =  ( 5  +  6 )
149 6p5e11 10432 . . . . . . . . . 10  |-  ( 6  +  5 )  = ; 1
1
150131, 95, 149addcomli 9258 . . . . . . . . 9  |-  ( 5  +  6 )  = ; 1
1
151148, 150eqtri 2456 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  6 )  = ; 1
1
15217, 35, 13, 28, 90, 143, 13, 13, 13, 146, 151decma2c 10422 . . . . . . 7  |-  ( ( 1  x. ; 4 5 )  +  (; 1 6  +  0 ) )  = ; 6 1
15376oveq1i 6091 . . . . . . . 8  |-  ( ( 1  x.  3 )  +  6 )  =  ( 3  +  6 )
154 6p3e9 10121 . . . . . . . . 9  |-  ( 6  +  3 )  =  9
155131, 49, 154addcomli 9258 . . . . . . . 8  |-  ( 3  +  6 )  =  9
15630dec0h 10398 . . . . . . . 8  |-  9  = ; 0 9
157153, 155, 1563eqtri 2460 . . . . . . 7  |-  ( ( 1  x.  3 )  +  6 )  = ; 0
9
15836, 11, 29, 28, 141, 124, 13, 30, 3, 152, 157decma2c 10422 . . . . . 6  |-  ( ( 1  x. ;; 4 5 3 )  + ;; 1 6 6 )  = ;; 6 1 9
15913, 38, 37, 140, 158gcdi 13409 . . . . 5  |-  (;; 6 1 9  gcd ;; 4 5 3 )  =  1
160 eqid 2436 . . . . . 6  |- ;; 6 1 9  = ;; 6 1 9
161 7nn0 10243 . . . . . . 7  |-  7  e.  NN0
162 eqid 2436 . . . . . . 7  |- ; 6 1  = ; 6 1
163 5p2e7 10116 . . . . . . . 8  |-  ( 5  +  2 )  =  7
16417, 35, 10, 90, 163decaddi 10426 . . . . . . 7  |-  (; 4 5  +  2 )  = ; 4 7
165105oveq2i 6092 . . . . . . . 8  |-  ( ( 2  x.  6 )  +  ( 4  +  0 ) )  =  ( ( 2  x.  6 )  +  4 )
16613, 10, 17, 133, 114decaddi 10426 . . . . . . . 8  |-  ( ( 2  x.  6 )  +  4 )  = ; 1
6
167165, 166eqtri 2456 . . . . . . 7  |-  ( ( 2  x.  6 )  +  ( 4  +  0 ) )  = ; 1
6
16864oveq1i 6091 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  7 )  =  ( 2  +  7 )
169 7nn 10138 . . . . . . . . . 10  |-  7  e.  NN
170169nncni 10010 . . . . . . . . 9  |-  7  e.  CC
171 7p2e9 10123 . . . . . . . . 9  |-  ( 7  +  2 )  =  9
172170, 63, 171addcomli 9258 . . . . . . . 8  |-  ( 2  +  7 )  =  9
173168, 172, 1563eqtri 2460 . . . . . . 7  |-  ( ( 2  x.  1 )  +  7 )  = ; 0
9
17428, 13, 17, 161, 162, 164, 10, 30, 3, 167, 173decma2c 10422 . . . . . 6  |-  ( ( 2  x. ; 6 1 )  +  (; 4 5  +  2 ) )  = ;; 1 6 9
175 9nn 10140 . . . . . . . . 9  |-  9  e.  NN
176175nncni 10010 . . . . . . . 8  |-  9  e.  CC
177 9t2e18 10477 . . . . . . . 8  |-  ( 9  x.  2 )  = ; 1
8
178176, 63, 177mulcomli 9097 . . . . . . 7  |-  ( 2  x.  9 )  = ; 1
8
17913, 2, 11, 178, 127, 13, 71decaddci 10427 . . . . . 6  |-  ( ( 2  x.  9 )  +  3 )  = ; 2
1
18033, 30, 36, 11, 160, 141, 10, 13, 10, 174, 179decma2c 10422 . . . . 5  |-  ( ( 2  x. ;; 6 1 9 )  + ;; 4 5 3 )  = ;;; 1 6 9 1
18110, 37, 34, 159, 180gcdi 13409 . . . 4  |-  (;;; 1 6 9 1  gcd ;; 6 1 9 )  =  1
182 eqid 2436 . . . . 5  |- ;;; 1 6 9 1  = ;;; 1 6 9 1
183 eqid 2436 . . . . . 6  |- ;; 1 6 9  = ;; 1 6 9
18428, 13, 127, 162decsuc 10405 . . . . . 6  |-  (; 6 1  +  1 )  = ; 6 2
185 6p1e7 10107 . . . . . . . 8  |-  ( 6  +  1 )  =  7
186161dec0h 10398 . . . . . . . 8  |-  7  = ; 0 7
187185, 186eqtri 2456 . . . . . . 7  |-  ( 6  +  1 )  = ; 0
7
18882, 24oveq12i 6093 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  ( 1  +  1 )
189188, 127eqtri 2456 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  2
190131mulid2i 9093 . . . . . . . . 9  |-  ( 1  x.  6 )  =  6
191190oveq1i 6091 . . . . . . . 8  |-  ( ( 1  x.  6 )  +  7 )  =  ( 6  +  7 )
192 7p6e13 10436 . . . . . . . . 9  |-  ( 7  +  6 )  = ; 1
3
193170, 131, 192addcomli 9258 . . . . . . . 8  |-  ( 6  +  7 )  = ; 1
3
194191, 193eqtri 2456 . . . . . . 7  |-  ( ( 1  x.  6 )  +  7 )  = ; 1
3
19513, 28, 3, 161, 125, 187, 13, 11, 13, 189, 194decma2c 10422 . . . . . 6  |-  ( ( 1  x. ; 1 6 )  +  ( 6  +  1 ) )  = ; 2 3
196176mulid2i 9093 . . . . . . . 8  |-  ( 1  x.  9 )  =  9
197196oveq1i 6091 . . . . . . 7  |-  ( ( 1  x.  9 )  +  2 )  =  ( 9  +  2 )
198 9p2e11 10444 . . . . . . 7  |-  ( 9  +  2 )  = ; 1
1
199197, 198eqtri 2456 . . . . . 6  |-  ( ( 1  x.  9 )  +  2 )  = ; 1
1
20029, 30, 28, 10, 183, 184, 13, 13, 13, 195, 199decma2c 10422 . . . . 5  |-  ( ( 1  x. ;; 1 6 9 )  +  (; 6 1  +  1 ) )  = ;; 2 3 1
20182oveq1i 6091 . . . . . 6  |-  ( ( 1  x.  1 )  +  9 )  =  ( 1  +  9 )
202 9p1e10 10110 . . . . . . 7  |-  ( 9  +  1 )  =  10
203176, 77, 202addcomli 9258 . . . . . 6  |-  ( 1  +  9 )  =  10
204201, 203, 983eqtri 2460 . . . . 5  |-  ( ( 1  x.  1 )  +  9 )  = ; 1
0
20531, 13, 33, 30, 182, 160, 13, 3, 13, 200, 204decma2c 10422 . . . 4  |-  ( ( 1  x. ;;; 1 6 9 1 )  + ;; 6 1 9 )  = ;;; 2 3 1 0
20613, 34, 32, 181, 205gcdi 13409 . . 3  |-  (;;; 2 3 1 0  gcd ;;; 1 6 9 1 )  =  1
207 eqid 2436 . . . . . 6  |- ;; 2 3 1  = ;; 2 3 1
20831nn0cni 10233 . . . . . . 7  |- ;; 1 6 9  e.  CC
209208addid1i 9253 . . . . . 6  |-  (;; 1 6 9  +  0 )  = ;; 1 6 9
210 eqid 2436 . . . . . . 7  |- ; 2 3  = ; 2 3
21113, 28, 185, 125decsuc 10405 . . . . . . 7  |-  (; 1 6  +  1 )  = ; 1 7
212111, 127oveq12i 6093 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
213212, 52eqtri 2456 . . . . . . 7  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  4
21476oveq1i 6091 . . . . . . . 8  |-  ( ( 1  x.  3 )  +  7 )  =  ( 3  +  7 )
215 7p3e10 10124 . . . . . . . . 9  |-  ( 7  +  3 )  =  10
216170, 49, 215addcomli 9258 . . . . . . . 8  |-  ( 3  +  7 )  =  10
217214, 216, 983eqtri 2460 . . . . . . 7  |-  ( ( 1  x.  3 )  +  7 )  = ; 1
0
21810, 11, 13, 161, 210, 211, 13, 3, 13, 213, 217decma2c 10422 . . . . . 6  |-  ( ( 1  x. ; 2 3 )  +  (; 1 6  +  1 ) )  = ; 4 0
21912, 13, 29, 30, 207, 209, 13, 3, 13, 218, 204decma2c 10422 . . . . 5  |-  ( ( 1  x. ;; 2 3 1 )  +  (;; 1 6 9  +  0 ) )  = ;; 4 0 0
22077mul01i 9256 . . . . . . 7  |-  ( 1  x.  0 )  =  0
221220oveq1i 6091 . . . . . 6  |-  ( ( 1  x.  0 )  +  1 )  =  ( 0  +  1 )
22213dec0h 10398 . . . . . 6  |-  1  = ; 0 1
223221, 24, 2223eqtri 2460 . . . . 5  |-  ( ( 1  x.  0 )  +  1 )  = ; 0
1
22414, 3, 31, 13, 25, 182, 13, 13, 3, 219, 223decma2c 10422 . . . 4  |-  ( ( 1  x. ;;; 2 3 1 0 )  + ;;; 1 6 9 1 )  = ;;; 4 0 0 1
225224, 16eqtr4i 2459 . . 3  |-  ( ( 1  x. ;;; 2 3 1 0 )  + ;;; 1 6 9 1 )  =  N
22613, 32, 15, 206, 225gcdi 13409 . 2  |-  ( N  gcd ;;; 2 3 1 0 )  =  1
2279, 15, 22, 27, 226gcdmodi 13410 1  |-  ( ( ( 2 ^;; 8 0 0 )  - 
1 )  gcd  N
)  =  1
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1652    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   NNcn 10000   2c2 10049   3c3 10050   4c4 10051   5c5 10052   6c6 10053   7c7 10054   8c8 10055   9c9 10056   10c10 10057   NN0cn0 10221   ZZcz 10282  ;cdc 10382   ^cexp 11382    || cdivides 12852    gcd cgcd 13006   Primecprime 13079
This theorem is referenced by:  4001prm  13464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-rp 10613  df-fz 11044  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080
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