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Theorem 2503lem3 13137
Description: Lemma for 2503prm 13138. Calculate the GCD of  2 ^ 1 8  -  1  ==  1 8 3 1 with  N  =  2 5 0 3. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
Hypothesis
Ref Expression
2503prm.1  |-  N  = ;;; 2 5 0 3
Assertion
Ref Expression
2503lem3  |-  ( ( ( 2 ^; 1 8 )  - 
1 )  gcd  N
)  =  1

Proof of Theorem 2503lem3
StepHypRef Expression
1 2nn 9877 . . . 4  |-  2  e.  NN
2 1nn0 9981 . . . . 5  |-  1  e.  NN0
3 8nn0 9988 . . . . 5  |-  8  e.  NN0
42, 3deccl 10138 . . . 4  |- ; 1 8  e.  NN0
5 nnexpcl 11116 . . . 4  |-  ( ( 2  e.  NN  /\ ; 1 8  e.  NN0 )  -> 
( 2 ^; 1 8 )  e.  NN )
61, 4, 5mp2an 653 . . 3  |-  ( 2 ^; 1 8 )  e.  NN
7 nnm1nn0 10005 . . 3  |-  ( ( 2 ^; 1 8 )  e.  NN  ->  ( (
2 ^; 1 8 )  - 
1 )  e.  NN0 )
86, 7ax-mp 8 . 2  |-  ( ( 2 ^; 1 8 )  - 
1 )  e.  NN0
9 3nn0 9983 . . . 4  |-  3  e.  NN0
104, 9deccl 10138 . . 3  |- ;; 1 8 3  e.  NN0
1110, 2deccl 10138 . 2  |- ;;; 1 8 3 1  e.  NN0
12 2503prm.1 . . 3  |-  N  = ;;; 2 5 0 3
13 2nn0 9982 . . . . . 6  |-  2  e.  NN0
14 5nn0 9985 . . . . . 6  |-  5  e.  NN0
1513, 14deccl 10138 . . . . 5  |- ; 2 5  e.  NN0
16 0nn0 9980 . . . . 5  |-  0  e.  NN0
1715, 16deccl 10138 . . . 4  |- ;; 2 5 0  e.  NN0
18 3nn 9878 . . . 4  |-  3  e.  NN
1917, 18decnncl 10137 . . 3  |- ;;; 2 5 0 3  e.  NN
2012, 19eqeltri 2353 . 2  |-  N  e.  NN
21122503lem1 13135 . . 3  |-  ( ( 2 ^; 1 8 )  mod 
N )  =  (;;; 1 8 3 2  mod 
N )
22 1p1e2 9840 . . . 4  |-  ( 1  +  1 )  =  2
23 eqid 2283 . . . 4  |- ;;; 1 8 3 1  = ;;; 1 8 3 1
2410, 2, 22, 23decsuc 10147 . . 3  |-  (;;; 1 8 3 1  +  1 )  = ;;; 1 8 3 2
2520, 6, 2, 11, 21, 24modsubi 13087 . 2  |-  ( ( ( 2 ^; 1 8 )  - 
1 )  mod  N
)  =  (;;; 1 8 3 1  mod  N )
26 6nn0 9986 . . . . 5  |-  6  e.  NN0
27 7nn0 9987 . . . . 5  |-  7  e.  NN0
2826, 27deccl 10138 . . . 4  |- ; 6 7  e.  NN0
2928, 13deccl 10138 . . 3  |- ;; 6 7 2  e.  NN0
30 4nn0 9984 . . . . . 6  |-  4  e.  NN0
3130, 3deccl 10138 . . . . 5  |- ; 4 8  e.  NN0
3231, 27deccl 10138 . . . 4  |- ;; 4 8 7  e.  NN0
334, 14deccl 10138 . . . . 5  |- ;; 1 8 5  e.  NN0
342, 2deccl 10138 . . . . . . 7  |- ; 1 1  e.  NN0
3534, 27deccl 10138 . . . . . 6  |- ;; 1 1 7  e.  NN0
3626, 3deccl 10138 . . . . . . 7  |- ; 6 8  e.  NN0
37 9nn0 9989 . . . . . . . . 9  |-  9  e.  NN0
3830, 37deccl 10138 . . . . . . . 8  |- ; 4 9  e.  NN0
392, 37deccl 10138 . . . . . . . . 9  |- ; 1 9  e.  NN0
4038nn0zi 10048 . . . . . . . . . . 11  |- ; 4 9  e.  ZZ
4139nn0zi 10048 . . . . . . . . . . 11  |- ; 1 9  e.  ZZ
42 gcdcom 12699 . . . . . . . . . . 11  |-  ( (; 4
9  e.  ZZ  /\ ; 1 9  e.  ZZ )  -> 
(; 4 9  gcd ; 1 9 )  =  (; 1 9  gcd ; 4 9 ) )
4340, 41, 42mp2an 653 . . . . . . . . . 10  |-  (; 4 9  gcd ; 1 9 )  =  (; 1 9  gcd ; 4 9 )
44 9nn 9884 . . . . . . . . . . . . 13  |-  9  e.  NN
452, 44decnncl 10137 . . . . . . . . . . . 12  |- ; 1 9  e.  NN
46 1nn 9757 . . . . . . . . . . . . 13  |-  1  e.  NN
472, 46decnncl 10137 . . . . . . . . . . . 12  |- ; 1 1  e.  NN
48 eqid 2283 . . . . . . . . . . . . 13  |- ; 1 9  = ; 1 9
49 eqid 2283 . . . . . . . . . . . . 13  |- ; 1 1  = ; 1 1
50 2cn 9816 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
5150mulid2i 8840 . . . . . . . . . . . . . . 15  |-  ( 1  x.  2 )  =  2
5251, 22oveq12i 5870 . . . . . . . . . . . . . 14  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
53 2p2e4 9842 . . . . . . . . . . . . . 14  |-  ( 2  +  2 )  =  4
5452, 53eqtri 2303 . . . . . . . . . . . . 13  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  4
55 8p1e9 9853 . . . . . . . . . . . . . 14  |-  ( 8  +  1 )  =  9
56 9t2e18 10219 . . . . . . . . . . . . . 14  |-  ( 9  x.  2 )  = ; 1
8
572, 3, 55, 56decsuc 10147 . . . . . . . . . . . . 13  |-  ( ( 9  x.  2 )  +  1 )  = ; 1
9
582, 37, 2, 2, 48, 49, 13, 37, 2, 54, 57decmac 10163 . . . . . . . . . . . 12  |-  ( (; 1
9  x.  2 )  + ; 1 1 )  = ; 4
9
59 1lt9 9921 . . . . . . . . . . . . 13  |-  1  <  9
602, 2, 44, 59declt 10145 . . . . . . . . . . . 12  |- ; 1 1  < ; 1 9
6145, 13, 47, 58, 60ndvdsi 12609 . . . . . . . . . . 11  |-  -. ; 1 9  || ; 4 9
62 19prm 13119 . . . . . . . . . . . 12  |- ; 1 9  e.  Prime
63 coprm 12779 . . . . . . . . . . . 12  |-  ( (; 1
9  e.  Prime  /\ ; 4 9  e.  ZZ )  ->  ( -. ; 1 9  || ; 4 9  <->  (; 1 9  gcd ; 4 9 )  =  1 ) )
6462, 40, 63mp2an 653 . . . . . . . . . . 11  |-  ( -. ; 1
9  || ; 4 9  <->  (; 1 9  gcd ; 4 9 )  =  1 )
6561, 64mpbi 199 . . . . . . . . . 10  |-  (; 1 9  gcd ; 4 9 )  =  1
6643, 65eqtri 2303 . . . . . . . . 9  |-  (; 4 9  gcd ; 1 9 )  =  1
67 eqid 2283 . . . . . . . . . 10  |- ; 4 9  = ; 4 9
68 4cn 9820 . . . . . . . . . . . . 13  |-  4  e.  CC
6968mulid2i 8840 . . . . . . . . . . . 12  |-  ( 1  x.  4 )  =  4
7069, 22oveq12i 5870 . . . . . . . . . . 11  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  ( 4  +  2 )
71 4p2e6 9857 . . . . . . . . . . 11  |-  ( 4  +  2 )  =  6
7270, 71eqtri 2303 . . . . . . . . . 10  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  6
7344nncni 9756 . . . . . . . . . . . . 13  |-  9  e.  CC
7473mulid2i 8840 . . . . . . . . . . . 12  |-  ( 1  x.  9 )  =  9
7574oveq1i 5868 . . . . . . . . . . 11  |-  ( ( 1  x.  9 )  +  9 )  =  ( 9  +  9 )
76 9p9e18 10193 . . . . . . . . . . 11  |-  ( 9  +  9 )  = ; 1
8
7775, 76eqtri 2303 . . . . . . . . . 10  |-  ( ( 1  x.  9 )  +  9 )  = ; 1
8
7830, 37, 2, 37, 67, 48, 2, 3, 2, 72, 77decma2c 10164 . . . . . . . . 9  |-  ( ( 1  x. ; 4 9 )  + ; 1
9 )  = ; 6 8
792, 39, 38, 66, 78gcdi 13088 . . . . . . . 8  |-  (; 6 8  gcd ; 4 9 )  =  1
80 eqid 2283 . . . . . . . . 9  |- ; 6 8  = ; 6 8
81 6nn 9881 . . . . . . . . . . . . 13  |-  6  e.  NN
8281nncni 9756 . . . . . . . . . . . 12  |-  6  e.  CC
8382mulid2i 8840 . . . . . . . . . . 11  |-  ( 1  x.  6 )  =  6
84 4p1e5 9849 . . . . . . . . . . 11  |-  ( 4  +  1 )  =  5
8583, 84oveq12i 5870 . . . . . . . . . 10  |-  ( ( 1  x.  6 )  +  ( 4  +  1 ) )  =  ( 6  +  5 )
86 6p5e11 10174 . . . . . . . . . 10  |-  ( 6  +  5 )  = ; 1
1
8785, 86eqtri 2303 . . . . . . . . 9  |-  ( ( 1  x.  6 )  +  ( 4  +  1 ) )  = ; 1
1
88 8nn 9883 . . . . . . . . . . . . 13  |-  8  e.  NN
8988nncni 9756 . . . . . . . . . . . 12  |-  8  e.  CC
9089mulid2i 8840 . . . . . . . . . . 11  |-  ( 1  x.  8 )  =  8
9190oveq1i 5868 . . . . . . . . . 10  |-  ( ( 1  x.  8 )  +  9 )  =  ( 8  +  9 )
92 9p8e17 10192 . . . . . . . . . . 11  |-  ( 9  +  8 )  = ; 1
7
9373, 89, 92addcomli 9004 . . . . . . . . . 10  |-  ( 8  +  9 )  = ; 1
7
9491, 93eqtri 2303 . . . . . . . . 9  |-  ( ( 1  x.  8 )  +  9 )  = ; 1
7
9526, 3, 30, 37, 80, 67, 2, 27, 2, 87, 94decma2c 10164 . . . . . . . 8  |-  ( ( 1  x. ; 6 8 )  + ; 4
9 )  = ;; 1 1 7
962, 38, 36, 79, 95gcdi 13088 . . . . . . 7  |-  (;; 1 1 7  gcd ; 6 8 )  =  1
97 eqid 2283 . . . . . . . 8  |- ;; 1 1 7  = ;; 1 1 7
98 6p1e7 9851 . . . . . . . . . 10  |-  ( 6  +  1 )  =  7
9927dec0h 10140 . . . . . . . . . 10  |-  7  = ; 0 7
10098, 99eqtri 2303 . . . . . . . . 9  |-  ( 6  +  1 )  = ; 0
7
101 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
102101mulid1i 8839 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
103 00id 8987 . . . . . . . . . . 11  |-  ( 0  +  0 )  =  0
104102, 103oveq12i 5870 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
105101addid1i 8999 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
106104, 105eqtri 2303 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
107102oveq1i 5868 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  7 )  =  ( 1  +  7 )
108 7nn 9882 . . . . . . . . . . . 12  |-  7  e.  NN
109108nncni 9756 . . . . . . . . . . 11  |-  7  e.  CC
110 7p1e8 9852 . . . . . . . . . . 11  |-  ( 7  +  1 )  =  8
111109, 101, 110addcomli 9004 . . . . . . . . . 10  |-  ( 1  +  7 )  =  8
1123dec0h 10140 . . . . . . . . . 10  |-  8  = ; 0 8
113107, 111, 1123eqtri 2307 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  7 )  = ; 0
8
1142, 2, 16, 27, 49, 100, 2, 3, 16, 106, 113decma2c 10164 . . . . . . . 8  |-  ( ( 1  x. ; 1 1 )  +  ( 6  +  1 ) )  = ; 1 8
115109mulid2i 8840 . . . . . . . . . 10  |-  ( 1  x.  7 )  =  7
116115oveq1i 5868 . . . . . . . . 9  |-  ( ( 1  x.  7 )  +  8 )  =  ( 7  +  8 )
117 8p7e15 10184 . . . . . . . . . 10  |-  ( 8  +  7 )  = ; 1
5
11889, 109, 117addcomli 9004 . . . . . . . . 9  |-  ( 7  +  8 )  = ; 1
5
119116, 118eqtri 2303 . . . . . . . 8  |-  ( ( 1  x.  7 )  +  8 )  = ; 1
5
12034, 27, 26, 3, 97, 80, 2, 14, 2, 114, 119decma2c 10164 . . . . . . 7  |-  ( ( 1  x. ;; 1 1 7 )  + ; 6
8 )  = ;; 1 8 5
1212, 36, 35, 96, 120gcdi 13088 . . . . . 6  |-  (;; 1 8 5  gcd ;; 1 1 7 )  =  1
122 eqid 2283 . . . . . . 7  |- ;; 1 8 5  = ;; 1 8 5
123 eqid 2283 . . . . . . . 8  |- ; 1 8  = ; 1 8
1242, 2, 22, 49decsuc 10147 . . . . . . . 8  |-  (; 1 1  +  1 )  = ; 1 2
12550mulid1i 8839 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
126125, 22oveq12i 5870 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
127126, 53eqtri 2303 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  4
128 8t2e16 10212 . . . . . . . . . 10  |-  ( 8  x.  2 )  = ; 1
6
12989, 50, 128mulcomli 8844 . . . . . . . . 9  |-  ( 2  x.  8 )  = ; 1
6
130 6p2e8 9864 . . . . . . . . 9  |-  ( 6  +  2 )  =  8
1312, 26, 13, 129, 130decaddi 10168 . . . . . . . 8  |-  ( ( 2  x.  8 )  +  2 )  = ; 1
8
1322, 3, 2, 13, 123, 124, 13, 3, 2, 127, 131decma2c 10164 . . . . . . 7  |-  ( ( 2  x. ; 1 8 )  +  (; 1 1  +  1 ) )  = ; 4 8
133 5nn 9880 . . . . . . . . . . 11  |-  5  e.  NN
134133nncni 9756 . . . . . . . . . 10  |-  5  e.  CC
135 5t2e10 9875 . . . . . . . . . 10  |-  ( 5  x.  2 )  =  10
136134, 50, 135mulcomli 8844 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
137 dec10 10154 . . . . . . . . 9  |-  10  = ; 1 0
138136, 137eqtri 2303 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
139109addid2i 9000 . . . . . . . 8  |-  ( 0  +  7 )  =  7
1402, 16, 27, 138, 139decaddi 10168 . . . . . . 7  |-  ( ( 2  x.  5 )  +  7 )  = ; 1
7
1414, 14, 34, 27, 122, 97, 13, 27, 2, 132, 140decma2c 10164 . . . . . 6  |-  ( ( 2  x. ;; 1 8 5 )  + ;; 1 1 7 )  = ;; 4 8 7
14213, 35, 33, 121, 141gcdi 13088 . . . . 5  |-  (;; 4 8 7  gcd ;; 1 8 5 )  =  1
143 eqid 2283 . . . . . 6  |- ;; 4 8 7  = ;; 4 8 7
144 eqid 2283 . . . . . . 7  |- ; 4 8  = ; 4 8
1452, 3, 55, 123decsuc 10147 . . . . . . 7  |-  (; 1 8  +  1 )  = ; 1 9
14630, 3, 2, 37, 144, 145, 2, 27, 2, 72, 94decma2c 10164 . . . . . 6  |-  ( ( 1  x. ; 4 8 )  +  (; 1 8  +  1 ) )  = ; 6 7
147115oveq1i 5868 . . . . . . 7  |-  ( ( 1  x.  7 )  +  5 )  =  ( 7  +  5 )
148 7p5e12 10177 . . . . . . 7  |-  ( 7  +  5 )  = ; 1
2
149147, 148eqtri 2303 . . . . . 6  |-  ( ( 1  x.  7 )  +  5 )  = ; 1
2
15031, 27, 4, 14, 143, 122, 2, 13, 2, 146, 149decma2c 10164 . . . . 5  |-  ( ( 1  x. ;; 4 8 7 )  + ;; 1 8 5 )  = ;; 6 7 2
1512, 33, 32, 142, 150gcdi 13088 . . . 4  |-  (;; 6 7 2  gcd ;; 4 8 7 )  =  1
152 eqid 2283 . . . . 5  |- ;; 6 7 2  = ;; 6 7 2
153 eqid 2283 . . . . . 6  |- ; 6 7  = ; 6 7
15430, 3, 55, 144decsuc 10147 . . . . . 6  |-  (; 4 8  +  1 )  = ; 4 9
15571oveq2i 5869 . . . . . . 7  |-  ( ( 2  x.  6 )  +  ( 4  +  2 ) )  =  ( ( 2  x.  6 )  +  6 )
156 6t2e12 10201 . . . . . . . . 9  |-  ( 6  x.  2 )  = ; 1
2
15782, 50, 156mulcomli 8844 . . . . . . . 8  |-  ( 2  x.  6 )  = ; 1
2
15882, 50, 130addcomli 9004 . . . . . . . 8  |-  ( 2  +  6 )  =  8
1592, 13, 26, 157, 158decaddi 10168 . . . . . . 7  |-  ( ( 2  x.  6 )  +  6 )  = ; 1
8
160155, 159eqtri 2303 . . . . . 6  |-  ( ( 2  x.  6 )  +  ( 4  +  2 ) )  = ; 1
8
161 7t2e14 10206 . . . . . . . 8  |-  ( 7  x.  2 )  = ; 1
4
162109, 50, 161mulcomli 8844 . . . . . . 7  |-  ( 2  x.  7 )  = ; 1
4
163 9p4e13 10188 . . . . . . . 8  |-  ( 9  +  4 )  = ; 1
3
16473, 68, 163addcomli 9004 . . . . . . 7  |-  ( 4  +  9 )  = ; 1
3
1652, 30, 37, 162, 22, 9, 164decaddci 10169 . . . . . 6  |-  ( ( 2  x.  7 )  +  9 )  = ; 2
3
16626, 27, 30, 37, 153, 154, 13, 9, 13, 160, 165decma2c 10164 . . . . 5  |-  ( ( 2  x. ; 6 7 )  +  (; 4 8  +  1 ) )  = ;; 1 8 3
167 2t2e4 9871 . . . . . . 7  |-  ( 2  x.  2 )  =  4
168167oveq1i 5868 . . . . . 6  |-  ( ( 2  x.  2 )  +  7 )  =  ( 4  +  7 )
169 7p4e11 10176 . . . . . . 7  |-  ( 7  +  4 )  = ; 1
1
170109, 68, 169addcomli 9004 . . . . . 6  |-  ( 4  +  7 )  = ; 1
1
171168, 170eqtri 2303 . . . . 5  |-  ( ( 2  x.  2 )  +  7 )  = ; 1
1
17228, 13, 31, 27, 152, 143, 13, 2, 2, 166, 171decma2c 10164 . . . 4  |-  ( ( 2  x. ;; 6 7 2 )  + ;; 4 8 7 )  = ;;; 1 8 3 1
17313, 32, 29, 151, 172gcdi 13088 . . 3  |-  (;;; 1 8 3 1  gcd ;; 6 7 2 )  =  1
174 eqid 2283 . . . . . 6  |- ;; 1 8 3  = ;; 1 8 3
17528nn0cni 9977 . . . . . . 7  |- ; 6 7  e.  CC
176175addid1i 8999 . . . . . 6  |-  (; 6 7  +  0 )  = ; 6 7
177101addid2i 9000 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
178102, 177oveq12i 5870 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  ( 1  +  1 )
179178, 22eqtri 2303 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  2
18090oveq1i 5868 . . . . . . . 8  |-  ( ( 1  x.  8 )  +  7 )  =  ( 8  +  7 )
181180, 117eqtri 2303 . . . . . . 7  |-  ( ( 1  x.  8 )  +  7 )  = ; 1
5
1822, 3, 16, 27, 123, 100, 2, 14, 2, 179, 181decma2c 10164 . . . . . 6  |-  ( ( 1  x. ; 1 8 )  +  ( 6  +  1 ) )  = ; 2 5
183 3cn 9818 . . . . . . . . 9  |-  3  e.  CC
184183mulid2i 8840 . . . . . . . 8  |-  ( 1  x.  3 )  =  3
185184oveq1i 5868 . . . . . . 7  |-  ( ( 1  x.  3 )  +  7 )  =  ( 3  +  7 )
186 7p3e10 9868 . . . . . . . 8  |-  ( 7  +  3 )  =  10
187109, 183, 186addcomli 9004 . . . . . . 7  |-  ( 3  +  7 )  =  10
188185, 187, 1373eqtri 2307 . . . . . 6  |-  ( ( 1  x.  3 )  +  7 )  = ; 1
0
1894, 9, 26, 27, 174, 176, 2, 16, 2, 182, 188decma2c 10164 . . . . 5  |-  ( ( 1  x. ;; 1 8 3 )  +  (; 6 7  +  0 ) )  = ;; 2 5 0
190102oveq1i 5868 . . . . . 6  |-  ( ( 1  x.  1 )  +  2 )  =  ( 1  +  2 )
191 2p1e3 9847 . . . . . . 7  |-  ( 2  +  1 )  =  3
19250, 101, 191addcomli 9004 . . . . . 6  |-  ( 1  +  2 )  =  3
1939dec0h 10140 . . . . . 6  |-  3  = ; 0 3
194190, 192, 1933eqtri 2307 . . . . 5  |-  ( ( 1  x.  1 )  +  2 )  = ; 0
3
19510, 2, 28, 13, 23, 152, 2, 9, 16, 189, 194decma2c 10164 . . . 4  |-  ( ( 1  x. ;;; 1 8 3 1 )  + ;; 6 7 2 )  = ;;; 2 5 0 3
196195, 12eqtr4i 2306 . . 3  |-  ( ( 1  x. ;;; 1 8 3 1 )  + ;; 6 7 2 )  =  N
1972, 29, 11, 173, 196gcdi 13088 . 2  |-  ( N  gcd ;;; 1 8 3 1 )  =  1
1988, 11, 20, 25, 197gcdmodi 13089 1  |-  ( ( ( 2 ^; 1 8 )  - 
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   10c10 9803   NN0cn0 9965   ZZcz 10024  ;cdc 10124   ^cexp 11104    || cdivides 12531    gcd cgcd 12685   Primecprime 12758
This theorem is referenced by:  2503prm  13138
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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