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Theorem 2503lem3 14498
Description: Lemma for 2503prm 14499. 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 10699 . . . 4  |-  2  e.  NN
2 1nn0 10817 . . . . 5  |-  1  e.  NN0
3 8nn0 10824 . . . . 5  |-  8  e.  NN0
42, 3deccl 10998 . . . 4  |- ; 1 8  e.  NN0
5 nnexpcl 12158 . . . 4  |-  ( ( 2  e.  NN  /\ ; 1 8  e.  NN0 )  -> 
( 2 ^; 1 8 )  e.  NN )
61, 4, 5mp2an 672 . . 3  |-  ( 2 ^; 1 8 )  e.  NN
7 nnm1nn0 10843 . . 3  |-  ( ( 2 ^; 1 8 )  e.  NN  ->  ( (
2 ^; 1 8 )  - 
1 )  e.  NN0 )
86, 7ax-mp 5 . 2  |-  ( ( 2 ^; 1 8 )  - 
1 )  e.  NN0
9 3nn0 10819 . . . 4  |-  3  e.  NN0
104, 9deccl 10998 . . 3  |- ;; 1 8 3  e.  NN0
1110, 2deccl 10998 . 2  |- ;;; 1 8 3 1  e.  NN0
12 2503prm.1 . . 3  |-  N  = ;;; 2 5 0 3
13 2nn0 10818 . . . . . 6  |-  2  e.  NN0
14 5nn0 10821 . . . . . 6  |-  5  e.  NN0
1513, 14deccl 10998 . . . . 5  |- ; 2 5  e.  NN0
16 0nn0 10816 . . . . 5  |-  0  e.  NN0
1715, 16deccl 10998 . . . 4  |- ;; 2 5 0  e.  NN0
18 3nn 10700 . . . 4  |-  3  e.  NN
1917, 18decnncl 10997 . . 3  |- ;;; 2 5 0 3  e.  NN
2012, 19eqeltri 2527 . 2  |-  N  e.  NN
21122503lem1 14496 . . 3  |-  ( ( 2 ^; 1 8 )  mod 
N )  =  (;;; 1 8 3 2  mod 
N )
22 1p1e2 10655 . . . 4  |-  ( 1  +  1 )  =  2
23 eqid 2443 . . . 4  |- ;;; 1 8 3 1  = ;;; 1 8 3 1
2410, 2, 22, 23decsuc 11007 . . 3  |-  (;;; 1 8 3 1  +  1 )  = ;;; 1 8 3 2
2520, 6, 2, 11, 21, 24modsubi 14435 . 2  |-  ( ( ( 2 ^; 1 8 )  - 
1 )  mod  N
)  =  (;;; 1 8 3 1  mod  N )
26 6nn0 10822 . . . . 5  |-  6  e.  NN0
27 7nn0 10823 . . . . 5  |-  7  e.  NN0
2826, 27deccl 10998 . . . 4  |- ; 6 7  e.  NN0
2928, 13deccl 10998 . . 3  |- ;; 6 7 2  e.  NN0
30 4nn0 10820 . . . . . 6  |-  4  e.  NN0
3130, 3deccl 10998 . . . . 5  |- ; 4 8  e.  NN0
3231, 27deccl 10998 . . . 4  |- ;; 4 8 7  e.  NN0
334, 14deccl 10998 . . . . 5  |- ;; 1 8 5  e.  NN0
342, 2deccl 10998 . . . . . . 7  |- ; 1 1  e.  NN0
3534, 27deccl 10998 . . . . . 6  |- ;; 1 1 7  e.  NN0
3626, 3deccl 10998 . . . . . . 7  |- ; 6 8  e.  NN0
37 9nn0 10825 . . . . . . . . 9  |-  9  e.  NN0
3830, 37deccl 10998 . . . . . . . 8  |- ; 4 9  e.  NN0
392, 37deccl 10998 . . . . . . . . 9  |- ; 1 9  e.  NN0
4038nn0zi 10895 . . . . . . . . . . 11  |- ; 4 9  e.  ZZ
4139nn0zi 10895 . . . . . . . . . . 11  |- ; 1 9  e.  ZZ
42 gcdcom 14035 . . . . . . . . . . 11  |-  ( (; 4
9  e.  ZZ  /\ ; 1 9  e.  ZZ )  -> 
(; 4 9  gcd ; 1 9 )  =  (; 1 9  gcd ; 4 9 ) )
4340, 41, 42mp2an 672 . . . . . . . . . 10  |-  (; 4 9  gcd ; 1 9 )  =  (; 1 9  gcd ; 4 9 )
44 9nn 10706 . . . . . . . . . . . . 13  |-  9  e.  NN
452, 44decnncl 10997 . . . . . . . . . . . 12  |- ; 1 9  e.  NN
46 1nn 10553 . . . . . . . . . . . . 13  |-  1  e.  NN
472, 46decnncl 10997 . . . . . . . . . . . 12  |- ; 1 1  e.  NN
48 eqid 2443 . . . . . . . . . . . . 13  |- ; 1 9  = ; 1 9
49 eqid 2443 . . . . . . . . . . . . 13  |- ; 1 1  = ; 1 1
50 2cn 10612 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
5150mulid2i 9602 . . . . . . . . . . . . . . 15  |-  ( 1  x.  2 )  =  2
5251, 22oveq12i 6293 . . . . . . . . . . . . . 14  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
53 2p2e4 10659 . . . . . . . . . . . . . 14  |-  ( 2  +  2 )  =  4
5452, 53eqtri 2472 . . . . . . . . . . . . 13  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  4
55 8p1e9 10672 . . . . . . . . . . . . . 14  |-  ( 8  +  1 )  =  9
56 9t2e18 11079 . . . . . . . . . . . . . 14  |-  ( 9  x.  2 )  = ; 1
8
572, 3, 55, 56decsuc 11007 . . . . . . . . . . . . 13  |-  ( ( 9  x.  2 )  +  1 )  = ; 1
9
582, 37, 2, 2, 48, 49, 13, 37, 2, 54, 57decmac 11023 . . . . . . . . . . . 12  |-  ( (; 1
9  x.  2 )  + ; 1 1 )  = ; 4
9
59 1lt9 10743 . . . . . . . . . . . . 13  |-  1  <  9
602, 2, 44, 59declt 11005 . . . . . . . . . . . 12  |- ; 1 1  < ; 1 9
6145, 13, 47, 58, 60ndvdsi 13945 . . . . . . . . . . 11  |-  -. ; 1 9  || ; 4 9
62 19prm 14480 . . . . . . . . . . . 12  |- ; 1 9  e.  Prime
63 coprm 14118 . . . . . . . . . . . 12  |-  ( (; 1
9  e.  Prime  /\ ; 4 9  e.  ZZ )  ->  ( -. ; 1 9  || ; 4 9  <->  (; 1 9  gcd ; 4 9 )  =  1 ) )
6462, 40, 63mp2an 672 . . . . . . . . . . 11  |-  ( -. ; 1
9  || ; 4 9  <->  (; 1 9  gcd ; 4 9 )  =  1 )
6561, 64mpbi 208 . . . . . . . . . 10  |-  (; 1 9  gcd ; 4 9 )  =  1
6643, 65eqtri 2472 . . . . . . . . 9  |-  (; 4 9  gcd ; 1 9 )  =  1
67 eqid 2443 . . . . . . . . . 10  |- ; 4 9  = ; 4 9
68 4cn 10619 . . . . . . . . . . . . 13  |-  4  e.  CC
6968mulid2i 9602 . . . . . . . . . . . 12  |-  ( 1  x.  4 )  =  4
7069, 22oveq12i 6293 . . . . . . . . . . 11  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  ( 4  +  2 )
71 4p2e6 10676 . . . . . . . . . . 11  |-  ( 4  +  2 )  =  6
7270, 71eqtri 2472 . . . . . . . . . 10  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  6
73 9cn 10629 . . . . . . . . . . . . 13  |-  9  e.  CC
7473mulid2i 9602 . . . . . . . . . . . 12  |-  ( 1  x.  9 )  =  9
7574oveq1i 6291 . . . . . . . . . . 11  |-  ( ( 1  x.  9 )  +  9 )  =  ( 9  +  9 )
76 9p9e18 11053 . . . . . . . . . . 11  |-  ( 9  +  9 )  = ; 1
8
7775, 76eqtri 2472 . . . . . . . . . 10  |-  ( ( 1  x.  9 )  +  9 )  = ; 1
8
7830, 37, 2, 37, 67, 48, 2, 3, 2, 72, 77decma2c 11024 . . . . . . . . 9  |-  ( ( 1  x. ; 4 9 )  + ; 1
9 )  = ; 6 8
792, 39, 38, 66, 78gcdi 14436 . . . . . . . 8  |-  (; 6 8  gcd ; 4 9 )  =  1
80 eqid 2443 . . . . . . . . 9  |- ; 6 8  = ; 6 8
81 6cn 10623 . . . . . . . . . . . 12  |-  6  e.  CC
8281mulid2i 9602 . . . . . . . . . . 11  |-  ( 1  x.  6 )  =  6
83 4p1e5 10668 . . . . . . . . . . 11  |-  ( 4  +  1 )  =  5
8482, 83oveq12i 6293 . . . . . . . . . 10  |-  ( ( 1  x.  6 )  +  ( 4  +  1 ) )  =  ( 6  +  5 )
85 6p5e11 11034 . . . . . . . . . 10  |-  ( 6  +  5 )  = ; 1
1
8684, 85eqtri 2472 . . . . . . . . 9  |-  ( ( 1  x.  6 )  +  ( 4  +  1 ) )  = ; 1
1
87 8cn 10627 . . . . . . . . . . . 12  |-  8  e.  CC
8887mulid2i 9602 . . . . . . . . . . 11  |-  ( 1  x.  8 )  =  8
8988oveq1i 6291 . . . . . . . . . 10  |-  ( ( 1  x.  8 )  +  9 )  =  ( 8  +  9 )
90 9p8e17 11052 . . . . . . . . . . 11  |-  ( 9  +  8 )  = ; 1
7
9173, 87, 90addcomli 9775 . . . . . . . . . 10  |-  ( 8  +  9 )  = ; 1
7
9289, 91eqtri 2472 . . . . . . . . 9  |-  ( ( 1  x.  8 )  +  9 )  = ; 1
7
9326, 3, 30, 37, 80, 67, 2, 27, 2, 86, 92decma2c 11024 . . . . . . . 8  |-  ( ( 1  x. ; 6 8 )  + ; 4
9 )  = ;; 1 1 7
942, 38, 36, 79, 93gcdi 14436 . . . . . . 7  |-  (;; 1 1 7  gcd ; 6 8 )  =  1
95 eqid 2443 . . . . . . . 8  |- ;; 1 1 7  = ;; 1 1 7
96 6p1e7 10670 . . . . . . . . . 10  |-  ( 6  +  1 )  =  7
9727dec0h 11000 . . . . . . . . . 10  |-  7  = ; 0 7
9896, 97eqtri 2472 . . . . . . . . 9  |-  ( 6  +  1 )  = ; 0
7
99 ax-1cn 9553 . . . . . . . . . . . 12  |-  1  e.  CC
10099mulid1i 9601 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
101 00id 9758 . . . . . . . . . . 11  |-  ( 0  +  0 )  =  0
102100, 101oveq12i 6293 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
10399addid1i 9770 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
104102, 103eqtri 2472 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
105100oveq1i 6291 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  7 )  =  ( 1  +  7 )
106 7cn 10625 . . . . . . . . . . 11  |-  7  e.  CC
107 7p1e8 10671 . . . . . . . . . . 11  |-  ( 7  +  1 )  =  8
108106, 99, 107addcomli 9775 . . . . . . . . . 10  |-  ( 1  +  7 )  =  8
1093dec0h 11000 . . . . . . . . . 10  |-  8  = ; 0 8
110105, 108, 1093eqtri 2476 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  7 )  = ; 0
8
1112, 2, 16, 27, 49, 98, 2, 3, 16, 104, 110decma2c 11024 . . . . . . . 8  |-  ( ( 1  x. ; 1 1 )  +  ( 6  +  1 ) )  = ; 1 8
112106mulid2i 9602 . . . . . . . . . 10  |-  ( 1  x.  7 )  =  7
113112oveq1i 6291 . . . . . . . . 9  |-  ( ( 1  x.  7 )  +  8 )  =  ( 7  +  8 )
114 8p7e15 11044 . . . . . . . . . 10  |-  ( 8  +  7 )  = ; 1
5
11587, 106, 114addcomli 9775 . . . . . . . . 9  |-  ( 7  +  8 )  = ; 1
5
116113, 115eqtri 2472 . . . . . . . 8  |-  ( ( 1  x.  7 )  +  8 )  = ; 1
5
11734, 27, 26, 3, 95, 80, 2, 14, 2, 111, 116decma2c 11024 . . . . . . 7  |-  ( ( 1  x. ;; 1 1 7 )  + ; 6
8 )  = ;; 1 8 5
1182, 36, 35, 94, 117gcdi 14436 . . . . . 6  |-  (;; 1 8 5  gcd ;; 1 1 7 )  =  1
119 eqid 2443 . . . . . . 7  |- ;; 1 8 5  = ;; 1 8 5
120 eqid 2443 . . . . . . . 8  |- ; 1 8  = ; 1 8
1212, 2, 22, 49decsuc 11007 . . . . . . . 8  |-  (; 1 1  +  1 )  = ; 1 2
12250mulid1i 9601 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
123122, 22oveq12i 6293 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
124123, 53eqtri 2472 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  4
125 8t2e16 11072 . . . . . . . . . 10  |-  ( 8  x.  2 )  = ; 1
6
12687, 50, 125mulcomli 9606 . . . . . . . . 9  |-  ( 2  x.  8 )  = ; 1
6
127 6p2e8 10683 . . . . . . . . 9  |-  ( 6  +  2 )  =  8
1282, 26, 13, 126, 127decaddi 11028 . . . . . . . 8  |-  ( ( 2  x.  8 )  +  2 )  = ; 1
8
1292, 3, 2, 13, 120, 121, 13, 3, 2, 124, 128decma2c 11024 . . . . . . 7  |-  ( ( 2  x. ; 1 8 )  +  (; 1 1  +  1 ) )  = ; 4 8
130 5cn 10621 . . . . . . . . . 10  |-  5  e.  CC
131 5t2e10 10696 . . . . . . . . . 10  |-  ( 5  x.  2 )  =  10
132130, 50, 131mulcomli 9606 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
133 dec10 11014 . . . . . . . . 9  |-  10  = ; 1 0
134132, 133eqtri 2472 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
135106addid2i 9771 . . . . . . . 8  |-  ( 0  +  7 )  =  7
1362, 16, 27, 134, 135decaddi 11028 . . . . . . 7  |-  ( ( 2  x.  5 )  +  7 )  = ; 1
7
1374, 14, 34, 27, 119, 95, 13, 27, 2, 129, 136decma2c 11024 . . . . . 6  |-  ( ( 2  x. ;; 1 8 5 )  + ;; 1 1 7 )  = ;; 4 8 7
13813, 35, 33, 118, 137gcdi 14436 . . . . 5  |-  (;; 4 8 7  gcd ;; 1 8 5 )  =  1
139 eqid 2443 . . . . . 6  |- ;; 4 8 7  = ;; 4 8 7
140 eqid 2443 . . . . . . 7  |- ; 4 8  = ; 4 8
1412, 3, 55, 120decsuc 11007 . . . . . . 7  |-  (; 1 8  +  1 )  = ; 1 9
14230, 3, 2, 37, 140, 141, 2, 27, 2, 72, 92decma2c 11024 . . . . . 6  |-  ( ( 1  x. ; 4 8 )  +  (; 1 8  +  1 ) )  = ; 6 7
143112oveq1i 6291 . . . . . . 7  |-  ( ( 1  x.  7 )  +  5 )  =  ( 7  +  5 )
144 7p5e12 11037 . . . . . . 7  |-  ( 7  +  5 )  = ; 1
2
145143, 144eqtri 2472 . . . . . 6  |-  ( ( 1  x.  7 )  +  5 )  = ; 1
2
14631, 27, 4, 14, 139, 119, 2, 13, 2, 142, 145decma2c 11024 . . . . 5  |-  ( ( 1  x. ;; 4 8 7 )  + ;; 1 8 5 )  = ;; 6 7 2
1472, 33, 32, 138, 146gcdi 14436 . . . 4  |-  (;; 6 7 2  gcd ;; 4 8 7 )  =  1
148 eqid 2443 . . . . 5  |- ;; 6 7 2  = ;; 6 7 2
149 eqid 2443 . . . . . 6  |- ; 6 7  = ; 6 7
15030, 3, 55, 140decsuc 11007 . . . . . 6  |-  (; 4 8  +  1 )  = ; 4 9
15171oveq2i 6292 . . . . . . 7  |-  ( ( 2  x.  6 )  +  ( 4  +  2 ) )  =  ( ( 2  x.  6 )  +  6 )
152 6t2e12 11061 . . . . . . . . 9  |-  ( 6  x.  2 )  = ; 1
2
15381, 50, 152mulcomli 9606 . . . . . . . 8  |-  ( 2  x.  6 )  = ; 1
2
15481, 50, 127addcomli 9775 . . . . . . . 8  |-  ( 2  +  6 )  =  8
1552, 13, 26, 153, 154decaddi 11028 . . . . . . 7  |-  ( ( 2  x.  6 )  +  6 )  = ; 1
8
156151, 155eqtri 2472 . . . . . 6  |-  ( ( 2  x.  6 )  +  ( 4  +  2 ) )  = ; 1
8
157 7t2e14 11066 . . . . . . . 8  |-  ( 7  x.  2 )  = ; 1
4
158106, 50, 157mulcomli 9606 . . . . . . 7  |-  ( 2  x.  7 )  = ; 1
4
159 9p4e13 11048 . . . . . . . 8  |-  ( 9  +  4 )  = ; 1
3
16073, 68, 159addcomli 9775 . . . . . . 7  |-  ( 4  +  9 )  = ; 1
3
1612, 30, 37, 158, 22, 9, 160decaddci 11029 . . . . . 6  |-  ( ( 2  x.  7 )  +  9 )  = ; 2
3
16226, 27, 30, 37, 149, 150, 13, 9, 13, 156, 161decma2c 11024 . . . . 5  |-  ( ( 2  x. ; 6 7 )  +  (; 4 8  +  1 ) )  = ;; 1 8 3
163 2t2e4 10691 . . . . . . 7  |-  ( 2  x.  2 )  =  4
164163oveq1i 6291 . . . . . 6  |-  ( ( 2  x.  2 )  +  7 )  =  ( 4  +  7 )
165 7p4e11 11036 . . . . . . 7  |-  ( 7  +  4 )  = ; 1
1
166106, 68, 165addcomli 9775 . . . . . 6  |-  ( 4  +  7 )  = ; 1
1
167164, 166eqtri 2472 . . . . 5  |-  ( ( 2  x.  2 )  +  7 )  = ; 1
1
16828, 13, 31, 27, 148, 139, 13, 2, 2, 162, 167decma2c 11024 . . . 4  |-  ( ( 2  x. ;; 6 7 2 )  + ;; 4 8 7 )  = ;;; 1 8 3 1
16913, 32, 29, 147, 168gcdi 14436 . . 3  |-  (;;; 1 8 3 1  gcd ;; 6 7 2 )  =  1
170 eqid 2443 . . . . . 6  |- ;; 1 8 3  = ;; 1 8 3
17128nn0cni 10813 . . . . . . 7  |- ; 6 7  e.  CC
172171addid1i 9770 . . . . . 6  |-  (; 6 7  +  0 )  = ; 6 7
17399addid2i 9771 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
174100, 173oveq12i 6293 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  ( 1  +  1 )
175174, 22eqtri 2472 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  2
17688oveq1i 6291 . . . . . . . 8  |-  ( ( 1  x.  8 )  +  7 )  =  ( 8  +  7 )
177176, 114eqtri 2472 . . . . . . 7  |-  ( ( 1  x.  8 )  +  7 )  = ; 1
5
1782, 3, 16, 27, 120, 98, 2, 14, 2, 175, 177decma2c 11024 . . . . . 6  |-  ( ( 1  x. ; 1 8 )  +  ( 6  +  1 ) )  = ; 2 5
179 3cn 10616 . . . . . . . . 9  |-  3  e.  CC
180179mulid2i 9602 . . . . . . . 8  |-  ( 1  x.  3 )  =  3
181180oveq1i 6291 . . . . . . 7  |-  ( ( 1  x.  3 )  +  7 )  =  ( 3  +  7 )
182 7p3e10 10687 . . . . . . . 8  |-  ( 7  +  3 )  =  10
183106, 179, 182addcomli 9775 . . . . . . 7  |-  ( 3  +  7 )  =  10
184181, 183, 1333eqtri 2476 . . . . . 6  |-  ( ( 1  x.  3 )  +  7 )  = ; 1
0
1854, 9, 26, 27, 170, 172, 2, 16, 2, 178, 184decma2c 11024 . . . . 5  |-  ( ( 1  x. ;; 1 8 3 )  +  (; 6 7  +  0 ) )  = ;; 2 5 0
186100oveq1i 6291 . . . . . 6  |-  ( ( 1  x.  1 )  +  2 )  =  ( 1  +  2 )
187 1p2e3 10666 . . . . . 6  |-  ( 1  +  2 )  =  3
1889dec0h 11000 . . . . . 6  |-  3  = ; 0 3
189186, 187, 1883eqtri 2476 . . . . 5  |-  ( ( 1  x.  1 )  +  2 )  = ; 0
3
19010, 2, 28, 13, 23, 148, 2, 9, 16, 185, 189decma2c 11024 . . . 4  |-  ( ( 1  x. ;;; 1 8 3 1 )  + ;; 6 7 2 )  = ;;; 2 5 0 3
191190, 12eqtr4i 2475 . . 3  |-  ( ( 1  x. ;;; 1 8 3 1 )  + ;; 6 7 2 )  =  N
1922, 29, 11, 169, 191gcdi 14436 . 2  |-  ( N  gcd ;;; 1 8 3 1 )  =  1
1938, 11, 20, 25, 192gcdmodi 14437 1  |-  ( ( ( 2 ^; 1 8 )  - 
1 )  gcd  N
)  =  1
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1383    e. wcel 1804   class class class wbr 4437  (class class class)co 6281   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500    - cmin 9810   NNcn 10542   2c2 10591   3c3 10592   4c4 10593   5c5 10594   6c6 10595   7c7 10596   8c8 10597   9c9 10598   10c10 10599   NN0cn0 10801   ZZcz 10870  ;cdc 10984   ^cexp 12145    || cdvds 13863    gcd cgcd 14021   Primecprime 14094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-rp 11230  df-fz 11682  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-dvds 13864  df-gcd 14022  df-prm 14095
This theorem is referenced by:  2503prm  14499
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