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Theorem 2503lem3 14623
Description: Lemma for 2503prm 14624. 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 10610 . . . 4  |-  2  e.  NN
2 1nn0 10728 . . . . 5  |-  1  e.  NN0
3 8nn0 10735 . . . . 5  |-  8  e.  NN0
42, 3deccl 10909 . . . 4  |- ; 1 8  e.  NN0
5 nnexpcl 12082 . . . 4  |-  ( ( 2  e.  NN  /\ ; 1 8  e.  NN0 )  -> 
( 2 ^; 1 8 )  e.  NN )
61, 4, 5mp2an 670 . . 3  |-  ( 2 ^; 1 8 )  e.  NN
7 nnm1nn0 10754 . . 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 10730 . . . 4  |-  3  e.  NN0
104, 9deccl 10909 . . 3  |- ;; 1 8 3  e.  NN0
1110, 2deccl 10909 . 2  |- ;;; 1 8 3 1  e.  NN0
12 2503prm.1 . . 3  |-  N  = ;;; 2 5 0 3
13 2nn0 10729 . . . . . 6  |-  2  e.  NN0
14 5nn0 10732 . . . . . 6  |-  5  e.  NN0
1513, 14deccl 10909 . . . . 5  |- ; 2 5  e.  NN0
16 0nn0 10727 . . . . 5  |-  0  e.  NN0
1715, 16deccl 10909 . . . 4  |- ;; 2 5 0  e.  NN0
18 3nn 10611 . . . 4  |-  3  e.  NN
1917, 18decnncl 10908 . . 3  |- ;;; 2 5 0 3  e.  NN
2012, 19eqeltri 2466 . 2  |-  N  e.  NN
21122503lem1 14621 . . 3  |-  ( ( 2 ^; 1 8 )  mod 
N )  =  (;;; 1 8 3 2  mod 
N )
22 1p1e2 10566 . . . 4  |-  ( 1  +  1 )  =  2
23 eqid 2382 . . . 4  |- ;;; 1 8 3 1  = ;;; 1 8 3 1
2410, 2, 22, 23decsuc 10918 . . 3  |-  (;;; 1 8 3 1  +  1 )  = ;;; 1 8 3 2
2520, 6, 2, 11, 21, 24modsubi 14560 . 2  |-  ( ( ( 2 ^; 1 8 )  - 
1 )  mod  N
)  =  (;;; 1 8 3 1  mod  N )
26 6nn0 10733 . . . . 5  |-  6  e.  NN0
27 7nn0 10734 . . . . 5  |-  7  e.  NN0
2826, 27deccl 10909 . . . 4  |- ; 6 7  e.  NN0
2928, 13deccl 10909 . . 3  |- ;; 6 7 2  e.  NN0
30 4nn0 10731 . . . . . 6  |-  4  e.  NN0
3130, 3deccl 10909 . . . . 5  |- ; 4 8  e.  NN0
3231, 27deccl 10909 . . . 4  |- ;; 4 8 7  e.  NN0
334, 14deccl 10909 . . . . 5  |- ;; 1 8 5  e.  NN0
342, 2deccl 10909 . . . . . . 7  |- ; 1 1  e.  NN0
3534, 27deccl 10909 . . . . . 6  |- ;; 1 1 7  e.  NN0
3626, 3deccl 10909 . . . . . . 7  |- ; 6 8  e.  NN0
37 9nn0 10736 . . . . . . . . 9  |-  9  e.  NN0
3830, 37deccl 10909 . . . . . . . 8  |- ; 4 9  e.  NN0
392, 37deccl 10909 . . . . . . . . 9  |- ; 1 9  e.  NN0
4038nn0zi 10806 . . . . . . . . . . 11  |- ; 4 9  e.  ZZ
4139nn0zi 10806 . . . . . . . . . . 11  |- ; 1 9  e.  ZZ
42 gcdcom 14160 . . . . . . . . . . 11  |-  ( (; 4
9  e.  ZZ  /\ ; 1 9  e.  ZZ )  -> 
(; 4 9  gcd ; 1 9 )  =  (; 1 9  gcd ; 4 9 ) )
4340, 41, 42mp2an 670 . . . . . . . . . 10  |-  (; 4 9  gcd ; 1 9 )  =  (; 1 9  gcd ; 4 9 )
44 9nn 10617 . . . . . . . . . . . . 13  |-  9  e.  NN
452, 44decnncl 10908 . . . . . . . . . . . 12  |- ; 1 9  e.  NN
46 1nn 10463 . . . . . . . . . . . . 13  |-  1  e.  NN
472, 46decnncl 10908 . . . . . . . . . . . 12  |- ; 1 1  e.  NN
48 eqid 2382 . . . . . . . . . . . . 13  |- ; 1 9  = ; 1 9
49 eqid 2382 . . . . . . . . . . . . 13  |- ; 1 1  = ; 1 1
50 2cn 10523 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
5150mulid2i 9510 . . . . . . . . . . . . . . 15  |-  ( 1  x.  2 )  =  2
5251, 22oveq12i 6208 . . . . . . . . . . . . . 14  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
53 2p2e4 10570 . . . . . . . . . . . . . 14  |-  ( 2  +  2 )  =  4
5452, 53eqtri 2411 . . . . . . . . . . . . 13  |-  ( ( 1  x.  2 )  +  ( 1  +  1 ) )  =  4
55 8p1e9 10583 . . . . . . . . . . . . . 14  |-  ( 8  +  1 )  =  9
56 9t2e18 10990 . . . . . . . . . . . . . 14  |-  ( 9  x.  2 )  = ; 1
8
572, 3, 55, 56decsuc 10918 . . . . . . . . . . . . 13  |-  ( ( 9  x.  2 )  +  1 )  = ; 1
9
582, 37, 2, 2, 48, 49, 13, 37, 2, 54, 57decmac 10934 . . . . . . . . . . . 12  |-  ( (; 1
9  x.  2 )  + ; 1 1 )  = ; 4
9
59 1lt9 10654 . . . . . . . . . . . . 13  |-  1  <  9
602, 2, 44, 59declt 10916 . . . . . . . . . . . 12  |- ; 1 1  < ; 1 9
6145, 13, 47, 58, 60ndvdsi 14070 . . . . . . . . . . 11  |-  -. ; 1 9  || ; 4 9
62 19prm 14605 . . . . . . . . . . . 12  |- ; 1 9  e.  Prime
63 coprm 14243 . . . . . . . . . . . 12  |-  ( (; 1
9  e.  Prime  /\ ; 4 9  e.  ZZ )  ->  ( -. ; 1 9  || ; 4 9  <->  (; 1 9  gcd ; 4 9 )  =  1 ) )
6462, 40, 63mp2an 670 . . . . . . . . . . 11  |-  ( -. ; 1
9  || ; 4 9  <->  (; 1 9  gcd ; 4 9 )  =  1 )
6561, 64mpbi 208 . . . . . . . . . 10  |-  (; 1 9  gcd ; 4 9 )  =  1
6643, 65eqtri 2411 . . . . . . . . 9  |-  (; 4 9  gcd ; 1 9 )  =  1
67 eqid 2382 . . . . . . . . . 10  |- ; 4 9  = ; 4 9
68 4cn 10530 . . . . . . . . . . . . 13  |-  4  e.  CC
6968mulid2i 9510 . . . . . . . . . . . 12  |-  ( 1  x.  4 )  =  4
7069, 22oveq12i 6208 . . . . . . . . . . 11  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  ( 4  +  2 )
71 4p2e6 10587 . . . . . . . . . . 11  |-  ( 4  +  2 )  =  6
7270, 71eqtri 2411 . . . . . . . . . 10  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  6
73 9cn 10540 . . . . . . . . . . . . 13  |-  9  e.  CC
7473mulid2i 9510 . . . . . . . . . . . 12  |-  ( 1  x.  9 )  =  9
7574oveq1i 6206 . . . . . . . . . . 11  |-  ( ( 1  x.  9 )  +  9 )  =  ( 9  +  9 )
76 9p9e18 10964 . . . . . . . . . . 11  |-  ( 9  +  9 )  = ; 1
8
7775, 76eqtri 2411 . . . . . . . . . 10  |-  ( ( 1  x.  9 )  +  9 )  = ; 1
8
7830, 37, 2, 37, 67, 48, 2, 3, 2, 72, 77decma2c 10935 . . . . . . . . 9  |-  ( ( 1  x. ; 4 9 )  + ; 1
9 )  = ; 6 8
792, 39, 38, 66, 78gcdi 14561 . . . . . . . 8  |-  (; 6 8  gcd ; 4 9 )  =  1
80 eqid 2382 . . . . . . . . 9  |- ; 6 8  = ; 6 8
81 6cn 10534 . . . . . . . . . . . 12  |-  6  e.  CC
8281mulid2i 9510 . . . . . . . . . . 11  |-  ( 1  x.  6 )  =  6
83 4p1e5 10579 . . . . . . . . . . 11  |-  ( 4  +  1 )  =  5
8482, 83oveq12i 6208 . . . . . . . . . 10  |-  ( ( 1  x.  6 )  +  ( 4  +  1 ) )  =  ( 6  +  5 )
85 6p5e11 10945 . . . . . . . . . 10  |-  ( 6  +  5 )  = ; 1
1
8684, 85eqtri 2411 . . . . . . . . 9  |-  ( ( 1  x.  6 )  +  ( 4  +  1 ) )  = ; 1
1
87 8cn 10538 . . . . . . . . . . . 12  |-  8  e.  CC
8887mulid2i 9510 . . . . . . . . . . 11  |-  ( 1  x.  8 )  =  8
8988oveq1i 6206 . . . . . . . . . 10  |-  ( ( 1  x.  8 )  +  9 )  =  ( 8  +  9 )
90 9p8e17 10963 . . . . . . . . . . 11  |-  ( 9  +  8 )  = ; 1
7
9173, 87, 90addcomli 9683 . . . . . . . . . 10  |-  ( 8  +  9 )  = ; 1
7
9289, 91eqtri 2411 . . . . . . . . 9  |-  ( ( 1  x.  8 )  +  9 )  = ; 1
7
9326, 3, 30, 37, 80, 67, 2, 27, 2, 86, 92decma2c 10935 . . . . . . . 8  |-  ( ( 1  x. ; 6 8 )  + ; 4
9 )  = ;; 1 1 7
942, 38, 36, 79, 93gcdi 14561 . . . . . . 7  |-  (;; 1 1 7  gcd ; 6 8 )  =  1
95 eqid 2382 . . . . . . . 8  |- ;; 1 1 7  = ;; 1 1 7
96 6p1e7 10581 . . . . . . . . . 10  |-  ( 6  +  1 )  =  7
9727dec0h 10911 . . . . . . . . . 10  |-  7  = ; 0 7
9896, 97eqtri 2411 . . . . . . . . 9  |-  ( 6  +  1 )  = ; 0
7
99 ax-1cn 9461 . . . . . . . . . . . 12  |-  1  e.  CC
10099mulid1i 9509 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
101 00id 9666 . . . . . . . . . . 11  |-  ( 0  +  0 )  =  0
102100, 101oveq12i 6208 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
10399addid1i 9678 . . . . . . . . . 10  |-  ( 1  +  0 )  =  1
104102, 103eqtri 2411 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
105100oveq1i 6206 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  +  7 )  =  ( 1  +  7 )
106 7cn 10536 . . . . . . . . . . 11  |-  7  e.  CC
107 7p1e8 10582 . . . . . . . . . . 11  |-  ( 7  +  1 )  =  8
108106, 99, 107addcomli 9683 . . . . . . . . . 10  |-  ( 1  +  7 )  =  8
1093dec0h 10911 . . . . . . . . . 10  |-  8  = ; 0 8
110105, 108, 1093eqtri 2415 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  7 )  = ; 0
8
1112, 2, 16, 27, 49, 98, 2, 3, 16, 104, 110decma2c 10935 . . . . . . . 8  |-  ( ( 1  x. ; 1 1 )  +  ( 6  +  1 ) )  = ; 1 8
112106mulid2i 9510 . . . . . . . . . 10  |-  ( 1  x.  7 )  =  7
113112oveq1i 6206 . . . . . . . . 9  |-  ( ( 1  x.  7 )  +  8 )  =  ( 7  +  8 )
114 8p7e15 10955 . . . . . . . . . 10  |-  ( 8  +  7 )  = ; 1
5
11587, 106, 114addcomli 9683 . . . . . . . . 9  |-  ( 7  +  8 )  = ; 1
5
116113, 115eqtri 2411 . . . . . . . 8  |-  ( ( 1  x.  7 )  +  8 )  = ; 1
5
11734, 27, 26, 3, 95, 80, 2, 14, 2, 111, 116decma2c 10935 . . . . . . 7  |-  ( ( 1  x. ;; 1 1 7 )  + ; 6
8 )  = ;; 1 8 5
1182, 36, 35, 94, 117gcdi 14561 . . . . . 6  |-  (;; 1 8 5  gcd ;; 1 1 7 )  =  1
119 eqid 2382 . . . . . . 7  |- ;; 1 8 5  = ;; 1 8 5
120 eqid 2382 . . . . . . . 8  |- ; 1 8  = ; 1 8
1212, 2, 22, 49decsuc 10918 . . . . . . . 8  |-  (; 1 1  +  1 )  = ; 1 2
12250mulid1i 9509 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
123122, 22oveq12i 6208 . . . . . . . . 9  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  ( 2  +  2 )
124123, 53eqtri 2411 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  ( 1  +  1 ) )  =  4
125 8t2e16 10983 . . . . . . . . . 10  |-  ( 8  x.  2 )  = ; 1
6
12687, 50, 125mulcomli 9514 . . . . . . . . 9  |-  ( 2  x.  8 )  = ; 1
6
127 6p2e8 10594 . . . . . . . . 9  |-  ( 6  +  2 )  =  8
1282, 26, 13, 126, 127decaddi 10939 . . . . . . . 8  |-  ( ( 2  x.  8 )  +  2 )  = ; 1
8
1292, 3, 2, 13, 120, 121, 13, 3, 2, 124, 128decma2c 10935 . . . . . . 7  |-  ( ( 2  x. ; 1 8 )  +  (; 1 1  +  1 ) )  = ; 4 8
130 5cn 10532 . . . . . . . . . 10  |-  5  e.  CC
131 5t2e10 10607 . . . . . . . . . 10  |-  ( 5  x.  2 )  =  10
132130, 50, 131mulcomli 9514 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
133 dec10 10925 . . . . . . . . 9  |-  10  = ; 1 0
134132, 133eqtri 2411 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
135106addid2i 9679 . . . . . . . 8  |-  ( 0  +  7 )  =  7
1362, 16, 27, 134, 135decaddi 10939 . . . . . . 7  |-  ( ( 2  x.  5 )  +  7 )  = ; 1
7
1374, 14, 34, 27, 119, 95, 13, 27, 2, 129, 136decma2c 10935 . . . . . 6  |-  ( ( 2  x. ;; 1 8 5 )  + ;; 1 1 7 )  = ;; 4 8 7
13813, 35, 33, 118, 137gcdi 14561 . . . . 5  |-  (;; 4 8 7  gcd ;; 1 8 5 )  =  1
139 eqid 2382 . . . . . 6  |- ;; 4 8 7  = ;; 4 8 7
140 eqid 2382 . . . . . . 7  |- ; 4 8  = ; 4 8
1412, 3, 55, 120decsuc 10918 . . . . . . 7  |-  (; 1 8  +  1 )  = ; 1 9
14230, 3, 2, 37, 140, 141, 2, 27, 2, 72, 92decma2c 10935 . . . . . 6  |-  ( ( 1  x. ; 4 8 )  +  (; 1 8  +  1 ) )  = ; 6 7
143112oveq1i 6206 . . . . . . 7  |-  ( ( 1  x.  7 )  +  5 )  =  ( 7  +  5 )
144 7p5e12 10948 . . . . . . 7  |-  ( 7  +  5 )  = ; 1
2
145143, 144eqtri 2411 . . . . . 6  |-  ( ( 1  x.  7 )  +  5 )  = ; 1
2
14631, 27, 4, 14, 139, 119, 2, 13, 2, 142, 145decma2c 10935 . . . . 5  |-  ( ( 1  x. ;; 4 8 7 )  + ;; 1 8 5 )  = ;; 6 7 2
1472, 33, 32, 138, 146gcdi 14561 . . . 4  |-  (;; 6 7 2  gcd ;; 4 8 7 )  =  1
148 eqid 2382 . . . . 5  |- ;; 6 7 2  = ;; 6 7 2
149 eqid 2382 . . . . . 6  |- ; 6 7  = ; 6 7
15030, 3, 55, 140decsuc 10918 . . . . . 6  |-  (; 4 8  +  1 )  = ; 4 9
15171oveq2i 6207 . . . . . . 7  |-  ( ( 2  x.  6 )  +  ( 4  +  2 ) )  =  ( ( 2  x.  6 )  +  6 )
152 6t2e12 10972 . . . . . . . . 9  |-  ( 6  x.  2 )  = ; 1
2
15381, 50, 152mulcomli 9514 . . . . . . . 8  |-  ( 2  x.  6 )  = ; 1
2
15481, 50, 127addcomli 9683 . . . . . . . 8  |-  ( 2  +  6 )  =  8
1552, 13, 26, 153, 154decaddi 10939 . . . . . . 7  |-  ( ( 2  x.  6 )  +  6 )  = ; 1
8
156151, 155eqtri 2411 . . . . . 6  |-  ( ( 2  x.  6 )  +  ( 4  +  2 ) )  = ; 1
8
157 7t2e14 10977 . . . . . . . 8  |-  ( 7  x.  2 )  = ; 1
4
158106, 50, 157mulcomli 9514 . . . . . . 7  |-  ( 2  x.  7 )  = ; 1
4
159 9p4e13 10959 . . . . . . . 8  |-  ( 9  +  4 )  = ; 1
3
16073, 68, 159addcomli 9683 . . . . . . 7  |-  ( 4  +  9 )  = ; 1
3
1612, 30, 37, 158, 22, 9, 160decaddci 10940 . . . . . 6  |-  ( ( 2  x.  7 )  +  9 )  = ; 2
3
16226, 27, 30, 37, 149, 150, 13, 9, 13, 156, 161decma2c 10935 . . . . 5  |-  ( ( 2  x. ; 6 7 )  +  (; 4 8  +  1 ) )  = ;; 1 8 3
163 2t2e4 10602 . . . . . . 7  |-  ( 2  x.  2 )  =  4
164163oveq1i 6206 . . . . . 6  |-  ( ( 2  x.  2 )  +  7 )  =  ( 4  +  7 )
165 7p4e11 10947 . . . . . . 7  |-  ( 7  +  4 )  = ; 1
1
166106, 68, 165addcomli 9683 . . . . . 6  |-  ( 4  +  7 )  = ; 1
1
167164, 166eqtri 2411 . . . . 5  |-  ( ( 2  x.  2 )  +  7 )  = ; 1
1
16828, 13, 31, 27, 148, 139, 13, 2, 2, 162, 167decma2c 10935 . . . 4  |-  ( ( 2  x. ;; 6 7 2 )  + ;; 4 8 7 )  = ;;; 1 8 3 1
16913, 32, 29, 147, 168gcdi 14561 . . 3  |-  (;;; 1 8 3 1  gcd ;; 6 7 2 )  =  1
170 eqid 2382 . . . . . 6  |- ;; 1 8 3  = ;; 1 8 3
17128nn0cni 10724 . . . . . . 7  |- ; 6 7  e.  CC
172171addid1i 9678 . . . . . 6  |-  (; 6 7  +  0 )  = ; 6 7
17399addid2i 9679 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
174100, 173oveq12i 6208 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  ( 1  +  1 )
175174, 22eqtri 2411 . . . . . . 7  |-  ( ( 1  x.  1 )  +  ( 0  +  1 ) )  =  2
17688oveq1i 6206 . . . . . . . 8  |-  ( ( 1  x.  8 )  +  7 )  =  ( 8  +  7 )
177176, 114eqtri 2411 . . . . . . 7  |-  ( ( 1  x.  8 )  +  7 )  = ; 1
5
1782, 3, 16, 27, 120, 98, 2, 14, 2, 175, 177decma2c 10935 . . . . . 6  |-  ( ( 1  x. ; 1 8 )  +  ( 6  +  1 ) )  = ; 2 5
179 3cn 10527 . . . . . . . . 9  |-  3  e.  CC
180179mulid2i 9510 . . . . . . . 8  |-  ( 1  x.  3 )  =  3
181180oveq1i 6206 . . . . . . 7  |-  ( ( 1  x.  3 )  +  7 )  =  ( 3  +  7 )
182 7p3e10 10598 . . . . . . . 8  |-  ( 7  +  3 )  =  10
183106, 179, 182addcomli 9683 . . . . . . 7  |-  ( 3  +  7 )  =  10
184181, 183, 1333eqtri 2415 . . . . . 6  |-  ( ( 1  x.  3 )  +  7 )  = ; 1
0
1854, 9, 26, 27, 170, 172, 2, 16, 2, 178, 184decma2c 10935 . . . . 5  |-  ( ( 1  x. ;; 1 8 3 )  +  (; 6 7  +  0 ) )  = ;; 2 5 0
186100oveq1i 6206 . . . . . 6  |-  ( ( 1  x.  1 )  +  2 )  =  ( 1  +  2 )
187 1p2e3 10577 . . . . . 6  |-  ( 1  +  2 )  =  3
1889dec0h 10911 . . . . . 6  |-  3  = ; 0 3
189186, 187, 1883eqtri 2415 . . . . 5  |-  ( ( 1  x.  1 )  +  2 )  = ; 0
3
19010, 2, 28, 13, 23, 148, 2, 9, 16, 185, 189decma2c 10935 . . . 4  |-  ( ( 1  x. ;;; 1 8 3 1 )  + ;; 6 7 2 )  = ;;; 2 5 0 3
191190, 12eqtr4i 2414 . . 3  |-  ( ( 1  x. ;;; 1 8 3 1 )  + ;; 6 7 2 )  =  N
1922, 29, 11, 169, 191gcdi 14561 . 2  |-  ( N  gcd ;;; 1 8 3 1 )  =  1
1938, 11, 20, 25, 192gcdmodi 14562 1  |-  ( ( ( 2 ^; 1 8 )  - 
1 )  gcd  N
)  =  1
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1399    e. wcel 1826   class class class wbr 4367  (class class class)co 6196   0cc0 9403   1c1 9404    + caddc 9406    x. cmul 9408    - cmin 9718   NNcn 10452   2c2 10502   3c3 10503   4c4 10504   5c5 10505   6c6 10506   7c7 10507   8c8 10508   9c9 10509   10c10 10510   NN0cn0 10712   ZZcz 10781  ;cdc 10895   ^cexp 12069    || cdvds 13988    gcd cgcd 14146   Primecprime 14219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-rp 11140  df-fz 11594  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-dvds 13989  df-gcd 14147  df-prm 14220
This theorem is referenced by:  2503prm  14624
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