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Theorem 317prm 13143
Description: 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
Assertion
Ref Expression
317prm  |- ;; 3 1 7  e.  Prime

Proof of Theorem 317prm
StepHypRef Expression
1 3nn0 9999 . . . 4  |-  3  e.  NN0
2 1nn0 9997 . . . 4  |-  1  e.  NN0
31, 2deccl 10154 . . 3  |- ; 3 1  e.  NN0
4 7nn 9898 . . 3  |-  7  e.  NN
53, 4decnncl 10153 . 2  |- ;; 3 1 7  e.  NN
6 8nn0 10004 . . . 4  |-  8  e.  NN0
7 4nn0 10000 . . . 4  |-  4  e.  NN0
86, 7deccl 10154 . . 3  |- ; 8 4  e.  NN0
9 7nn0 10003 . . 3  |-  7  e.  NN0
10 7lt10 9940 . . 3  |-  7  <  10
11 1lt10 9946 . . . 4  |-  1  <  10
12 3lt8 9927 . . . 4  |-  3  <  8
131, 6, 2, 7, 11, 12decltc 10162 . . 3  |- ; 3 1  < ; 8 4
143, 8, 9, 2, 10, 13decltc 10162 . 2  |- ;; 3 1 7  < ;; 8 4 1
15 1nn 9773 . . . 4  |-  1  e.  NN
161, 15decnncl 10153 . . 3  |- ; 3 1  e.  NN
1716, 9, 2, 11declti 10165 . 2  |-  1  < ;; 3 1 7
18 3t2e6 9888 . . 3  |-  ( 3  x.  2 )  =  6
19 df-7 9825 . . 3  |-  7  =  ( 6  +  1 )
203, 1, 18, 19dec2dvds 13094 . 2  |-  -.  2  || ;; 3 1 7
21 3nn 9894 . . 3  |-  3  e.  NN
22 10nn0 10006 . . . 4  |-  10  e.  NN0
23 5nn0 10001 . . . 4  |-  5  e.  NN0
2422, 23deccl 10154 . . 3  |- ; 10 5  e.  NN0
25 2nn 9893 . . 3  |-  2  e.  NN
26 0nn0 9996 . . . 4  |-  0  e.  NN0
27 2nn0 9998 . . . 4  |-  2  e.  NN0
28 eqid 2296 . . . 4  |- ; 10 5  = ; 10 5
2927dec0h 10156 . . . 4  |-  2  = ; 0 2
30 dec10 10170 . . . . 5  |-  10  = ; 1 0
31 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
3231addid2i 9016 . . . . . 6  |-  ( 0  +  1 )  =  1
332dec0h 10156 . . . . . 6  |-  1  = ; 0 1
3432, 33eqtri 2316 . . . . 5  |-  ( 0  +  1 )  = ; 0
1
35 3cn 9834 . . . . . . . 8  |-  3  e.  CC
3635mulid1i 8855 . . . . . . 7  |-  ( 3  x.  1 )  =  3
37 00id 9003 . . . . . . 7  |-  ( 0  +  0 )  =  0
3836, 37oveq12i 5886 . . . . . 6  |-  ( ( 3  x.  1 )  +  ( 0  +  0 ) )  =  ( 3  +  0 )
3935addid1i 9015 . . . . . 6  |-  ( 3  +  0 )  =  3
4038, 39eqtri 2316 . . . . 5  |-  ( ( 3  x.  1 )  +  ( 0  +  0 ) )  =  3
4135mul01i 9018 . . . . . . . 8  |-  ( 3  x.  0 )  =  0
4241oveq1i 5884 . . . . . . 7  |-  ( ( 3  x.  0 )  +  1 )  =  ( 0  +  1 )
4342, 32eqtri 2316 . . . . . 6  |-  ( ( 3  x.  0 )  +  1 )  =  1
4443, 33eqtri 2316 . . . . 5  |-  ( ( 3  x.  0 )  +  1 )  = ; 0
1
452, 26, 26, 2, 30, 34, 1, 2, 26, 40, 44decma2c 10180 . . . 4  |-  ( ( 3  x.  10 )  +  ( 0  +  1 ) )  = ; 3
1
46 5nn 9896 . . . . . . 7  |-  5  e.  NN
4746nncni 9772 . . . . . 6  |-  5  e.  CC
48 5t3e15 10214 . . . . . 6  |-  ( 5  x.  3 )  = ; 1
5
4947, 35, 48mulcomli 8860 . . . . 5  |-  ( 3  x.  5 )  = ; 1
5
50 5p2e7 9876 . . . . 5  |-  ( 5  +  2 )  =  7
512, 23, 27, 49, 50decaddi 10184 . . . 4  |-  ( ( 3  x.  5 )  +  2 )  = ; 1
7
5222, 23, 26, 27, 28, 29, 1, 9, 2, 45, 51decma2c 10180 . . 3  |-  ( ( 3  x. ; 10 5 )  +  2 )  = ;; 3 1 7
53 2lt3 9903 . . 3  |-  2  <  3
5421, 24, 25, 52, 53ndvdsi 12625 . 2  |-  -.  3  || ;; 3 1 7
55 2lt5 9910 . . 3  |-  2  <  5
563, 25, 55, 50dec5dvds2 13096 . 2  |-  -.  5  || ;; 3 1 7
577, 23deccl 10154 . . 3  |- ; 4 5  e.  NN0
58 eqid 2296 . . . 4  |- ; 4 5  = ; 4 5
5935addid2i 9016 . . . . . 6  |-  ( 0  +  3 )  =  3
6059oveq2i 5885 . . . . 5  |-  ( ( 7  x.  4 )  +  ( 0  +  3 ) )  =  ( ( 7  x.  4 )  +  3 )
61 7t4e28 10224 . . . . . 6  |-  ( 7  x.  4 )  = ; 2
8
62 2p1e3 9863 . . . . . 6  |-  ( 2  +  1 )  =  3
63 8p3e11 10196 . . . . . 6  |-  ( 8  +  3 )  = ; 1
1
6427, 6, 1, 61, 62, 2, 63decaddci 10185 . . . . 5  |-  ( ( 7  x.  4 )  +  3 )  = ; 3
1
6560, 64eqtri 2316 . . . 4  |-  ( ( 7  x.  4 )  +  ( 0  +  3 ) )  = ; 3
1
66 7t5e35 10225 . . . . 5  |-  ( 7  x.  5 )  = ; 3
5
671, 23, 27, 66, 50decaddi 10184 . . . 4  |-  ( ( 7  x.  5 )  +  2 )  = ; 3
7
687, 23, 26, 27, 58, 29, 9, 9, 1, 65, 67decma2c 10180 . . 3  |-  ( ( 7  x. ; 4 5 )  +  2 )  = ;; 3 1 7
69 2lt7 9921 . . 3  |-  2  <  7
704, 57, 25, 68, 69ndvdsi 12625 . 2  |-  -.  7  || ;; 3 1 7
712, 15decnncl 10153 . . 3  |- ; 1 1  e.  NN
7227, 6deccl 10154 . . 3  |- ; 2 8  e.  NN0
73 9nn 9900 . . 3  |-  9  e.  NN
74 9nn0 10005 . . . 4  |-  9  e.  NN0
75 eqid 2296 . . . 4  |- ; 2 8  = ; 2 8
7674dec0h 10156 . . . 4  |-  9  = ; 0 9
772, 2deccl 10154 . . . 4  |- ; 1 1  e.  NN0
78 eqid 2296 . . . . 5  |- ; 1 1  = ; 1 1
7973nncni 9772 . . . . . . 7  |-  9  e.  CC
8079addid2i 9016 . . . . . 6  |-  ( 0  +  9 )  =  9
8180, 76eqtri 2316 . . . . 5  |-  ( 0  +  9 )  = ; 0
9
82 2cn 9832 . . . . . . . 8  |-  2  e.  CC
8382mulid2i 8856 . . . . . . 7  |-  ( 1  x.  2 )  =  2
8483, 32oveq12i 5886 . . . . . 6  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
8584, 62eqtri 2316 . . . . 5  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  3
8683oveq1i 5884 . . . . . 6  |-  ( ( 1  x.  2 )  +  9 )  =  ( 2  +  9 )
87 9p2e11 10202 . . . . . . 7  |-  ( 9  +  2 )  = ; 1
1
8879, 82, 87addcomli 9020 . . . . . 6  |-  ( 2  +  9 )  = ; 1
1
8986, 88eqtri 2316 . . . . 5  |-  ( ( 1  x.  2 )  +  9 )  = ; 1
1
902, 2, 26, 74, 78, 81, 27, 2, 2, 85, 89decmac 10179 . . . 4  |-  ( (; 1
1  x.  2 )  +  ( 0  +  9 ) )  = ; 3
1
91 8nn 9899 . . . . . . . . 9  |-  8  e.  NN
9291nncni 9772 . . . . . . . 8  |-  8  e.  CC
9392mulid2i 8856 . . . . . . 7  |-  ( 1  x.  8 )  =  8
9493, 32oveq12i 5886 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  1 ) )  =  ( 8  +  1 )
95 8p1e9 9869 . . . . . 6  |-  ( 8  +  1 )  =  9
9694, 95eqtri 2316 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  1 ) )  =  9
9793oveq1i 5884 . . . . . 6  |-  ( ( 1  x.  8 )  +  9 )  =  ( 8  +  9 )
98 9p8e17 10208 . . . . . . 7  |-  ( 9  +  8 )  = ; 1
7
9979, 92, 98addcomli 9020 . . . . . 6  |-  ( 8  +  9 )  = ; 1
7
10097, 99eqtri 2316 . . . . 5  |-  ( ( 1  x.  8 )  +  9 )  = ; 1
7
1012, 2, 26, 74, 78, 76, 6, 9, 2, 96, 100decmac 10179 . . . 4  |-  ( (; 1
1  x.  8 )  +  9 )  = ; 9
7
10227, 6, 26, 74, 75, 76, 77, 9, 74, 90, 101decma2c 10180 . . 3  |-  ( (; 1
1  x. ; 2 8 )  +  9 )  = ;; 3 1 7
103 9lt10 9938 . . . 4  |-  9  <  10
10415, 2, 74, 103declti 10165 . . 3  |-  9  < ; 1
1
10571, 72, 73, 102, 104ndvdsi 12625 . 2  |-  -. ; 1 1  || ;; 3 1 7
1062, 21decnncl 10153 . . 3  |- ; 1 3  e.  NN
10727, 7deccl 10154 . . 3  |- ; 2 4  e.  NN0
108 eqid 2296 . . . 4  |- ; 2 4  = ; 2 4
10923dec0h 10156 . . . 4  |-  5  = ; 0 5
1102, 1deccl 10154 . . . 4  |- ; 1 3  e.  NN0
111 eqid 2296 . . . . 5  |- ; 1 3  = ; 1 3
11247addid2i 9016 . . . . . 6  |-  ( 0  +  5 )  =  5
113112, 109eqtri 2316 . . . . 5  |-  ( 0  +  5 )  = ; 0
5
11418oveq1i 5884 . . . . . 6  |-  ( ( 3  x.  2 )  +  5 )  =  ( 6  +  5 )
115 6p5e11 10190 . . . . . 6  |-  ( 6  +  5 )  = ; 1
1
116114, 115eqtri 2316 . . . . 5  |-  ( ( 3  x.  2 )  +  5 )  = ; 1
1
1172, 1, 26, 23, 111, 113, 27, 2, 2, 85, 116decmac 10179 . . . 4  |-  ( (; 1
3  x.  2 )  +  ( 0  +  5 ) )  = ; 3
1
118 4cn 9836 . . . . . . . 8  |-  4  e.  CC
119118mulid2i 8856 . . . . . . 7  |-  ( 1  x.  4 )  =  4
120119, 32oveq12i 5886 . . . . . 6  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
121 4p1e5 9865 . . . . . 6  |-  ( 4  +  1 )  =  5
122120, 121eqtri 2316 . . . . 5  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  5
123 4t3e12 10212 . . . . . . 7  |-  ( 4  x.  3 )  = ; 1
2
124118, 35, 123mulcomli 8860 . . . . . 6  |-  ( 3  x.  4 )  = ; 1
2
12547, 82, 50addcomli 9020 . . . . . 6  |-  ( 2  +  5 )  =  7
1262, 27, 23, 124, 125decaddi 10184 . . . . 5  |-  ( ( 3  x.  4 )  +  5 )  = ; 1
7
1272, 1, 26, 23, 111, 109, 7, 9, 2, 122, 126decmac 10179 . . . 4  |-  ( (; 1
3  x.  4 )  +  5 )  = ; 5
7
12827, 7, 26, 23, 108, 109, 110, 9, 23, 117, 127decma2c 10180 . . 3  |-  ( (; 1
3  x. ; 2 4 )  +  5 )  = ;; 3 1 7
129 5lt10 9942 . . . 4  |-  5  <  10
13015, 1, 23, 129declti 10165 . . 3  |-  5  < ; 1
3
131106, 107, 46, 128, 130ndvdsi 12625 . 2  |-  -. ; 1 3  || ;; 3 1 7
1322, 4decnncl 10153 . . 3  |- ; 1 7  e.  NN
1332, 6deccl 10154 . . 3  |- ; 1 8  e.  NN0
134 eqid 2296 . . . 4  |- ; 1 8  = ; 1 8
1352, 9deccl 10154 . . . 4  |- ; 1 7  e.  NN0
136 eqid 2296 . . . . 5  |- ; 1 7  = ; 1 7
137 3p1e4 9864 . . . . . . 7  |-  ( 3  +  1 )  =  4
13835, 31, 137addcomli 9020 . . . . . 6  |-  ( 1  +  3 )  =  4
13926, 2, 2, 1, 33, 111, 32, 138decadd 10181 . . . . 5  |-  ( 1  + ; 1 3 )  = ; 1
4
14031mulid1i 8855 . . . . . . 7  |-  ( 1  x.  1 )  =  1
141 1p1e2 9856 . . . . . . 7  |-  ( 1  +  1 )  =  2
142140, 141oveq12i 5886 . . . . . 6  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  ( 1  +  2 )
14382, 31, 62addcomli 9020 . . . . . 6  |-  ( 1  +  2 )  =  3
144142, 143eqtri 2316 . . . . 5  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  3
1454nncni 9772 . . . . . . . 8  |-  7  e.  CC
146145mulid1i 8855 . . . . . . 7  |-  ( 7  x.  1 )  =  7
147146oveq1i 5884 . . . . . 6  |-  ( ( 7  x.  1 )  +  4 )  =  ( 7  +  4 )
148 7p4e11 10192 . . . . . 6  |-  ( 7  +  4 )  = ; 1
1
149147, 148eqtri 2316 . . . . 5  |-  ( ( 7  x.  1 )  +  4 )  = ; 1
1
1502, 9, 2, 7, 136, 139, 2, 2, 2, 144, 149decmac 10179 . . . 4  |-  ( (; 1
7  x.  1 )  +  ( 1  + ; 1
3 ) )  = ; 3
1
15193, 112oveq12i 5886 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  =  ( 8  +  5 )
152 8p5e13 10198 . . . . . 6  |-  ( 8  +  5 )  = ; 1
3
153151, 152eqtri 2316 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  = ; 1
3
154 6nn0 10002 . . . . . 6  |-  6  e.  NN0
155 6p1e7 9867 . . . . . 6  |-  ( 6  +  1 )  =  7
156 8t7e56 10233 . . . . . . 7  |-  ( 8  x.  7 )  = ; 5
6
15792, 145, 156mulcomli 8860 . . . . . 6  |-  ( 7  x.  8 )  = ; 5
6
15823, 154, 155, 157decsuc 10163 . . . . 5  |-  ( ( 7  x.  8 )  +  1 )  = ; 5
7
1592, 9, 26, 2, 136, 33, 6, 9, 23, 153, 158decmac 10179 . . . 4  |-  ( (; 1
7  x.  8 )  +  1 )  = ;; 1 3 7
1602, 6, 2, 2, 134, 78, 135, 9, 110, 150, 159decma2c 10180 . . 3  |-  ( (; 1
7  x. ; 1 8 )  + ; 1
1 )  = ;; 3 1 7
161 1lt7 9922 . . . 4  |-  1  <  7
1622, 2, 4, 161declt 10161 . . 3  |- ; 1 1  < ; 1 7
163132, 133, 71, 160, 162ndvdsi 12625 . 2  |-  -. ; 1 7  || ;; 3 1 7
1642, 73decnncl 10153 . . 3  |- ; 1 9  e.  NN
1652, 154deccl 10154 . . 3  |- ; 1 6  e.  NN0
166 eqid 2296 . . . 4  |- ; 1 6  = ; 1 6
1672, 74deccl 10154 . . . 4  |- ; 1 9  e.  NN0
168 eqid 2296 . . . . 5  |- ; 1 9  = ; 1 9
16926, 2, 2, 2, 33, 78, 32, 141decadd 10181 . . . . 5  |-  ( 1  + ; 1 1 )  = ; 1
2
17079mulid1i 8855 . . . . . . 7  |-  ( 9  x.  1 )  =  9
171170oveq1i 5884 . . . . . 6  |-  ( ( 9  x.  1 )  +  2 )  =  ( 9  +  2 )
172171, 87eqtri 2316 . . . . 5  |-  ( ( 9  x.  1 )  +  2 )  = ; 1
1
1732, 74, 2, 27, 168, 169, 2, 2, 2, 144, 172decmac 10179 . . . 4  |-  ( (; 1
9  x.  1 )  +  ( 1  + ; 1
1 ) )  = ; 3
1
1741dec0h 10156 . . . . 5  |-  3  = ; 0 3
175 6nn 9897 . . . . . . . . 9  |-  6  e.  NN
176175nncni 9772 . . . . . . . 8  |-  6  e.  CC
177176mulid2i 8856 . . . . . . 7  |-  ( 1  x.  6 )  =  6
178177, 112oveq12i 5886 . . . . . 6  |-  ( ( 1  x.  6 )  +  ( 0  +  5 ) )  =  ( 6  +  5 )
179178, 115eqtri 2316 . . . . 5  |-  ( ( 1  x.  6 )  +  ( 0  +  5 ) )  = ; 1
1
180 9t6e54 10239 . . . . . 6  |-  ( 9  x.  6 )  = ; 5
4
181 4p3e7 9874 . . . . . 6  |-  ( 4  +  3 )  =  7
18223, 7, 1, 180, 181decaddi 10184 . . . . 5  |-  ( ( 9  x.  6 )  +  3 )  = ; 5
7
1832, 74, 26, 1, 168, 174, 154, 9, 23, 179, 182decmac 10179 . . . 4  |-  ( (; 1
9  x.  6 )  +  3 )  = ;; 1 1 7
1842, 154, 2, 1, 166, 111, 167, 9, 77, 173, 183decma2c 10180 . . 3  |-  ( (; 1
9  x. ; 1 6 )  + ; 1
3 )  = ;; 3 1 7
185 3lt9 9935 . . . 4  |-  3  <  9
1862, 1, 73, 185declt 10161 . . 3  |- ; 1 3  < ; 1 9
187164, 165, 106, 184, 186ndvdsi 12625 . 2  |-  -. ; 1 9  || ;; 3 1 7
18827, 21decnncl 10153 . . 3  |- ; 2 3  e.  NN
189106nnnn0i 9989 . . 3  |- ; 1 3  e.  NN0
1902, 91decnncl 10153 . . 3  |- ; 1 8  e.  NN
19127, 1deccl 10154 . . . 4  |- ; 2 3  e.  NN0
192 eqid 2296 . . . . 5  |- ; 2 3  = ; 2 3
193 7p1e8 9868 . . . . . . 7  |-  ( 7  +  1 )  =  8
194145, 31, 193addcomli 9020 . . . . . 6  |-  ( 1  +  7 )  =  8
1956dec0h 10156 . . . . . 6  |-  8  = ; 0 8
196194, 195eqtri 2316 . . . . 5  |-  ( 1  +  7 )  = ; 0
8
19782mulid1i 8855 . . . . . . 7  |-  ( 2  x.  1 )  =  2
198197, 32oveq12i 5886 . . . . . 6  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
199198, 62eqtri 2316 . . . . 5  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  3
20036oveq1i 5884 . . . . . 6  |-  ( ( 3  x.  1 )  +  8 )  =  ( 3  +  8 )
20192, 35, 63addcomli 9020 . . . . . 6  |-  ( 3  +  8 )  = ; 1
1
202200, 201eqtri 2316 . . . . 5  |-  ( ( 3  x.  1 )  +  8 )  = ; 1
1
20327, 1, 26, 6, 192, 196, 2, 2, 2, 199, 202decmac 10179 . . . 4  |-  ( (; 2
3  x.  1 )  +  ( 1  +  7 ) )  = ; 3
1
20435, 82, 18mulcomli 8860 . . . . . . 7  |-  ( 2  x.  3 )  =  6
205204, 32oveq12i 5886 . . . . . 6  |-  ( ( 2  x.  3 )  +  ( 0  +  1 ) )  =  ( 6  +  1 )
206205, 155eqtri 2316 . . . . 5  |-  ( ( 2  x.  3 )  +  ( 0  +  1 ) )  =  7
207 3t3e9 9889 . . . . . . 7  |-  ( 3  x.  3 )  =  9
208207oveq1i 5884 . . . . . 6  |-  ( ( 3  x.  3 )  +  8 )  =  ( 9  +  8 )
209208, 98eqtri 2316 . . . . 5  |-  ( ( 3  x.  3 )  +  8 )  = ; 1
7
21027, 1, 26, 6, 192, 195, 1, 9, 2, 206, 209decmac 10179 . . . 4  |-  ( (; 2
3  x.  3 )  +  8 )  = ; 7
7
2112, 1, 2, 6, 111, 134, 191, 9, 9, 203, 210decma2c 10180 . . 3  |-  ( (; 2
3  x. ; 1 3 )  + ; 1
8 )  = ;; 3 1 7
212 8lt10 9939 . . . 4  |-  8  <  10
213 1lt2 9902 . . . 4  |-  1  <  2
2142, 27, 6, 1, 212, 213decltc 10162 . . 3  |- ; 1 8  < ; 2 3
215188, 189, 190, 211, 214ndvdsi 12625 . 2  |-  -. ; 2 3  || ;; 3 1 7
2165, 14, 17, 20, 54, 56, 70, 105, 131, 163, 187, 215prmlem2 13137 1  |- ;; 3 1 7  e.  Prime
Colors of variables: wff set class
Syntax hints:    e. wcel 1696  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   2c2 9811   3c3 9812   4c4 9813   5c5 9814   6c6 9815   7c7 9816   8c8 9817   9c9 9818   10c10 9819  ;cdc 10140   Primecprime 12774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-prm 12775
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